text:
\section{The Formulas} \label{sec:Formulas} One of the selling points for this
article is that the formulas in it are concise. Thus we start by running
through these formulas for a knot invariant $\rho_1$ as quickly as we
can. In Section~\ref{sec:Implementation} we turn the formulas into a
short yet very fast computer program, in Section~\ref{sec:Proofs} we
give a partial interpretation of the formulas in terms of car traffic
on a knot diagram and use it to prove the invariance of $\rho_1$, and
in Section~\ref{sec:Context} we quickly sketch the context: Alexander,
Burau, Jones, Melvin, Morton, Rozansky, Overbay, and our own prior
work. This article accompanies a talk,~\cite{Talk:Waco-2203} (video
and handout available), and a future talk,~\cite{Talk:Geneva-2206}
(registration link and handout available).

\parpic[r]{\input{figs/SampleKnot.pdf_t}}
Given an oriented $n$-crossing knot $K$, we draw it in the plane as a
long knot diagram $D$ in such a way that the two strands intersecting
at each crossing are pointed up (that's always possible because we
can always rotate crossings as needed), and so that at its beginning
and at its end the knot is oriented upward. We call such a diagram an
{\em upright knot diagram.} An example of an upright knot diagram $D_3$
is shown on the right.

We then label each edge of the diagram with two integer labels: a running index $k$ which runs from 1 to $2n+1$, and a ``rotation number'' $\varphi_k$, the geometric rotation number of that edge (the signed number of times the tangent to the edge is horizontal and heading right, with cups counted with $+1$ signs and caps with $-1$; this number is well defined because at their ends, all edges are headed up). On the right the running index runs from $1$ to $7$, and the rotation numbers for all edges are $0$ (and hence are omitted) except for $\varphi_4$, which is $-1$.

{\em A Technicality.} Some Reidemeister moves create or lose an edge and to avoid the need for renumbering it is beneficial to also allow labelling the edges with non-consecutive labels. Hence we allow that, and write ${i^+}} \def\jp{{j^+}} \def\kp{{k^+}$ for the successor of the label $i$ along the knot, and ${i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}}$ for the successor of ${i^+}} \def\jp{{j^+}} \def\kp{{k^+}$ (these are $i+1$ and $i+2$ if the labelling is by consecutive integers). Also, ``$1$'' will always refer to the label of the first edge, and ``$2n+1$'' will always refer to the label of the last.

We let $A$ be the $(2n+1)\times(2n+1)$ matrix of Laurent polynomials in the formal variable $T$ defined by
\[ A \coloneqq I - \sum_c \left( T^sE_{i,{i^+}} \def\jp{{j^+}} \def\kp{{k^+}} + (1-T^s)E_{i,\jp} + E_{j,\jp} \right), \]
where $I$ is the identity matrix, $E_{\alpha\beta}$ denotes the elementary
matrix with $1$ in row $\alpha$ and column $\beta$ and zeros elsewhere.
The summation is over the crossings $c$ of the diagram $D$, and once $c$
is chosen, $s$ denotes its sign and $i$ and $j$ denote the labels below
the crossing where the label $i$ belongs to the over-strand and $j$
to the under-strand.

Alternatively, $A=I + \sum_c A_c$, where $A_c$ is a matrix of zeros except for the blocks as follows:
\begin{equation} \label{eq:A}
  \begin{array}{c}\input{figs/Xings.pdf_t}\end{array}
  \qquad\longrightarrow\qquad
  \begin{array}{c|cccc}
    A_c &   \text{column }{i^+}} \def\jp{{j^+}} \def\kp{{k^+}  &  \text{column }\jp \\
    \hline
    \text{row }i & -T^s  & T^s-1 \\
    \text{row }j & 0  & -1
  \end{array}
\end{equation}

