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# Lexicographic degrees of two-bridge knots with 12 crossings

E. Brugallé, P. -V. Koseleff, D. Pecker

The two-brige knots are knots that admit trigonal form diagrams. They admit a polynomial parametrisation of degree $(3,b,c)$, where $3 < b < c$.
We proved in [BKP3]that the lexicographic degree of a two-bridge knot with $N \leq 11$ crossings or fewer, is $(3,b,c)$ where $b+c=3N$, see also details. When $N\geq 12$, we have $b+c \geq 3N$.
We list here the degree $b$ corresponding to the lexicographic degree $(3,b,c)$ for two-bridge knots with 12 crossings.
For each knot we show that there exists a diagram with $s$ sign changes in its Gauss sequence, such that there exists $c$ such that $b+c=3N=36$.
We list all (simple) diagrams that could have a smaller degree if the braid condition hold (quasipositity of the associated braid, see [BKP2]).

NameSchub.Conway Not.Lex. Deg.Details
$12a_{38}$ 71/33 $C(2,6,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(2,6,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(2,6,1,1,2)$ $b \geq 16$ $\quad D(2,7,2,2)$ $b \geq 17$ $\quad D(3,2,6,2)$ $b \geq 19$ $\quad D(2,5,1,2,1,2)$ $b \geq 16$
$12a_{169}$ 49/23 $C(2,7,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(2,6,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(2,7,1,2)$ $b \geq 16$ $\quad D(3,8,2)$ $b \geq 19$ $\quad D(2,6,1,2,2)$ $b \geq 16$
$12a_{197}$ 69/32 $C(2,6,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 149 diagrams of degree $(3,17,19)$. For example $\quad C(2,7,-1,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 4 simple diagrams with 15 crossings or fewer $\quad D(2,6,2,2)$ $b \geq 17$ $\quad D(2,5,1,3,2)$ $b \geq 17$ $\quad D(3,2,7,2)$ $b \geq 19$ $\quad D(2,4,1,2,3,2)$ $b \geq 17$
$12a_{204}$ 173/76 $C(2,3,1,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,14,22)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,1,1,1,1,1,2)$ $b \geq 14$
$12a_{206}$ 105/47 $C(2,4,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(2,4,2,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(2,4,3,1,2)$ $b \geq 16$ $\quad D(3,4,4,2)$ $b \geq 19$ $\quad D(2,3,1,4,1,2)$ $b \geq 16$ $\quad D(2,4,2,1,2,2)$ $b \geq 16$
$12a_{221}$ 169/66 $C(2,1,1,3,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,14,22)$ 8 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,1,3,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,1,3,1,1,1,2)$ $b \geq 14$
$12a_{226}$ 181/75 $C(2,2,2,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 52 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,1,-3,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 6 simple diagrams with 13 crossings or fewer $\quad D(2,2,3,2,2,2)$ $b \geq 19$ $\quad D(2,2,2,2,1,1,2)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(2,1,1,3,2,1,1,2)$ $b \geq 16$ $\quad D(2,2,1,1,3,1,1,2)$ $b \geq 16$ $\quad D(2,2,2,1,1,2,1,2)$ $b \geq 16$
$12a_{239}$ 87/40 $C(2,5,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,5,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,5,1,2,2)$ $b \geq 14$
$12a_{241}$ 127/57 $C(2,4,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(2,4,2,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(2,4,2,1,1,2)$ $b \geq 16$ $\quad D(2,4,3,2,2)$ $b \geq 16$ $\quad D(3,2,2,4,2)$ $b \geq 19$ $\quad D(2,3,1,3,1,1,2)$ $b \geq 16$ $\quad D(2,4,1,1,2,1,2)$ $b \geq 16$
$12a_{243}$ 133/60 $C(2,4,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,4,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,4,1,1,1,1,2)$ $b \geq 14$
$12a_{247}$ 163/71 $C(2,3,2,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,-3,-1,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,2,1,1,1,2)$ $b \geq 16$
$12a_{251}$ 159/59 $C(2,1,2,3,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,2,3,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,2,3,1,1,2)$ $b \geq 14$
$12a_{254}$ 97/23 $C(4,4,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-5,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,4,1,1,2)$ $b \geq 16$ $\quad D(4,4,2,3)$ $b \geq 19$ $\quad D(4,5,2,2)$ $b \geq 16$ $\quad D(4,3,1,2,1,2)$ $b \geq 16$
$12a_{255}$ 107/28 $C(3,1,4,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,4,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,4,1,1,2)$ $b \geq 14$
$12a_{257}$ 191/80 $C(2,2,1,1,2,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,1,1,2,1,1,2)$ $b \geq 14$
$12a_{259}$ 115/52 $C(2,4,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,4,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,4,1,2,1,2)$ $b \geq 14$
$12a_{300}$ 155/68 $C(2,3,1,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,1,1,2,1,2)$ $b \geq 14$
$12a_{302}$ 147/61 $C(2,2,2,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 50 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,1,-4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(2,2,2,3,1,2)$ $b \geq 16$ $\quad D(3,4,2,2,2)$ $b \geq 19$ $\quad D(2,2,1,1,4,1,2)$ $b \geq 16$ $\quad D(2,2,2,2,1,2,2)$ $b \geq 16$
$12a_{303}$ 153/64 $C(2,2,1,1,3,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,1,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,1,1,3,1,2)$ $b \geq 14$
$12a_{306}$ 147/64 $C(2,3,2,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,2,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,2,1,2,2)$ $b \geq 14$
$12a_{307}$ 157/69 $C(2,3,1,1,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,-2,-1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,1,1,1,2,2)$ $b \geq 16$
$12a_{330}$ 95/43 $C(2,4,1,3,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,2,-1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,4,1,3,2)$ $b \geq 16$
$12a_{378}$ 127/45 $C(2,1,4,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,4,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,4,1,1,1,2)$ $b \geq 14$
$12a_{379}$ 71/17 $C(4,5,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-6,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,5,1,2)$ $b \geq 16$ $\quad D(4,6,3)$ $b \geq 19$ $\quad D(4,4,1,2,2)$ $b \geq 16$
$12a_{380}$ 77/20 $C(3,1,5,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,5,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,5,1,2)$ $b \geq 14$
$12a_{384}$ 151/62 $C(2,2,3,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,2,1,-3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,3,2,1,2)$ $b \geq 16$
$12a_{385}$ 161/66 $C(2,2,3,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(3,-2,-1,-3,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,3,1,1,1,2)$ $b \geq 16$
$12a_{406}$ 179/74 $C(2,2,2,1,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,2,-1,-1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,2,1,1,2,2)$ $b \geq 16$
$12a_{425}$ 81/37 $C(2,5,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 67 diagrams of degree $(3,16,20)$. For example $\quad C(2,4,-1,-4,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 1 simple diagrams with 13 crossings or fewer $\quad D(2,5,3,2)$ $b \geq 16$
$12a_{437}$ 149/65 $C(2,3,2,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,-3,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,2,2,1,2)$ $b \geq 16$
$12a_{447}$ 121/43 $C(2,1,4,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,4,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,4,2,1,2)$ $b \geq 14$
$12a_{454}$ 103/27 $C(3,1,4,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,3,1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,4,2,2)$ $b \geq 16$
$12a_{471}$ 85/38 $C(2,4,4,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 71 diagrams of degree $(3,17,19)$. For example $\quad C(2,5,-1,-4,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(2,4,4,2)$ $b \geq 17$
$12a_{477}$ 169/70 $C(2,2,2,2,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 74 diagrams of degree $(3,17,19)$. For example $\quad C(2,-3,-3,1,1,2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 5 simple diagrams with 15 crossings or fewer $\quad D(2,2,2,2,2,2)$ $b \geq 17$ $\quad D(2,3,2,3,2,2)$ $b \geq 19$ $\quad D(3,2,3,2,2,2)$ $b \geq 19$ $\quad D(3,2,3,1,1,3,2)$ $b \geq 19$ $\quad D(3,2,3,2,1,1,3)$ $b \geq 19$
$12a_{482}$ 93/22 $C(4,4,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 154 diagrams of degree $(3,17,19)$. For example $\quad C(3,1,-5,-3,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 6 simple diagrams with 15 crossings or fewer $\quad D(4,4,2,2)$ $b \geq 17$ $\quad D(4,3,1,3,2)$ $b \geq 17$ $\quad D(4,4,1,1,3)$ $b \geq 17$ $\quad D(4,5,2,3)$ $b \geq 17$ $\quad D(4,2,1,2,3,2)$ $b \geq 17$ $\quad D(4,1,1,2,2,3,2)$ $b \geq 19$
$12a_{497}$ 209/81 $C(2,1,1,2,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,1,2,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{498}$ 207/76 $C(2,1,2,1,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,2,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{499}$ 233/89 $C(2,1,1,1,1,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,13,23)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,1,1,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{500}$ 167/60 $C(2,1,3,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,14,22)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,3,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{501}$ 199/55 $C(3,1,1,1,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,14,22)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,1,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{502}$ 91/37 $C(2,2,5,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(3,-1,-1,-5,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(2,2,5,1,2)$ $b \geq 16$ $\quad D(3,6,2,2)$ $b \geq 19$ $\quad D(2,2,4,1,2,2)$ $b \geq 16$
$12a_{506}$ 185/68 $C(2,1,2,1,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,2,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{508}$ 129/56 $C(2,3,3,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,2,1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,3,2,2)$ $b \geq 16$