\parpic[r]{\input{figs/D2.pdf_t}}
For example, if $D=D_1=\ \uparrow$ is the diagram with no crossings, the resulting matrix $A$ is the $1\times 1$ identity matrix $(1)$. If $D=D_2$ is the diagram on the right (here $s=+1$, $(i,j)=(2,1)$, and $({i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp)=(3,2)$), then
\[ A =
  \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}
  + \begin{pmatrix}0&-1&0\\0&T-1&-T\\0&0&0\end{pmatrix}
  = \begin{pmatrix}1&-1&0\\0&T&-T\\0&0&1\end{pmatrix},
\]
and for $D_3$ as on the first page, we have
\[ A = \footnotesize \left(
  \begin{array}{ccccccc}
   1 & -T & 0 & 0 & T-1 & 0 & 0 \\
   0 & 1 & -1 & 0 & 0 & 0 & 0 \\
   0 & 0 & 1 & -T & 0 & 0 & T-1 \\
   0 & 0 & 0 & 1 & -1 & 0 & 0 \\
   0 & 0 & T-1 & 0 & 1 & -T & 0 \\
   0 & 0 & 0 & 0 & 0 & 1 & -1 \\
   0 & 0 & 0 & 0 & 0 & 0 & 1 \\
  \end{array}
\right). \]

\picskip{0}
We note without supplying details that the matrix $A$ comes in a
straightforward way from Fox calculus as it is applied to the Wirtinger
presentation of the fundamental group of the complement of $K$ (using
the diagram $D$). Hence the determinant of $A$ is equal up to a unit
to the normalized Alexander polynomial $\Delta$ of $K$ (which satisfies
$\Delta(T)=\Delta(T^{-1})$ and $\Delta(1)=1$). In fact, we have that
\begin{equation} \label{eq:Delta} \Delta = T^{(-\varphi(D)-w(D))/2}\det(A), \end{equation}
where $\varphi(D)\coloneqq\sum_k\varphi_k$ is the total rotation number of $D$ and where $w(D)=\sum_cs_c$ is the writhe of $D$, namely the sum of the signs $s_c$ of all the crossings $c$ in $D$.

For our example $D_2$, $\det(A)=T$, $\varphi(D)=1$, and $w(D)=1$, so $\Delta=T^{(-1-1)/2}\cdot T=1$, as expected for a diagram of the unknot.
For $D_3$, $\det(A)=1-T+T^2$, $\varphi(D)=-1$, and $w(D)=3$, so $\Delta=T^{(1-3)/2}(1-T+T^2)=T-1+T^{-1}$, as expected for the trefoil knot.

We set\footnote{At $T=1$ the matrix $A$ has $1$'s on the main diagonal, $(-1)$'s on the diagonal above it, and $0$'s everywhere else. Hence $A$ is invertible at $T=1$ and hence over the field of rational functions.} $G=(g_{\alpha\beta})=A^{-1}$, and taking our inspiration from physics, we name $g_{\alpha\beta}$ the {\em Green function} for the diagram $D$. For our three examples $D_1$, $D_2$, and $D_3$, the Green function $G$ is respectively
\begin{equation} \label{eq:GExamples}
  \begin{pmatrix} 1 \end{pmatrix},
  \quad \begin{pmatrix} 1&T^{-1}&1\\0&T^{-1}&1\\0&0&1 \end{pmatrix},
  \quad \footnotesize \left(
    \begin{array}{ccccccc}
     1 & \frac{T^3-T^2+T}{T^2-T+1} & 1 &
       \frac{T^3-T^2+T}{T^2-T+1} & 1 &
       \frac{T^3-T^2+T}{T^2-T+1} & 1 \\
     0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} &
       \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\
     0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} &
       \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\
     0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} &
       \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\
     0 & 0 & \frac{1-T}{T^2-T+1} & \frac{T-T^2}{T^2-T+1}
       & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\
     0 & 0 & 0 & 0 & 0 & 1 & 1 \\
     0 & 0 & 0 & 0 & 0 & 0 & 1 \\
    \end{array}
  \right).
\end{equation}