$12a_{510}$ 193/81 $C(2,2,1,1,1,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,1,1,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{511}$ 125/51 $C(2,2,4,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 48 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,3,1,-2,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(2,2,4,1,1,2)$ $b \geq 16$ $\quad D(3,2,4,2,2)$ $b \geq 19$ $\quad D(2,2,3,1,2,1,2)$ $b \geq 16$
$12a_{512}$ 151/64 $C(2,2,1,3,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,3,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{514}$ 187/79 $C(2,2,1,2,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,2,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{517}$ 145/52 $C(2,1,3,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,3,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{518}$ 157/34 $C(4,1,1,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,14,22)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,1,1,1,1,1,2)$ $b \geq 14$
$12a_{519}$ 111/25 $C(4,2,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 61 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,-4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,2,3,1,2)$ $b \geq 16$ $\quad D(4,2,4,3)$ $b \geq 19$ $\quad D(4,1,1,4,1,2)$ $b \geq 16$ $\quad D(4,2,2,1,2,2)$ $b \geq 16$
$12a_{520}$ 133/36 $C(3,1,2,3,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{521}$ 113/48 $C(2,2,1,4,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,4,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{522}$ 173/73 $C(2,2,1,2,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,2,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{528}$ 183/67 $C(2,1,2,1,2,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,2,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{532}$ 125/33 $C(3,1,3,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,3,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{533}$ 137/31 $C(4,2,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 49 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-3,-2,-1,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(4,2,2,1,1,2)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(4,2,3,2,2)$ $b \geq 16$ $\quad D(4,1,1,3,1,1,2)$ $b \geq 16$ $\quad D(4,2,1,1,2,1,2)$ $b \geq 16$
$12a_{534}$ 163/44 $C(3,1,2,2,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{535}$ 175/47 $C(3,1,2,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{536}$ 137/29 $C(4,1,2,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,2,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{537}$ 179/50 $C(3,1,1,2,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(3,-2,-1,-2,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,1,2,1,1,1,2)$ $b \geq 14$
$12a_{538}$ 83/13 $C(6,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,1,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(6,2,1,1,2)$ $b \geq 16$ $\quad D(6,2,2,3)$ $b \geq 19$ $\quad D(6,3,2,2)$ $b \geq 16$ $\quad D(6,1,1,2,1,2)$ $b \geq 16$
$12a_{539}$ 145/44 $C(3,3,2,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,-3,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,2,1,1,2)$ $b \geq 16$
$12a_{540}$ 165/49 $C(3,2,1,2,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,2,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{541}$ 153/41 $C(3,1,2,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{545}$ 143/63 $C(2,3,1,2,2,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,3,1,2,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{549}$ 111/26 $C(4,3,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(4,3,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,3,1,2,2)$ $b \geq 14$
$12a_{550}$ 149/34 $C(4,2,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,2,1,1,1,1,2)$ $b \geq 14$
$12a_{551}$ 103/18 $C(5,1,2,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(5,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,1,2,1,1,2)$ $b \geq 14$
$12a_{552}$ 131/30 $C(4,2,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,2,1,2,1,2)$ $b \geq 14$
$12a_{579}$ 177/49 $C(3,1,1,1,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,1,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{580}$ 69/11 $C(6,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(7,-1,-2,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(6,3,1,2)$ $b \geq 16$ $\quad D(6,4,3)$ $b \geq 19$ $\quad D(6,2,1,2,2)$ $b \geq 16$
$12a_{581}$ 119/36 $C(3,3,3,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,-4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,3,1,2)$ $b \geq 16$
$12a_{582}$ 131/39 $C(3,2,1,3,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,2,1,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{583}$ 161/45 $C(3,1,1,2,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,1,2,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{584}$ 143/31 $C(4,1,1,1,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,1,1,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{585}$ 181/50 $C(3,1,1,1,1,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,1,1,1,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,1,1,1,1,2,2)$ $b \geq 14$
$12a_{595}$ 139/30 $C(4,1,1,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,1,1,2,1,2)$ $b \geq 14$
$12a_{596}$ 81/14 $C(5,1,3,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(5,1,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,1,3,1,2)$ $b \geq 14$
$12a_{597}$ 123/26 $C(4,1,2,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,2,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,2,1,2,2)$ $b \geq 14$
$12a_{600}$ 109/25 $C(4,2,1,3,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,2,1,3,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{601}$ 127/34 $C(3,1,2,1,3,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,2,1,3,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,2,1,3,2)$ $b \geq 14$
$12a_{643}$ 99/23 $C(4,3,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 63 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-4,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,3,3,2)$ $b \geq 16$ $\quad D(4,2,1,4,2)$ $b \geq 16$ $\quad D(4,3,2,1,3)$ $b \geq 16$
$12a_{644}$ 113/30 $C(3,1,3,3,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,3,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,3,3,2)$ $b \geq 16$
$12a_{649}$ 127/27 $C(4,1,2,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,2,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{650}$ 165/46 $C(3,1,1,2,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,1,2,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,1,2,2,1,2)$ $b \geq 14$
$12a_{651}$ 97/17 $C(5,1,2,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(5,1,2,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{652}$ 155/46 $C(3,2,1,2,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,2,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,2,1,2,2,2)$ $b \geq 14$
$12a_{682}$ 107/29 $C(3,1,2,4,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,2,4,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,2,4,2)$ $b \geq 14$
$12a_{684}$ 135/41 $C(3,3,2,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,-3,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,2,2,2)$ $b \geq 16$
$12a_{690}$ 89/20 $C(4,2,4,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 155 diagrams of degree $(3,17,19)$. For example $\quad C(3,-1,-3,-3,1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 6 simple diagrams with 15 crossings or fewer $\quad D(4,2,4,2)$ $b \geq 17$ $\quad D(4,1,1,5,2)$ $b \geq 17$ $\quad D(4,2,3,1,3)$ $b \geq 17$ $\quad D(5,2,5,2)$ $b \geq 19$ $\quad D(4,2,2,1,2,3)$ $b \geq 17$ $\quad D(4,2,1,1,2,2,3)$ $b \geq 19$
$12a_{691}$ 77/12 $C(6,2,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 161 diagrams of degree $(3,17,19)$. For example $\quad C(6,3,-1,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 5 simple diagrams with 15 crossings or fewer $\quad D(6,2,2,2)$ $b \geq 17$ $\quad D(6,1,1,3,2)$ $b \geq 17$ $\quad D(6,2,1,1,3)$ $b \geq 17$ $\quad D(6,3,2,3)$ $b \geq 17$ $\quad D(7,2,3,2)$ $b \geq 19$
$12a_{713}$ 139/39 $C(3,1,1,3,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(3,-1,-1,-3,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,3,2,2)$ $b \geq 16$ $\quad D(3,2,4,2,2)$ $b \geq 19$ $\quad D(4,2,3,2,2)$ $b \geq 16$ $\quad D(3,1,1,2,1,3,2)$ $b \geq 16$
$12a_{714}$ 107/19 $C(5,1,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(6,-2,-1,-3,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,1,2,2)$ $b \geq 16$ $\quad D(5,1,2,3,2)$ $b \geq 16$ $\quad D(5,2,2,2,2)$ $b \geq 19$ $\quad D(6,2,1,2,2)$ $b \geq 16$ $\quad D(5,1,1,1,1,1,3)$ $b \geq 16$
$12a_{715}$ 169/50 $C(3,2,1,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(3,2,1,2,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 6 simple diagrams with 13 crossings or fewer $\quad D(3,2,1,1,1,2,2)$ $b \geq 16$ $\quad D(3,2,1,2,3,2)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(3,3,2,1,2,2)$ $b \geq 16$ $\quad D(3,1,1,2,1,1,2,2)$ $b \geq 16$ $\quad D(3,2,1,1,1,1,1,3)$ $b \geq 16$
$12a_{716}$ 43/5 $C(8,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(8,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(8,1,1,2)$ $b \geq 16$ $\quad D(8,2,3)$ $b \geq 19$ $\quad D(9,2,2)$ $b \geq 19$
$12a_{717}$ 89/28 $C(3,5,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(2,-2,-1,-6,1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,5,1,1,2)$ $b \geq 16$ $\quad D(3,5,2,3)$ $b \geq 16$ $\quad D(3,6,2,2)$ $b \geq 19$ $\quad D(3,4,1,2,1,2)$ $b \geq 16$
$12a_{718}$ 141/41 $C(3,2,3,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(3,3,-1,-2,-2,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,2,3,1,1,2)$ $b \geq 16$ $\quad D(3,2,4,2,2)$ $b \geq 19$ $\quad D(3,1,1,4,1,1,2)$ $b \geq 16$ $\quad D(3,2,2,1,2,1,2)$ $b \geq 16$
$12a_{720}$ 113/21 $C(5,2,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(5,2,1,2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(5,2,1,1,1,2)$ $b \geq 16$ $\quad D(5,2,1,2,3)$ $b \geq 16$ $\quad D(5,2,2,2,2)$ $b \geq 19$ $\quad D(5,3,2,1,2)$ $b \geq 16$ $\quad D(5,1,1,2,1,1,2)$ $b \geq 16$
$12a_{721}$ 171/50 $C(3,2,2,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(3,-3,2,1,-3,-3)$, 15 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 6 simple diagrams with 13 crossings or fewer $\quad D(3,2,2,1,1,1,2)$ $b \geq 16$ $\quad D(3,2,2,1,2,3)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(3,2,3,2,1,2)$ $b \geq 17$ $\quad D(3,1,1,3,1,1,1,2)$ $b \geq 16$ $\quad D(3,2,1,1,2,1,1,2)$ $b \geq 16$
$12a_{722}$ 29/3 $C(9,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 79 diagrams of degree $(3,16,20)$. For example $\quad C(8,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(9,1,2)$ $b \geq 16$ $\quad D(10,3)$ $b \geq 19$
$12a_{723}$ 63/20 $C(3,6,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 52 diagrams of degree $(3,16,20)$. For example $\quad C(2,1,-7,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(3,6,1,2)$ $b \geq 16$ $\quad D(3,5,1,2,2)$ $b \geq 16$
$12a_{724}$ 107/31 $C(3,2,4,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,-5,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,5,2,3)$ $b \geq 16$ $\quad D(3,2,4,1,2)$ $b \geq 16$ $\quad D(3,1,1,5,1,2)$ $b \geq 16$ $\quad D(3,2,3,1,2,2)$ $b \geq 16$
$12a_{726}$ 103/19 $C(5,2,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,1,-3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,2,2,1,2)$ $b \geq 16$ $\quad D(5,2,3,3)$ $b \geq 17$ $\quad D(5,1,1,3,1,2)$ $b \geq 16$ $\quad D(5,2,1,1,2,2)$ $b \geq 16$
$12a_{727}$ 157/46 $C(3,2,2,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 43 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,-3,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,3,2,2,3)$ $b \geq 16$ $\quad D(3,2,2,2,1,2)$ $b \geq 16$ $\quad D(3,1,1,3,2,1,2)$ $b \geq 16$ $\quad D(3,2,1,1,3,1,2)$ $b \geq 16$ $\quad D(3,2,2,1,1,2,2)$ $b \geq 17$
$12a_{728}$ 133/29 $C(4,1,1,2,2,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,2,1,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,2,2,2)$ $b \geq 16$ $\quad D(4,2,3,2,2)$ $b \geq 16$ $\quad D(5,2,2,2,2)$ $b \geq 19$ $\quad D(4,1,1,1,1,3,2)$ $b \geq 16$ $\quad D(4,1,1,2,1,1,3)$ $b \geq 16$
$12a_{729}$ 167/46 $C(3,1,1,1,2,2,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(4,-2,-1,-2,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,1,2,2,2)$ $b \geq 16$ $\quad D(3,1,2,3,2,2)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(4,2,1,2,2,2)$ $b \geq 16$ $\quad D(3,1,1,1,1,1,3,2)$ $b \geq 16$
$12a_{731}$ 105/22 $C(4,1,3,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,2,1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,1,3,2,2)$ $b \geq 16$ $\quad D(5,4,2,2)$ $b \geq 19$ $\quad D(4,1,2,1,3,2)$ $b \geq 16$ $\quad D(4,1,3,1,1,3)$ $b \geq 16$
$12a_{732}$ 95/18 $C(5,3,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 61 diagrams of degree $(3,16,20)$. For example $\quad C(5,3,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,3,1,1,2)$ $b \geq 16$ $\quad D(5,3,2,3)$ $b \geq 16$ $\quad D(5,4,2,2)$ $b \geq 19$ $\quad D(5,2,1,2,1,2)$ $b \geq 16$
$12a_{733}$ 73/14 $C(5,4,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 47 diagrams of degree $(3,16,20)$. For example $\quad C(5,3,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,4,1,2)$ $b \geq 16$ $\quad D(5,5,3)$ $b \geq 16$ $\quad D(5,3,1,2,2)$ $b \geq 16$
$12a_{736}$ 141/43 $C(3,3,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,1,4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,3,1,1,2,2)$ $b \geq 16$ $\quad D(3,3,2,3,2)$ $b \geq 16$ $\quad D(3,4,2,2,2)$ $b \geq 19$ $\quad D(3,2,1,2,1,2,2)$ $b \geq 16$ $\quad D(3,3,1,1,1,1,3)$ $b \geq 16$
$12a_{738}$ 119/37 $C(3,4,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 55 diagrams of degree $(3,16,20)$. For example $\quad C(2,1,-5,-1,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,4,1,1,1,2)$ $b \geq 16$ $\quad D(3,4,1,2,3)$ $b \geq 16$ $\quad D(3,4,2,2,2)$ $b \geq 19$ $\quad D(3,5,2,1,2)$ $b \geq 16$ $\quad D(3,3,1,2,1,1,2)$ $b \geq 16$
$12a_{740}$ 113/35 $C(3,4,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 47 diagrams of degree $(3,16,20)$. For example $\quad C(2,1,-5,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,4,2,1,2)$ $b \geq 16$ $\quad D(3,4,3,3)$ $b \geq 16$ $\quad D(3,3,1,3,1,2)$ $b \geq 16$ $\quad D(3,4,1,1,2,2)$ $b \geq 16$
$12a_{743}$ 79/12 $C(6,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,-2,-1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(6,1,1,2,2)$ $b \geq 16$ $\quad D(6,2,3,2)$ $b \geq 17$ $\quad D(7,2,2,2)$ $b \geq 19$ $\quad D(6,1,1,1,1,3)$ $b \geq 16$
$12a_{744}$ 61/8 $C(7,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(7,1,2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(7,1,1,1,2)$ $b \geq 16$ $\quad D(7,1,2,3)$ $b \geq 16$ $\quad D(7,2,2,2)$ $b \geq 19$ $\quad D(8,2,1,2)$ $b \geq 17$
$12a_{745}$ 59/8 $C(7,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 33 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,-3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(7,2,1,2)$ $b \geq 16$ $\quad D(7,3,3)$ $b \geq 17$ $\quad D(7,1,1,2,2)$ $b \geq 17$
$12a_{758}$ 113/31 $C(3,1,1,1,4,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,2,-5,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,1,4,2)$ $b \geq 16$ $\quad D(3,1,2,5,2)$ $b \geq 16$ $\quad D(3,2,2,4,2)$ $b \geq 19$ $\quad D(4,2,1,4,2)$ $b \geq 16$
$12a_{759}$ 61/9 $C(6,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 61 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(6,1,3,2)$ $b \geq 16$ $\quad D(7,4,2)$ $b \geq 19$ $\quad D(6,1,2,1,3)$ $b \geq 16$
$12a_{760}$ 111/34 $C(3,3,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(3,2,1,-2,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,3,1,3,2)$ $b \geq 16$ $\quad D(3,4,4,2)$ $b \geq 19$ $\quad D(3,2,1,2,3,2)$ $b \geq 16$ $\quad D(3,3,1,2,1,3)$ $b \geq 16$
$12a_{761}$ 139/41 $C(3,2,1,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(2,3,1,1,3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,2,1,1,3,2)$ $b \geq 16$ $\quad D(3,2,2,4,2)$ $b \geq 19$ $\quad D(3,3,2,3,2)$ $b \geq 16$ $\quad D(3,1,1,2,1,3,2)$ $b \geq 16$ $\quad D(3,2,1,1,2,1,3)$ $b \geq 16$
$12a_{762}$ 51/7 $C(7,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 66 diagrams of degree $(3,16,20)$. For example $\quad C(7,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(7,3,2)$ $b \geq 16$ $\quad D(7,2,1,3)$ $b \geq 16$
$12a_{763}$ 97/30 $C(3,4,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 40 diagrams of degree $(3,16,20)$. For example $\quad C(3,-1,-2,-5,1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,4,3,2)$ $b \geq 16$ $\quad D(3,3,1,4,2)$ $b \geq 17$ $\quad D(3,4,2,1,3)$ $b \geq 16$
$12a_{764}$ 133/39 $C(3,2,2,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 44 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,-3,-3,1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,2,2,3,2)$ $b \geq 16$ $\quad D(3,1,1,3,3,2)$ $b \geq 16$ $\quad D(3,2,1,1,4,2)$ $b \geq 16$ $\quad D(3,2,2,2,1,3)$ $b \geq 16$
$12a_{773}$ 91/20 $C(4,1,1,4,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-2,-1,-4,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,4,2)$ $b \geq 16$ $\quad D(4,2,5,2)$ $b \geq 17$ $\quad D(5,2,4,2)$ $b \geq 19$ $\quad D(4,1,1,3,1,3)$ $b \geq 16$
$12a_{774}$ 89/16 $C(5,1,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,-2,-1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,3,2)$ $b \geq 16$ $\quad D(5,2,4,2)$ $b \geq 19$ $\quad D(6,2,3,2)$ $b \geq 17$ $\quad D(5,1,1,2,1,3)$ $b \geq 16$
$12a_{775}$ 87/16 $C(5,2,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 33 diagrams of degree $(3,16,20)$. For example $\quad C(5,2,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,2,3,2)$ $b \geq 16$ $\quad D(5,1,1,4,2)$ $b \geq 17$ $\quad D(5,2,2,1,3)$ $b \geq 16$
$12a_{791}$ 63/13 $C(4,1,5,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 63 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-2,-5,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,1,5,2)$ $b \geq 16$ $\quad D(5,6,2)$ $b \geq 19$ $\quad D(4,1,4,1,3)$ $b \geq 16$
$12a_{792}$ 85/24 $C(3,1,1,5,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(2,5,1,2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,5,2)$ $b \geq 16$ $\quad D(3,2,6,2)$ $b \geq 19$ $\quad D(4,2,5,2)$ $b \geq 17$
$12a_{796}$ 57/11 $C(5,5,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 46 diagrams of degree $(3,16,20)$. For example $\quad C(6,-1,-3,-1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(5,5,2)$ $b \geq 16$ $\quad D(5,4,1,3)$ $b \geq 16$
$12a_{797}$ 83/24 $C(3,2,5,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 25 diagrams of degree $(3,16,20)$. For example $\quad C(3,3,-1,-3,-1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,2,5,2)$ $b \geq 16$ $\quad D(3,1,1,6,2)$ $b \geq 17$ $\quad D(3,2,4,1,3)$ $b \geq 17$
$12a_{802}$ 47/15 $C(3,7,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 30 diagrams of degree $(3,16,20)$. For example $\quad C(4,-2,1,4,1,-3)$, 15 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(3,7,2)$ $b \geq 16$ $\quad D(3,6,1,3)$ $b \geq 16$
$12a_{803}$ 21/2 $C(10,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 142 diagrams of degree $(3,17,19)$. For example $\quad C(3,-1,-7,-1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(10,2)$ $b \geq 17$
$12a_{1023}$ 127/29 $C(4,2,1,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 67 diagrams of degree $(3,16,20)$. For example $\quad C(4,-2,-1,-2,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,2,1,1,1,3)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(4,3,2,1,3)$ $b \geq 16$ $\quad D(4,1,1,2,1,1,3)$ $b \geq 16$