We can now define our invariant $\rho_1$. It is the sum of two sums. The first is a sum of a term $R_1(c)$ over all crossings $c$ in $D$, where for such a crossing we let $s$ denote its sign and we let $i$ and $j$ denote the edge labels of the incoming over- and under-strands, respectively and where
\begin{equation} \label{eq:R1}
  R_1(c) \coloneqq s \left(
    g_{ji} \left(g_{\jp,j}+g_{j,\jp}-g_{ij}\right)
    -g_{ii} \left(g_{j,\jp}-1\right)
    -1/2
  \right).
\end{equation}
The second sum is a sum over the edges $k$ of $D$ of a correction term dependent on the rotation number $\varphi_k$. We multiply the result by $\Delta^2$ to ``clear the denominators''\footnote{$R_1(s)$ is quadratic in the entries of $G$ and hence it has denominators proportional to $\Delta^2$.}:
\begin{equation} \label{eq:rho1}
  \rho_1 \coloneqq \Delta^2\left(\sum_c R_1(c) - \sum_k\varphi_k\left(g_{kk}-1/2\right)\right),
\end{equation}

Direct calculations show that $\rho_1(D_1)=0$ (as the sums are empty), $\rho_1(D_2)=0$, and $\rho_1(D_3) = -T^2 +2T - 2 + 2T^{-1} - T^{-2}$.

\needspace{12mm}
\begin{theorem}[``Invariance'', proofs in Section~\ref{sec:Proofs}] \label{thm:Main} The quantity $\rho_1$ is a knot invariant.
\end{theorem}

As we shall see in the next section, $\rho_1$ has more separation power than the Jones polynomial, yet it is closer to the more topologically-meaningful Alexander polynomial $\Delta$: it is cooked up from the same matrix $A$ and in terms of computational complexity, computing $\rho_1$ is not very different from computing $\Delta$. In order to compute $\Delta$ we need to compute the determinant of $A$, while to compute $\rho_1$ we need to invert $A$ and then compute a sum of $O(n)$ terms that are quadratic in the entries of $A^{-1}$.\footnote{We prefer not to be more specific about the complexity of computing $\rho_1$. It is the same as the complexity of inverting $A$, and matrix inversion is poly-time, with a rather small exponent, even for matrices with entries in a ring of polynomials (e.g.~\cite{Storjohann:ComplexityOfInversion}). We have not explored how much one can further gain by exploiting the fact that $A$ is very sparse.} We have computed $\rho_1$ for knots with over 200 crossings using the unsophisticated implementation presented in Section~\ref{sec:Implementation}.

Topologists should be intrigued! $\rho_1$ is derived from the same matrix as the Alexander polynomial $\Delta$, yet we have no topological interpretation for $\rho_1$.

\section{Implementation and Power} \label{sec:Implementation}

\input APAINB.tex

\section{Proofs of Theorem~\ref{thm:Main}, the Invariance Theorem} \label{sec:Proofs}

We tell the proof of the Invariance Theorem (Theorem~\ref{thm:Main}) in two ways: an elegant and intuitive though slightly lacking telling in Section~\ref{ssec:Cars}, and a complete though slightly dull telling in Section~\ref{ssec:dull}. But first, a few common elements.

\subsection{Common Elements}

Two upright knot diagrams are considered the same (as diagrams) if their underlying knot diagrams are the same, and if the respective rotation numbers of their edges are all the same. It is clear that $\rho_1$ is well defined on upright knot diagrams. To prove Theorem~\ref{thm:Main} we need to know what to prove. Namely, when do two upright knot diagrams represent the same knot? This is answered in the spirit of the classical Reidemeister theorem by the following:

\begin{figure}
\[ \input{figs/UprightRMoves.pdf_t} \]
\caption{
  The upright Reidemeister moves: Reidemeister 1 left and right,
  Reidemeister 2 braid-like and cyclic, Reidemeister 3, and (the $+$)
  Swirl.
} \label{fig:UprightRMoves}
\end{figure}

\begin{theorem}[``Upright Reidemeister''] \label{thm:UprightRMoves}
Two upright knot diagrams represent the same knot if and only if they
differ by a sequence of R1l, R2r, R2b, R2c, R3, and Sw$^+$ moves as in
Figure~\ref{fig:UprightRMoves}.
\end{theorem}

\noindent{\em Sketch of the proof.} In the case of round knots (i.e.,~ not ``long''), knot diagrams can be turned upright by rotating individual crossings. The only ambiguity here is by powers of the full rotation, the {\em swirls} Sw$^+$ and Sw$^-$ (where Sw$^-$ is the same as Sw$^+$ except with a negative crossing, and we don't need to impose it separately as it follows from Sw$^+$ and R2). Hence we have a well-defined map $\{\text{knot diagrams}\}\to\{\text{upright knot diagrams}\}/$Sw$^\pm$. It remains to write the usual Reidemeister moves between knot diagrams as moves between upright knot diagrams. The result are the moves R1l, R2r, R2b, R2c, and R3. Note that unoriented knot theory is presented with just three Reidemeister moves, but these split into several versions in the oriented case. The sufficiency of the versions we picked can be found in~\cite{Polyak:RMoves}. In the case of long knots a minor further complication arises, regarding the rotation numbers of the initial and final edges. We leave the details of the problem and its resolution to the reader. {\linebreak[1]\null\hfill\text{$\Box$}}

Our key formulas,~\eqref{eq:R1} and~\eqref{eq:rho1} involve the Green function $g_{\alpha\beta}$. We need to know that it is subject to some relations, the {\em $g$-rules} of Lemma~\ref{lem:gRules} below, whose proof is so easy that it comes first:

\noindent{\em Proof of Lemma~\ref{lem:gRules}.} The first set of $g$-rules
reads out column $\beta$ of the equality $AG=I$, and the second set of
$g$-rules reads out row $\alpha$ of the equality $GA=I$. {\linebreak[1]\null\hfill\text{$\Box$}}

\begin{lemma}[``$g$-rules''] \label{lem:gRules} Given a fixed upright knot diagram $D$, its corresponding matrix $A$, and its inverse $G=(g_{\alpha\beta})$, and given a crossing $c=(s,i,j)$ in $D$ (with $s$, $i$, and $j$ as before), the following two sets of relations (the $g$-rules) hold (with $\delta$ denoting the Kronecker delta):
\begin{equation} \label{eq:CarRules}
  g_{i\beta} = \delta_{i\beta}+T^sg_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}+(1-T^s)g_{\jp,\beta},
  \qquad g_{j\beta} = \delta_{j\beta}+g_{\jp,\beta},
  \qquad g_{2n+1,\beta} = \delta_{2n+1,\beta}
\end{equation}
and
\begin{equation} \label{eq:CounterRules}
  g_{\alpha i} = T^{-s}(g_{\alpha,{i^+}} \def\jp{{j^+}} \def\kp{{k^+}}-\delta_{\alpha,{i^+}} \def\jp{{j^+}} \def\kp{{k^+}}),
  \qquad g_{\alpha j} = g_{\alpha,\jp} - (1-T^s)g_{\alpha i} - \delta_{\alpha,\jp},
  \qquad g_{\alpha,1} = \delta_{\alpha,1}.
\end{equation}
Furthermore, for each fixed $\beta$ there are $2n+1$ $g$-rules of type~\eqref{eq:CarRules} (the first two depend on a choice of one of $n$ crossings, and the third is fixed, to a total of $2n+1$ rules). These fully determine the $2n+1$ scalars $g_{\alpha\beta}$ corresponding to varying $\alpha$. Similarly, for each fixed $\alpha$ there are $2n+1$ $g$-rules of type~\eqref{eq:CounterRules}. These fully determine the $2n+1$ scalars $g_{\alpha\beta}$ corresponding to varying $\beta$.
\end{lemma}

For use later, we teach our computer about $g$-rules:

\input{gRules.tex}

\subsection{Cars, Traffic Counters, and Interchanges} \label{ssec:Cars}
Our first proof of Theorem~\ref{thm:Main} is slightly informal as it
uses the language and intuition of probability theory even though our
``probabilities'' are merely algebraic formulae and not numbers between
$0$ and $1$. Seasoned mathematicians should see that there is no real
problem here. Yet just to be safe, we also include a fully formal proof
in Section~\ref{ssec:dull}.