$12a_{1024}$ 149/40 $C(3,1,2,1,1,1,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,1,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1029}$ 81/19 $C(4,3,1,4)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 75 diagrams of degree $(3,16,20)$. For example $\quad C(4,2,-1,-2,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(4,3,1,4)$ $b \geq 16$ $\quad D(5,4,4)$ $b \geq 19$
$12a_{1030}$ 91/19 $C(4,1,3,1,3)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,3,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1033}$ 107/25 $C(4,3,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 64 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-4,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,3,1,1,3)$ $b \geq 16$ $\quad D(4,3,2,4)$ $b \geq 17$ $\quad D(4,4,2,3)$ $b \geq 19$ $\quad D(4,2,1,2,1,3)$ $b \geq 16$
$12a_{1034}$ 121/32 $C(3,1,3,1,1,3)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 8 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,3,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,3,1,1,3)$ $b \geq 14$
$12a_{1039}$ 137/37 $C(3,1,2,2,1,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,2,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1040}$ 115/26 $C(4,2,2,1,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,2,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,2,2,1,3)$ $b \geq 14$
$12a_{1125}$ 101/23 $C(4,2,1,1,4)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,3,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,2,1,1,4)$ $b \geq 16$ $\quad D(4,3,2,4)$ $b \geq 17$ $\quad D(5,2,2,4)$ $b \geq 19$
$12a_{1126}$ 119/26 $C(4,1,1,2,1,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,1,2,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,1,2,1,3)$ $b \geq 14$
$12a_{1127}$ 97/22 $C(4,2,2,4)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 83 diagrams of degree $(3,17,19)$. For example $\quad C(3,1,-3,-3,1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 2 simple diagrams with 15 crossings or fewer $\quad D(4,2,2,4)$ $b \geq 17$ $\quad D(5,2,3,4)$ $b \geq 19$
$12a_{1128}$ 59/9 $C(6,1,1,4)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 75 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,-2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(6,1,1,4)$ $b \geq 16$ $\quad D(6,2,5)$ $b \geq 19$ $\quad D(7,2,4)$ $b \geq 19$
$12a_{1129}$ 105/23 $C(4,1,1,3,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 60 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,3,3)$ $b \geq 16$ $\quad D(4,2,4,3)$ $b \geq 19$ $\quad D(5,2,3,3)$ $b \geq 17$
$12a_{1130}$ 125/27 $C(4,1,1,1,2,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 63 diagrams of degree $(3,16,20)$. For example $\quad C(5,-2,-1,-2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,1,2,3)$ $b \geq 16$ $\quad D(4,1,2,3,3)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(5,2,1,2,3)$ $b \geq 16$
$12a_{1131}$ 73/11 $C(6,1,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 64 diagrams of degree $(3,16,20)$. For example $\quad C(7,-2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(6,1,1,1,3)$ $b \geq 16$ $\quad D(6,1,2,4)$ $b \geq 16$ $\quad D(6,2,2,3)$ $b \geq 19$ $\quad D(7,2,1,3)$ $b \geq 16$
$12a_{1132}$ 131/40 $C(3,3,1,1,1,3)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(4,-2,-1,-3,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,1,1,1,3)$ $b \geq 16$
$12a_{1133}$ 159/47 $C(3,2,1,1,1,1,3)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,2,1,1,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1134}$ 53/7 $C(7,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 76 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,-2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(7,1,1,3)$ $b \geq 16$ $\quad D(7,2,4)$ $b \geq 19$ $\quad D(8,2,3)$ $b \geq 19$
$12a_{1135}$ 103/32 $C(3,4,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,5,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,4,1,1,3)$ $b \geq 16$ $\quad D(3,5,2,3)$ $b \geq 16$ $\quad D(4,2,4,3)$ $b \geq 19$ $\quad D(3,3,1,2,1,3)$ $b \geq 16$
$12a_{1136}$ 147/43 $C(3,2,2,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,-3,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,2,2,1,1,3)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(3,2,1,1,2,1,3)$ $b \geq 16$
$12a_{1138}$ 79/14 $C(5,1,1,1,4)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 65 diagrams of degree $(3,16,20)$. For example $\quad C(6,-2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,2,1,5)$ $b \geq 16$ $\quad D(5,1,1,1,4)$ $b \geq 16$ $\quad D(5,2,2,4)$ $b \geq 19$ $\quad D(6,2,1,4)$ $b \geq 16$
$12a_{1139}$ 101/18 $C(5,1,1,1,1,3)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(5,1,1,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,1,1,1,1,3)$ $b \geq 14$
$12a_{1140}$ 97/18 $C(5,2,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,1,-2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,2,1,1,3)$ $b \geq 16$ $\quad D(5,2,2,4)$ $b \geq 19$ $\quad D(5,3,2,3)$ $b \geq 16$ $\quad D(5,1,1,2,1,3)$ $b \geq 16$
$12a_{1145}$ 79/15 $C(5,3,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 64 diagrams of degree $(3,16,20)$. For example $\quad C(5,2,1,-2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,3,1,3)$ $b \geq 16$ $\quad D(5,4,4)$ $b \geq 19$ $\quad D(5,2,1,2,3)$ $b \geq 16$
$12a_{1146}$ 117/34 $C(3,2,3,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,4,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,2,3,1,3)$ $b \geq 16$ $\quad D(4,4,2,3)$ $b \geq 19$ $\quad D(3,2,2,1,2,3)$ $b \geq 16$
$12a_{1148}$ 73/23 $C(3,5,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,6,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,5,1,3)$ $b \geq 16$ $\quad D(4,6,3)$ $b \geq 19$ $\quad D(3,4,1,2,3)$ $b \geq 16$
$12a_{1149}$ 35/4 $C(8,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 52 diagrams of degree $(3,16,20)$. For example $\quad C(7,1,-2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(8,1,3)$ $b \geq 16$ $\quad D(9,4)$ $b \geq 19$
$12a_{1157}$ 39/5 $C(7,1,4)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 68 diagrams of degree $(3,16,20)$. For example $\quad C(7,2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(7,1,4)$ $b \geq 16$ $\quad D(8,5)$ $b \geq 19$
$12a_{1158}$ 77/16 $C(4,1,4,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(4,-1,-3,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,4,3)$ $b \geq 16$
$12a_{1159}$ 113/24 $C(4,1,2,2,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,2,2,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1161}$ 75/14 $C(5,2,1,4)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(5,2,1,4)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,2,1,4)$ $b \geq 14$
$12a_{1162}$ 69/13 $C(5,3,4)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(5,-1,-3,1,4)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(5,3,4)$ $b \geq 16$ $\quad D(5,2,1,5)$ $b \geq 16$
$12a_{1163}$ 103/24 $C(4,3,2,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,1,-3,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,3,2,3)$ $b \geq 16$
$12a_{1165}$ 67/16 $C(4,5,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 36 diagrams of degree $(3,16,20)$. For example $\quad C(5,-1,-3,-1,4)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(4,5,3)$ $b \geq 16$ $\quad D(4,4,1,4)$ $b \geq 16$
$12a_{1166}$ 33/4 $C(8,4)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 172 diagrams of degree $(3,17,19)$. For example $\quad C(3,1,-7,-1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(8,4)$ $b \geq 17$
$12a_{1273}$ 61/11 $C(5,1,1,5)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 36 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,5)$ $b \geq 16$ $\quad D(6,2,5)$ $b \geq 19$
$12a_{1274}$ 95/17 $C(5,1,1,2,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,1,3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,2,3)$ $b \geq 16$ $\quad D(5,2,3,3)$ $b \geq 17$ $\quad D(6,2,2,3)$ $b \geq 19$ $\quad D(5,1,1,1,1,4)$ $b \geq 16$
$12a_{1275}$ 149/44 $C(3,2,1,1,2,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 2 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,2,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,2,1,1,2,3)$ $b \geq 16$
$12a_{1276}$ 75/13 $C(5,1,3,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 66 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,2,1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,1,3,3)$ $b \geq 16$ $\quad D(6,4,3)$ $b \geq 19$ $\quad D(5,1,2,1,4)$ $b \geq 16$
$12a_{1277}$ 121/37 $C(3,3,1,2,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,3,1,2,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1278}$ 41/6 $C(6,1,5)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 65 diagrams of degree $(3,16,20)$. For example $\quad C(6,2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(6,1,5)$ $b \geq 16$ $\quad D(7,6)$ $b \geq 19$
$12a_{1279}$ 67/10 $C(6,1,2,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(6,1,2,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(6,1,2,3)$ $b \geq 14$
$12a_{1281}$ 109/33 $C(3,3,3,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 28 diagrams of degree $(3,16,20)$. For example $\quad C(4,-1,-3,1,2,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 1 simple diagrams with 13 crossings or fewer $\quad D(3,3,3,3)$ $b \geq 16$
$12a_{1282}$ 63/10 $C(6,3,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 48 diagrams of degree $(3,16,20)$. For example $\quad C(6,2,1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(6,3,3)$ $b \geq 16$ $\quad D(6,2,1,4)$ $b \geq 16$
$12a_{1287}$ 37/6 $C(6,6)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 81 diagrams of degree $(3,17,19)$. For example $\quad C(7,-1,-3,-1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(6,6)$ $b \geq 17$


text:
# Why the Spectrogram can't plot correctly?

The same data lead two different graph.

Plot[E^(-Pi*(t - 5)^2), {t, -10, 10}, PlotRange -> All]

The method to plot the spectrogram use the build-in function:

 Samplerate = 20;data = Table[E^(-Pi*(t - 5)^2), {t, 0, 10, 1/Samplerate}];
Spectrogram[data, 10, 2, SampleRate -> Samplerate]

You can see the center is not 5.but if you use the following code(sorry,I don't know how to add a link here,thanks Sjord C. De Vries). The center is 5.

sndData = data;
sndSampleRate = 20;
min = Min[Abs[Fourier[sndData]]];
partSize = 10;
offset = 2;
spectroGramData = Take[20*Log10[Abs[Fourier[#]]/min],{2, partSize/2 // Floor}] & /@
Partition[sndData, partSize, offset]; MatrixPlot[
Reverse[spectroGramData\[Transpose]], ColorFunction -> "Rainbow",
DataRange ->Round[{{0, Length[sndData]/sndSampleRate}, {sndSampleRate/partSize,
sndSampleRate/2}}, 0.1], AspectRatio -> 1/2, ImageSize -> 800,
Frame -> True, FrameLabel -> {"Frequency (Hz)", "Time (s)", "", ""}, BaseStyle -> {FontFamily -> "Arial", FontWeight -> Bold, 12}]

Do I misunderstand something?

I don't really know if what you calculate in your code is the same as what Spectrogram is calculating, even aside from the log scale you use. Just compare the dimensions of the data being plotted,

Dimensions@spectroGramData
Dimensions@SpectrogramArray[data, 10, 2]
(* {96, 4} *)
(* {101, 10} *)

I can't quite figure out exactly how SpectrogramArray is calculating things, but I can say that if you want to have the center of the spectrogram show up in the right spot, you need to increase your sampling rate:

plotGrid[
Transpose@{Table[
Spectrogram[
Table[E^(-π*(t - 5.0)^2), {t, 0, 10, 1/Samplerate}], 10, 2,
SampleRate -> Samplerate], {Samplerate, {1, 2, 5, 10, 20, 30, 50,
100}}]}
, 400, 600]

where I'm using Jens's plotGrid function to line the plots up nicely.

format: plain


text:
Charged Current $\nu_\mu+e^-$ interaction?

Basically I want to know if there is something that forbids an interaction like $$\nu_\mu+e^-\rightarrow\mu^-+\nu_e$$ Having a look at the effective fermi interaction charged current lagrangian $$\mathcal{L}=-\sum_{\alpha,\beta}\frac{G_F}{\sqrt{2}} \left[\overline{\nu}_\alpha\gamma^\mu (1-\gamma^5)l_\alpha\right] \left[\overline{\nu}_\beta\gamma_\mu(1-\gamma^5)l_\beta\right]+\text{h.c.}$$ it should allow such an interaction? However in my textbook (Fundamentals of neutrino physics by Giunti) only electron charged current interactions are displayed. Is this only due to the larger energy needed to produce a muon or is there something I forgot that forbids this interaction?

• @Katermickie Yes. Even lower, really. For instance, the threshold for muon production via $\rm \nu_\mu e\rightarrow\mu\nu_e$ is around $10~\rm GeV$ in vacuum, and CC interactions in matter can occur at energies as low as $m_\mu$ in general. – Chris Aug 16 '18 at 16:18


text:
» » » Choose the option nearest in meaning to the word(s) or phrase in italicsHe saw m...

Choose the option nearest in meaning to the word(s) or phrase in italicsHe saw m...

Question

Choose the option nearest in meaning to the word(s) or phrase in italics

He saw me in camera

Options

A)
secretly
B)
openly
C)
in the television
D)
hatefully


text:
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Tags

# Signatures of anisotropic sources in the squeezed-limit bispectrum of the cosmic microwave background

(13 Feb 2013)  Key: citeulike:12147965

## Likes (beta)

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### Abstract

The bispectrum of primordial curvature perturbations in the squeezed configuration, in which one wavenumber, $k_3$, is much smaller than the other two, $k_3\ll k_1≈ k_2$, plays a special role in constraining the physics of inflation. In this paper we study a new phenomenological signature in the squeezed-limit bispectrum: namely, the amplitude of the squeezed-limit bispectrum depends on an angle between $k_1$ and $k_3$ such that $B_ζ(k_1, k_2, k_3) \to 2 ∑_L c_L P_L( k_1 ⋅ k_3) P_ζ(k_1)P_ζ(k_3)$, where $P_L$ are the Legendre polynomials. While $c_0$ is related to the usual local-form $f_ NL$ parameter as $c_0=6f_ NL/5$, the higher-multipole coefficients, $c_1$, $c_2$, etc., have not been constrained by the data. Primordial curvature perturbations sourced by large-scale magnetic fields generate non-vanishing $c_0$, $c_1$, and $c_2$. Inflation models whose action contains a term like $I(φ)^2 F^2$ generate $c_2=c_0/2$. A recently proposed "solid inflation" model generates $c_2\gg c_0$. A cosmic-variance-limited experiment measuring temperature anisotropy of the cosmic microwave background up to $\ell_ max=2000$ is able to measure these coefficients down to $δ c_0=4.4$, $δ c_1=61$, and $δ c_2=13$ (68% CL). We also find that $c_0$ and $c_1$, and $c_0$ and $c_2$, are nearly uncorrelated. Measurements of these coefficients will open up a new window into the physics of inflation such as the existence of vector fields during inflation or non-trivial symmetry structure of inflaton fields. Finally, we show that the original form of the Suyama-Yamaguchi inequality does not apply to the case involving higher-spin fields, but a generalized form does.


text:
# Inverse а Hermitian Matrix

Dear experts!

I have a small Hermitian matrix (6*6). I need to inverse this matrix. The program memory is bounded.
What method is optimal in this case?

Inversion codes and algorithm options.

You aren't clear on how memory bounded your program is,
so it's hard to say what algorithms are best.

That is a tiny matrix and, being 6x6 it is composed of
only 2x2 size 3x3 sub-blocks, or alternatively 3x3 size 2x2
sub-blocks. Thus you could use optimized 2x2 or 3x3 block
algorithms to help with your memory constraint.

Certainly such a target 6x6 complex
double precision COMPLEX*8 matrix takes only 576 bytes
of storage, though since it's Hermitian you can store its
lower or upper triangle in 288 bytes of memory.

Given that a few hundred bytes of code memory or data
memory is easy to expend whether you use a
code-intensive storage-reduced algorithm, or a
storage-intensive code-reduced algorithm, it's perhaps
application dependent as to what trade-off is best for you.

Furthermore, the optimum algorithm will depend on
whether you know of any additional structure of the
matrix besides it being Hermitian (e.g. sparseness,
is it positive definite, et. al.).

I'll provide some links and source codes for algorithms
to factor and invert either an indefinite or positive
definite Hermitian matrix using either packed or normal
storage and given either the upper or lower triangular
portion of the input Hermitian matrix as input to the
algorithm.

If you're going to port or rewrite the algorithms in some
memory conserving language like DSP code, C, assembler,
you'll probably find the LINPACK codes to be more explicit
since they make less use of subroutines, and the LAPACK
codes more algorithmically modular. Obviously one
would tend to just reimplement the code for one's
particular language and use case given your memory
constraints and for such a small and fixed problem size.

In general most of the work involves factoring the
input matrix, then simple algorithms can use the factored
matrix representations to calculate the inverse.

Generally the algorithms iterate over portions / blocks
of the input matrix, so, again, expanding the code for
the blocks / structure of your particular input will
yield some simplicity since it's a small and obviously
nicely blocked Hermitian matrix.

Of course if you're just trying to solve linear equations,
eigenproblems, get eigenvalues/eigenvectors, there
are other and usually better algorithms (e.g. in LAPACK/LINPACK) for such problem areas that don't
generally involve calculating the specific matrix inverse.

http://www.netlib.org/lapack/

LAPACK routines.

hermitian indefinite

http://www.netlib.org/lapack/explore-html/zhetrf.f.html [Broken]
Code:
	*  ZHETRF computes the factorization of a complex Hermitian matrix A
*  using the Bunch-Kaufman diagonal pivoting method.  The form of the
*  factorization is
*
*     A = U*D*U**H  or  A = L*D*L**H
*
*  where U (or L) is a product of permutation and unit upper (lower)
*  triangular matrices, and D is Hermitian and block diagonal with
*  1-by-1 and 2-by-2 diagonal blocks.
*
*  This is the blocked version of the algorithm, calling Level 3 BLAS.