Cars ($\reflectbox{\usym{1F697}}$, $\usym{1F697}$) travel on knot diagrams subject to the following three rules, inspired by Jones' ``bowling balls''~\cite{Jones:Hecke} and by Lin, Tian, and Wang's ``random walks''~\cite{LinTianWang:RandomWalk}):
\begin{itemize}
\item On plain roads (edges) they travel following the orientation of the edge.
\item When reaching an underpass (the lower strand of a crossing), cars just pass through.
\item When reaching an overpass cars pass through with probability $T^s$
  (where $s=\pm 1$ is the sign of the crossing), yet drop over to the
  lower strand with the complementary probability of $1-T^s$.
\end{itemize}

These rules can be summarized by the following pictures:
\[ \resizebox{\linewidth}{!}{\input{figs/CarRules2.pdf_t}} \]

In these pictures the horizontal struts represent ``traffic counters''
which measure the amount of traffic that passes through their respective
roads, and the output reading of these counters is printed above
them. Thus for example, the last interchange picture indicates that if
a unit stream of cars is injected into the diagram on the bottom right
and two traffic counters are placed at the top, then the first of these
will read a car intensity of $T^{-1}$ and the second $(1-T^{-1})$.

Note that our probabilities aren't really probabilities, if only because
$T$ and $T^{-1}$ cannot both be between $0$ and $1$ simultaneously. Yet
we will manipulate them algebraically as if they are probabilities,
restricting ourselves to equalities and avoiding inequalities. With
this restriction, we can use intuition from probability theory. We will
pretend that $T^s\sim 1$, or equivalently, that $1-T^s\sim 0$. This
has an algebraic meaning that does not refer to inequalities. Namely,
certain series can be deemed summable. For example,
\[ \sum_{r\geq 0}(1-T^s)^r = \frac{1}{1-(1-T^s)} = T^{-s}. \]

\parpic[r]{\input{figs/D2Traffic.pdf_T}}
\begin{example} Cars are injected on edge $\#1$ of the diagram $D_2$ of Section~\ref{sec:Formulas} as indicated on the right. What does the indicated traffic counter on edge $\#2$ measure?

\picskip{2}
\noindent{\em Solution.} Every car coming through the interchange from
$\#1$ passes through the underpass and comes to $\#2$, so the counter
reads ``$1$'' just for this traffic. But then these cars continue and
pass on the overpass, and $(1-T)$ of them fall down and continue through
edge $\#2$ and get counted again. But then these fallen cars continue and
pass on the overpass once again, and $(1-T)$ of them, meaning $(1-T)^2$
of the original traffic, fall once more and contribute a further reading
of $(1-T)^2$. This process continues and the overall counter reading is
\[ 1+(1-T)+(1-T)^2+(1-T)^3+\ldots = \frac{1}{1-(1-T)} = T^{-1}. \]
Note that this is exactly the row $1$ column $2$ entry of the matrix $G$ computed for this tangle in~\eqref{eq:GExamples}.
\end{example}

We claim that this is general:

\begin{proposition} For a general knot diagram $D$, the entry
$g_{\alpha\beta}$ of its Green function is equal to the reading of a
traffic counter placed at $\beta$ given that traffic is injected into $D$
at $\alpha$. (In the case where $\alpha=\beta$, the counter is placed
after where the traffic is injected, not before).
\end{proposition}