*   CHETRF, ZHETRF:  Computes the factorization of a complex Hermitian
indefinite matrix, using the diagonal pivoting method.

*   CHETRI, ZHETRI:  Computes the inverse of a complex Hermitian
indefinite matrix, using the factorization computed by CHETRF.

hermitian indefinite packed storage

*   CHPTRF, ZHPTRF:  Computes the factorization of a complex Hermitian
indefinite matrix in packed storage, using the diagonal pivoting
method.

*   CHPTRI, ZHPTRI:  Computes the inverse of a complex Hermitian
indefinite matrix in packed storage, using the factorization computed
by CHPTRF.

hermitian positive definite

http://www.netlib.org/lapack/explore-html/zpotrf.f.html [Broken]

Code:
	*  ZPOTRF computes the Cholesky factorization of a complex Hermitian
*  positive definite matrix A.
*
*  The factorization has the form
*     A = U**H * U,  if UPLO = 'U', or
*     A = L  * L**H,  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular.
*
*  This is the block version of the algorithm, calling Level 3 BLAS.

*   SPOTRF, DPOTRF, CPOTRF, ZPOTRF:  Computes the Cholesky factorization
of a symmetric or Hermitian positive definite matrix.

*   SPOTRI, DPOTRI, CPOTRI, ZPOTRI:  Computes the inverse of a symmetric
or Hermitian positive definite matrix, using the Cholesky
factorization computed by SPOTRF or CPOTRF.

hermitian positive definite packed storage

*   SPPTRF, DPPTRF, CPPTRF, ZPPTRF:  Computes the Cholesky factorization
of a symmetric or Hermitian positive definite matrix in packed
storage.

*   SPPTRI, DPPTRI, CPPTRI, ZPPTRI:  Computes the inverse of a symmetric
or Hermitian positive definite matrix in packed storage, using the
Cholesky factorization computed by SPPTRF or CPPTRF.

........................................

http://www.netlib.org/linpack/

LINPACK routines

Code:
hermitian indefinite
single precision

file	chifa.f  chifa.f plus dependencies
gams	D2d1a
for	factors a complex Hermitian matrix
,	by elimination with symmetric pivoting
prec	complex

file	chidi.f  chidi.f plus dependencies
gams	D2d1a, D3d1a
for	computes the determinant, inertia and inverse of a complex
,	Hermitian matrix using the factors from linpack/chifa
prec	complex

hermitian indefinite
double precision

file	zhifa.f  zhifa.f plus dependencies
gams	D2d1a
for	factors a complex Hermitian matrix
,	by elimination with symmetric pivoting
prec	doublecomplex

file	zhidi.f  zhidi.f plus dependencies
gams	D2d1a, D3d1a
for	computes the determinant, inertia and inverse of a complex
,	Hermitian matrix using the factors from linpack/zhifa)
prec	doublecomplex

hermitian indefinite
single precision packed storage

file	chpfa.f  chpfa.f plus dependencies
gams	D2d1a
for	factors a complex Hermitian matrix
,	stored in packed form by elimination with symmetric pivoting
prec	complex

file	chpdi.f  chpdi.f plus dependencies
gams	D2d1a, D3d1a
for	computes the determinant, inertia and inverse of a
,	complex Hermitian matrix using the factors from linpack/chpfa,
,	where the matrix is stored in packed form
prec	complex

hermitian indefinite
double precision packed storage

file	zhpfa.f  zhpfa.f plus dependencies
gams	D2d1a
for	factors a complex Hermitian matrix
,	stored in packed form by elimination with symmetric pivoting
prec	doublecomplex

file	zhpdi.f  zhpdi.f plus dependencies
gams	D2d1a, D3d1a
for	computes the determinant, inertia and inverse of a
,	complex Hermitian matrix using the factors from linpack/zhpfa,
,	where the matrix is stored in packed form
prec	doublecomplex

Last edited by a moderator:


text:
# Why are gravitons tensorial?

1. Dec 8, 2012

### DrDu

Why are gravitons tensorial spin 2 particles while Newtonian gravity is a scalar?

2. Dec 8, 2012

### Dickfore

Because the spacetime metric is a tensor. And spin-2 comes from the fact that
(1/2,1/2) x (1/2, 1/2) = (0,0) + (1,0) + (0,1) + (1,1)

so a symmetric traceless tensor is in the (1,1) representation, whose spatial part corresponds to S=2 rep. of the so(3) ~ su(2) Lie algebra of spatial rotations.

EDIT:
Scalar gravitational potential from Newton's theory is the g00 perturbation to the Minkowski metric.

Last edited: Dec 8, 2012
3. Dec 8, 2012

### DrDu

Thank you Dick!

4. Dec 8, 2012

### haushofer

The newton potential is not a scalar under general coordinate transformations, only under the galilei group. More precisely, as soon as you make boosts quadratic in time or more, which are (time dependent) accelerations, the potential transforms inhomogeneously. You can derive this from the fact that the potential comes from the i00 component of the connection.

5. Dec 9, 2012

### atyy

I don't think there is a "why". It's just like that.

Nordstrom's second theory is a scalar relativistic theory of gravity that preceded general relativity. It satisfies the equivalence principle, and has a geometric formulation. It happens to predict the wrong perihelion precession.

http://arxiv.org/abs/gr-qc/0611100
http://arxiv.org/abs/gr-qc/0405030

Also relevant is exercise 24.1c in http://www.pma.caltech.edu/Courses/ph136/yr2006/0424.1.K.pdf about a scalar theory of gravity: "Explain why this prediction implies that there will be no deflection of light around the limb of the sun, which conflicts severely with experiments that were done after Einstein formulated his general theory of relativity. (There was no way, experimentally, to rule out the above theory in the epoch, ca. 1914, when Einstein was doing battle with his colleagues over whether gravity should be treated within the framework of special relativity or should be treated as a geometric extension of special relativity.)"

6. Dec 10, 2012

### dextercioby

The why is answered in a simple manner: Gravitons per definition are excitations of the quantized graviton field* (in exactly the same manner in which photons are excitations, i.e. 1-particle states of the quantized electromagnetic field in vacuum) which is described through a tensorial object carrying 2 flat space-time indices on the same position, covariant, and which is symmetric wrt interchange of the indices. An object with 2 tensorial indices <downstairs> automatically contains a spin 2 field (the spin 1 content is removed through symmetrization and the spin 0 content by redefining the field to be traceless) as it follows from the theory of non-unitary finite dimensional vectorial representations of the connected component of the Lorentz group.

*The graviton field is a first perturbation of the curved spacetime metric around the flat Minkowski background is conventionally called <the Pauli-Fierz field> and obeys the same quantum dynamics as a linearized vierbein (the antisymmetric components of the linearized vierbein do not propagate) which is used in supergravity theories.

7. Dec 11, 2012

### haushofer

What exactly do you mean by "antisymmetric components of the vierbein"? The vierbein has two different components, namely a flat and a curved one, so it doesn't really make sense to take the antisymmetric part of it; then you're already talking about the metric, isn't it? :)

8. Dec 11, 2012

### andrien

I don't think it is appropriate to say that because of two indices on some stress-energy tensor type thing,one can say that it is spin 2.One say that with electromagnetic field the origin is charge current which has one index so it represents spin-1.this is really not right.If I remember it, then in feynman lectures on gravitation it is pointed out that spin 2 is the lowest possible spin which can be chosen,and is satisfactory.

9. Dec 11, 2012

### haael

The question was about the difference between Newton and Einstein theories of gravity. In Newton's the gravitational potential is scalar. Why is it a tensor in Einstein's?

One simple answer might be like this:
In Newton's gravity the source of gravitational interaction is mass. Mass is a scalar. To describe an interaction intensity between two scalars, we need one scalar.

In Einstein's gravity the source of interaction is the whole energy-momentum 4-vector. To describe interaction between components of two 4-vectors we need a 4-tensor. Some other arguments are needed to show that the tensor must be symmetric.

This is not a mathematical derivation, rather an intuitive explanation. The very same line may be answered to a question why interactions of spinors are described by gauge-covariant vector fields.

10. Dec 11, 2012

### dextercioby

The linearized vierbein (the quotation left the essential word <linearized> out) is a genuine 2nd rank tensor in flat spacetime. It has a 0 + 1 + 2 spin content wrt both the connected component of O(1,3) and SL(2,C).


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

I’m back in business! After almost 5 months of hiatus, during which I was very busy with my new job, and new house, I’m in a position where I can write again. To celebrate my comeback, I’ll introduce to you the {tidytable} package, which I learned about this week on Twitter.

{tidytable} is a package that allows you to manipulate data.table objects with the speed of {data.table} and the convenience of the {tidyverse} syntax. My first reaction when I heard about {tidytable} was how is that different from {dtplyr}? Well, {dtplyr} focuses on providing a {data.table} backend for {dplyr}, while {tidytable} also allows you to use other {tidyverse} verbs on data.table objects, for instance some {tidyr} and {purrr} verbs.

Another very interesting feature of {tidytable} is that it supports {rlang}, which means that you can program with {tidytable}, which, as far as I know, is not possible with {dtplyr} (but fact-check me on that please).

So to summarise, the speed of {data.table} and the syntax of the {tidyverse}, plus verbs for {tidyr} and {purrr}? Sign me up!

To illustrate, I have downloaded a data set and wrote a function in both a {tidyverse} version and a {tidytable} version. Even though it is true that {tidytable}’s syntax is very much, almost the same as the regular {tidyverse} syntax, there are some minor differences. But more on that later. First, let’s get the data, which you can find here. Then, let’s load the needed packages:

library(tidyverse)
library(data.table)
library(tidytable)
library(readr)

and let’s take a look at the data a little bit:

energy <- read.csv("~/Downloads/energydata_complete.csv")

##                  date Appliances lights    T1     RH_1   T2     RH_2    T3
## 1 2016-01-11 17:00:00         60     30 19.89 47.59667 19.2 44.79000 19.79
## 2 2016-01-11 17:10:00         60     30 19.89 46.69333 19.2 44.72250 19.79
## 3 2016-01-11 17:20:00         50     30 19.89 46.30000 19.2 44.62667 19.79
## 4 2016-01-11 17:30:00         50     40 19.89 46.06667 19.2 44.59000 19.79
## 5 2016-01-11 17:40:00         60     40 19.89 46.33333 19.2 44.53000 19.79
## 6 2016-01-11 17:50:00         50     40 19.89 46.02667 19.2 44.50000 19.79
##       RH_3       T4     RH_4       T5  RH_5       T6     RH_6       T7     RH_7
## 1 44.73000 19.00000 45.56667 17.16667 55.20 7.026667 84.25667 17.20000 41.62667
## 2 44.79000 19.00000 45.99250 17.16667 55.20 6.833333 84.06333 17.20000 41.56000
## 3 44.93333 18.92667 45.89000 17.16667 55.09 6.560000 83.15667 17.20000 41.43333
## 4 45.00000 18.89000 45.72333 17.16667 55.09 6.433333 83.42333 17.13333 41.29000
## 5 45.00000 18.89000 45.53000 17.20000 55.09 6.366667 84.89333 17.20000 41.23000
## 6 44.93333 18.89000 45.73000 17.13333 55.03 6.300000 85.76667 17.13333 41.26000
##     T8     RH_8       T9  RH_9    T_out Press_mm_hg RH_out Windspeed Visibility
## 1 18.2 48.90000 17.03333 45.53 6.600000       733.5     92  7.000000   63.00000
## 2 18.2 48.86333 17.06667 45.56 6.483333       733.6     92  6.666667   59.16667
## 3 18.2 48.73000 17.00000 45.50 6.366667       733.7     92  6.333333   55.33333
## 4 18.1 48.59000 17.00000 45.40 6.250000       733.8     92  6.000000   51.50000
## 5 18.1 48.59000 17.00000 45.40 6.133333       733.9     92  5.666667   47.66667
## 6 18.1 48.59000 17.00000 45.29 6.016667       734.0     92  5.333333   43.83333
##   Tdewpoint      rv1      rv2
## 1       5.3 13.27543 13.27543
## 2       5.2 18.60619 18.60619
## 3       5.1 28.64267 28.64267
## 4       5.0 45.41039 45.41039
## 5       4.9 10.08410 10.08410
## 6       4.8 44.91948 44.91948

As you can see, this data is wide, and not long. Variables, or features, T1 to T9 provide the temperature of 9 rooms, and RH_1 to RH_9 provide the humidity of the same 9 rooms.

What if I’d like to make a plot of each room’s temperature throughout the year? In this format, it is not possible. So let’s reshape this a little bit:

flat_energy <- energy %>%
pivot_longer(cols = matches("T\\d{1}"), names_to = "temperature", values_to = "temp_value") %>%
pivot_longer(cols = matches("RH_\\d{1}"), names_to = "humidity", values_to = "hum_value") %>%
mutate(temperature = case_when(temperature == "T1" ~ "kitchen",
temperature == "T2" ~ "living",
temperature == "T3" ~ "laundry",
temperature == "T4" ~ "office",
temperature == "T5" ~ "bathroom",
temperature == "T6" ~ "north",
temperature == "T7" ~ "ironing",
temperature == "T8" ~ "teenager",
temperature == "T9" ~ "parents")) %>%
mutate(humidity = case_when(humidity == "RH_1" ~ "kitchen",
humidity == "RH_2" ~ "living",
humidity == "RH_3" ~ "laundry",
humidity == "RH_4" ~ "office",
humidity == "RH_5" ~ "bathroom",
humidity == "RH_6" ~ "north",
humidity == "RH_7" ~ "ironing",
humidity == "RH_8" ~ "teenager",
humidity == "RH_9" ~ "parents"))

As explained above, there are two variables that need this treatment; the temperature, and the humidity levels. In order to plot the average monthly temperature in each room, I need to use tidyr::pivot_longer() (a little side note, I could have used names_to = "room", instead of "temperature" and "humidity", but there’s a reason for that. More on it below).

Now let’s plot it:

flat_energy %>%
mutate(month = month(date)) %>%
group_by(month, temperature) %>%
summarise(avg_temp = mean(temp_value)) %>%
ggplot() +
geom_line(aes(y = avg_temp, x = month, col = temperature)) +
brotools::theme_blog()
## summarise() regrouping output by 'month' (override with .groups argument)

Ok great. But what if I had such a dataset per house for a whole city? How many datasets would that be? And how long would these operations take? The first step I would take if I were in this situation, would be to write a function. I would make it general enough to work with temperature or humidity. Below is this function:

prepare_data <- function(energy, variable){

variable <- enquo(variable)

variable_label <- as_label(variable)

regex_selector <- ifelse(variable_label == "temperature",
"T\\d{1}",
"RH_\\d{1}")
energy %>%
pivot_longer(cols = matches(regex_selector),
names_to = variable_label,
values_to = paste0(variable_label, "_value")) %>%
mutate(!!(variable) := case_when(grepl("1$", !!(variable)) ~ "kitchen", grepl("2$", !!(variable)) ~ "living",
grepl("3$", !!(variable)) ~ "laundry", grepl("4$", !!(variable)) ~ "office",
grepl("5$", !!(variable)) ~ "bathroom", grepl("6$", !!(variable)) ~ "outside",
grepl("7$", !!(variable)) ~ "ironing", grepl("8$", !!(variable)) ~ "teenager",
grepl("9$", !!(variable)) ~ "parents")) %>% mutate(month = month(date)) %>% group_by(month, !!(variable)) %>% summarise(across(.cols = ends_with("_value"), .fns = mean), .groups = "drop") } This function does exactly the same thing as above: prepare_data(energy, temperature) %>% ggplot() + geom_line(aes(y = temperature_value, x = month, col = temperature)) + brotools::theme_blog() As you can see, I have the exact same plot. What’s nice with this function, is that it uses many verbs from the {tidyverse} as well as the tidy eval framework for non-standard evaluation ( which is why I did not use names_to = "room", I wanted to use the variable label defined with as_label() and see if it works with {tidytable} as well). Ok, so now let’s imagine that I’m happy with this function, but I’d like it to run faster, and because I’m lazy, the less I have to modify it, the happier I am. This is where {tidytable} looks very promising. Let’s rewrite the function to make it work with {tidytable}: prepare_data_dt <- function(energy, variable){ variable <- enquo(variable) variable_label <- as_label(variable) regex_selector <- ifelse(variable_label == "temperature", "T\\d{1}", "RH_\\d{1}") energy %>% pivot_longer.(cols = matches(regex_selector), names_to = variable_label, values_to = paste0(variable_label, "_value")) %>% mutate.(!!(variable) := case_when(grepl("1$", !!(variable)) ~ "kitchen",
grepl("2$", !!(variable)) ~ "living", grepl("3$", !!(variable)) ~ "laundry",
grepl("4$", !!(variable)) ~ "office", grepl("5$", !!(variable)) ~ "bathroom",
grepl("6$", !!(variable)) ~ "outside", grepl("7$", !!(variable)) ~ "ironing",
grepl("8$", !!(variable)) ~ "teenager", grepl("9$", !!(variable)) ~ "parents")) %>%
mutate.(month = month(date)) %>%
summarise_across.(.cols = ends_with("_value"),
.fns = mean,
.by = c(month, !!(variable))) %>%
ungroup()
}

As you can see, it’s almost the same thing. {tidytable} verbs end with a '.' and that’s it. Well almost (again), the biggest difference is how {tidytable} groups by a variable. It’s very similar to how it’s done in {data.table}, by using a .by = argument to verbs that support it, such as summarise_across() (which is also, by the way, another difference with standard {tidyverse} syntax). While I’ll have to remember these, I’d argue that they’re minor differences and if it can make my function run faster, I don’t mind!

Now let’s run a little benchmark. But first, let’s define our data as a tidytable object:

energy_tidytable <- as_tidytable(energy)

Now we’re good to go:

microbenchmark::microbenchmark(
energy %>%
prepare_data(temperature),
energy_tidytable %>%
prepare_data_dt(temperature),
times = 10
)
## Unit: milliseconds
##                                               expr      min       lq     mean
##               energy %>% prepare_data(temperature) 847.9709 849.6671 868.6524
##  energy_tidytable %>% prepare_data_dt(temperature) 820.2051 838.6647 861.9685
##    median       uq      max neval
##  861.0652 880.8200 914.4685    10
##  858.9454 873.3268 936.0147    10

That is nice! It does indeed run faster, and with only some minor changes to the function! And how about using some more cores to run this function? This can be done using data.table::setDTthreads(n_cores) where n_cores is the number of cores you want to use:

data.table::setDTthreads(12)
microbenchmark::microbenchmark(
energy %>%
prepare_data(temperature),
energy_tidytable %>%
prepare_data_dt(temperature),
times = 10
)
## Unit: milliseconds
##                                               expr      min       lq     mean
##               energy %>% prepare_data(temperature) 832.9876 840.8000 874.3047
##  energy_tidytable %>% prepare_data_dt(temperature) 829.7937 831.2868 866.4383
##    median       uq      max neval
##  889.2684 898.6861 914.7178    10
##  836.8712 893.0613 997.8511    10

Maybe surprisingly, it did not run faster. It could very well be that my function does not really lend itself to running in parallel, and the overhead induced by distributing the work to the cpu cores cancels out the gains from running it in parallel. But in any case, this is really looking very interesting. I have not tested the whole package yet, but since the syntax is so similar to the {tidyverse}, you can try really quickly to see if the {tidytable} version of the function runs faster, and if yes, I don’t really see a reason not to use it!

Check out the project’s website here, and follow the author’s twitter here.

Hope you enjoyed! If you found this blog post useful, you might want to follow me on twitter for blog post updates and buy me an espresso or paypal.me, or buy my ebook on Leanpub.


text:
# Is there a general relativistic relativistic rocket equation?

I am not to familiar with rocket equations but I know that if you combine special relativity with classical mechanics you get "c" as a speed limit. Basically the closer you get to "c" the more you increase in relativistic momenta without increasing so much in velocity.

Now in general relativity there are two other effects that I can think of. We consider the case where the rocket is moving in a spherically symmetric gravitational field.

1. The speed of light is anisotropic and varies with altitude. In the Scharzschild solution in Schwarzschild coordinates in the radial direction in coordinate time you have $$v_{light}=c(1-\frac{2GM}{rc^2})$$ and in the direction transverse to the radial direction you have $$v_{light}=c\sqrt{1-\frac{2GM}{rc^2}}$$. The fact that the speed limit varies in this way should affect the rocket equation somehow.
2. The gravitational time dilation/gravitational red shift I think that in coordinate time the gravitationl acceleration for an object standing on the face of a planet such as the Earth is the same as classically, $$\frac{dv}{dt}=-\frac{GM}{r^2}$$. However in proper time you have $$\frac{dv}{dt}=-\frac{GM}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{rc^2}}}$$. I think this is correct. Equivalently the energy of the fuel in the rocket will be "redshifted" so the rockets will basically not contains as much energy as classically. This might somehow affect the rocket equation. This is sometimes a confusing subject.

Question: Is there a general relativistic, relativistic rocket equation and what does it look like?

I guess effects such as those mentioned might be to miniscule to bother with in the solar system but at least you should be able to prove that you can disregard them.

Edit: While not exactly a "rocket-equation" a common way to include effects of acceleration due to relativity is to use the post-Newtonian expansion. I wrote down the 3PN-accelerations for the most simple case of one test body in a spherical field on the physics site. At the 1PN level the acceleration becomes:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}+\frac{v^2}{c^2}\right)\hat{r} +\frac{4GM}{r^2}\left(\hat{r}\cdot \hat{v}\right)\frac{v^2}{c^2}\hat{v}$$

This is an expression that JPL always uses for orbit computation. When you do not have only one massive body of spherical symmetry there are more terms. For a rocket standing still, $$v=0$$, on the face of the earth you get:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}\right)\hat{r}$$

and for a rocket travelling radially upwards you get:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}-3\frac{v^2}{c^2}\right)\hat{r}$$

Question: How do these two extra terms make it harder/easier for a rocket to escape the gravitational field of the Earth?

• Yikes, great question :-) See the last paragraph in @DavidHammen's answer. Yikes again!
– uhoh
May 14 '19 at 0:40
• In which way is the relativistic rocket equation as given by wikipedia unsatisfactory? Your point (2) is explicitly mentioned in the derivation and I am not sure (1) matters. May 14 '19 at 16:08
• @Polygnome Can you point me to it? The first sentence in the article says "Relativistic rocket refers to any spacecraft that travels at a velocity close enough to light speed for relativistic effects to become significant." but the second effect should maybe be there even if you go very slow as compared to the speed of light. You might need more fuel in your rocket if you try to escape from a planet governed by GR and not by Classical Newton. If there is no gravitational field present or it can be neglected then I guess SR should be enough. May 14 '19 at 16:31
• If you rocket is traveling radially upward from a planet your velocity as a fraction of the speed of light can decrease even if you speed up, as measured in coordinate time. In "proper time" of the rocket things might be different. I do not know how much 1 matters or not, but if it do not matter, it would be great if someone could prove it. May 14 '19 at 16:42
• @Polygnome: The article presents a SRT rocket, OP asks for a GRT extension. May 18 '19 at 11:39


text:
# Star Shining Bright

A cepheid variable star's brightness can be predicted by observing the period over which its luminosity varies.

A particular cepheid variable is found by this method to have a luminosity of $4\times 10 ^ {31} \text{ W}$. The intensity of its light on Earth is $0.00025 \text{ W/m}^2$. How far away is the star in metres?

×


text:
# How to show that $f$ and $\mathcal{F}(f)$ can't both be compactly supported?

Let $f : \mathbb{R}\to \mathbb{R}$ be continuous, I want to show that it can't happen that both $f$ and its Fourier transform $\mathcal{F}(f)$ are compactly supported unless $f = 0$.

That means that I want to show that if both $f$ and $\mathcal{F}(f)$ are compactly supported, $f = 0$.

I've seem some questions like this here, but I believe this is not duplicate. That because on those questions I've seem people using the Fourier series, while here I'm talking about the Fourier transform.

In the Fourier series we require $f$ to be periodic. Here $f$ needs not to be periodic.

In that case, I can't restrict $f$ to $[-\pi,\pi]$ and talk about its Fourier series, because it need not be the case that $f(-\pi)=f(\pi)$, since $f$ is arbitrary.

Now, if $f$ is compactly supported, there's an interval $[a,b]$ such that $f(x) = 0$ for all $x\notin [a,b]$. If $\mathcal{F}(f)$ also is compactly supported there's $[c,d]$ such that $\mathcal{F}(f)(\xi)=0$ if $\xi\notin [c,d]$.

I've then thought of two approaches:

1. Try to use this together with the Fourier inversion formula, to show that $f(x) = 0$.

2. Try to use this together with Plancherel's theorem to show that $\|f\|_2 =0$ and hence $f = 0$.

Now, in the first case I get

$$f(x)=\int_{c}^{d}\mathcal{F}(f)(\xi)e^{2\pi i x\xi}d\xi = \int_c^d \int_a^b f(y)e^{-2\pi i y\xi}e^{2\pi ix\xi} dyd\xi$$

Or yet

$$f(x)=\int_a^b f(y)\int_c^d e^{2\pi i\xi (x-y)}d\xi dy,$$

but this doesn't seem to lead anywhere.

Plancherel's theorem also doesn't seem of great aid here. We have

$$\int_a^b |f(x)|^2 dx = \int_c^d |\mathcal{F}(f)(\xi)|^2 d\xi,$$

which doesn't help much indeed.

How can I show that if both $f$ and its Fourier transform are compactly supporetd then $f = 0$ in the context of the Fourier transform and not Fourier series?

• Do you know a Paley-Wiener-theorem that states that a $\mathcal F(f)$ is the restriction of an entire function if $f$ is compactly supported? – Tim B. Dec 11 '16 at 15:05
• No, although I've heard of this result, I'm trying to prove the claim without using it. – Gold Dec 11 '16 at 15:27

If $f$ is compactly supported then \begin{align} \hat{f}(\xi) & =\frac{1}{\sqrt{2\pi}}\int_{-M}^{M}f(x)e^{-ix\xi}dx \\ & = \frac{1}{\sqrt{2\pi}}\int_{-M}^{M}f(x)\sum_{n=0}^{\infty}\frac{(-ix\xi)^n}{n!}dx \\ & = \sum_{n=0}^{\infty}\left[\frac{(-i)^n}{\sqrt{2\pi}n!}\int_{-M}^{M}f(x)x^{n}dx\right]\xi^n \end{align} This power series converges everywhere. A power series cannot have an infinite set of zeros with a finite accumulation point, unless it is identically $0$. So $\hat{f}$ cannot be compactly supported if $f$ is compactly supported, unless $f$ is identically $0$.
• @user1620696 : If you expand the power series about a zero, then there's a first coefficient that is non-zero, allowing you to factor out $(\xi-\xi_0)^{n}$ and end up with a power series that is not zero at $\xi=\xi_0$. Then, by continuity, the power series cannot vanish in a neighborhood of $\xi_0$, which proves that zeros of a power series are isolated in the interior of the region of convergence. – Disintegrating By Parts Dec 11 '16 at 22:25
• @user1620696 : Yes, you are correct that the power series converges uniformly, which allows you to interchange integration and summation over the finite interval $[-M,M]$. – Disintegrating By Parts Dec 11 '16 at 22:27


text: Electrons fly straight out of an electron gun with velocity $$v$$. However, since the electron is located in a homogeneous magnetic field $$B$$, which is directed towards the screen plane, the electron is deflected due to Lorentz force. A charged particle like the electron is subjected to a magnetic force $$F$$ (magnetic component of the Lorentz force) when it flies through a magnetic field. The magnetic force is always orthogonal to the velocity, which causes the electron to move circular.