\noindent{\em Proof.} Consider the $g$-rules of type~\eqref{eq:CarRules}. The third, $g_{2n+1,\beta} = \delta_{2n+1,\beta}$ is the statement that if traffic is injected on the outgoing edge of $D$, it can only be measured on the outgoing edge of $D$ (so traffic never flows backwards). The second, $g_{j\beta} = \delta_{j\beta}+g_{\jp,\beta}$, is the statement that traffic goes through underpasses undisturbed, so $g_{j\beta} = g_{\jp,\beta}$ unless the traffic counter $\beta$ is placed between $j$ and $\jp$, in which case it measures one unit more if the cars are injected before it, at $j$, rather than after it, at $\jp$. Similarly the first of these $g$-rules, $g_{i\beta} = \delta_{i\beta}+T^sg_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}+(1-T^s)g_{\jp,\beta}$, is the statement of the behaviour of traffic at overpasses. Thus the rules in~\eqref{eq:CarRules} are obeyed by cars and traffic counters, and as the rules in~\eqref{eq:CarRules} determine $g_{\alpha\beta}$, the proposition follows. {\linebreak[1]\null\hfill\text{$\Box$}}

\begin{proposition} The quantity $\rho_1$ is invariant under R3.
\end{proposition}

\noindent{\em Proof.} We first show that cars entering a multiple
interchange styled as the left hand side of the R3 move, exit it with
the exact same distribution as cars entering the multiple interchange
styled as the right hand side. The hardest part of that computation is
when cars enter at the bottom left (at $i$) and it boils down to the
equality $1\!-\!T = (1\!-\!T)^2\!+\!T(1\!-\!T)$:
\[
  \def{$(1\!-\!T)^2\!+\!T(1\!-\!T)$}{{$(1\!-\!T)^2\!+\!T(1\!-\!T)$}}
  \def{$(1\!-\!T)T$}{{$(1\!-\!T)T$}}
  \def{$T(1\!-\!T)$}{{$T(1\!-\!T)$}}
  \def{$1\!-\!T$}{{$1\!-\!T$}}
  \input{figs/R3Traffic.pdf_t}
\]
If cars enter in the middle or at the bottom right (at $j$ or at $k$), the computation is even easier.\footnote{Note that this computation is exactly the one that proves that the Burau representation~\cite{Burau:Zopfgruppen} respects R3.}

The conclusion is that performing the R3 move does not affect traffic patterns outside the area of the move itself; namely, the Green function $g_{\alpha\beta}$ is unchanged if both $\alpha$ and $\beta$ are outside the area of the move.

Thus the only contribution to $\rho_1$ that may change (see~\eqref{eq:rho1}) is the contribution coming from the three $R_1$ terms corresponding to the crossings that move, and we need to know if the following equality holds:
\[
  R_1(+1,j,k) + R_1(+1,i,\kp) + R_1(+1,{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp)
  \overset{?}{=}
  R_1(+1,i,j) + R_1(+1,{i^+}} \def\jp{{j^+}} \def\kp{{k^+},k) + R_1(+1,\jp,\kp)
\]

Both sides here are messy quadratics involving the $g_{\alpha\beta}$'s of both sides, evaluated at $\alpha,\beta\in\{i,j,k,{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp,\kp,{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\jpp,\kpp\}$. But we can use the traffic rules (aka the $g$-rules) to rewrite these quadratics in terms of the $g_{\alpha\beta}$'s with $\alpha,\beta\in\{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\jpp,\kpp\}$, and these are unchanged between the sides. So we simply need to know whether the above equality holds {\em after} the relevant $g$-rules have been applied to both sides. We could do that by hand, but it's simpler to appeal to a higher wisdom:

\input{Invariance-R3-Short.tex} \ {\linebreak[1]\null\hfill\text{$\Box$}}

\noindent{\em First Proof of Theorem~\ref{thm:Main}, ``Invariance''.} We've shown invariance under R3. Invariance under the other moves is shown in a similar way: first one shows that overall traffic patterns are unchanged by each of the moves, and then one verifies that the local contributions to $\rho_1$ coming from the area changed by each move are equal once the $g$-rules are used to rewrite them in terms of $g_{\alpha\beta}$'s that are unaffected by the moves. This is shown in greater detail in the following section. {\linebreak[1]\null\hfill\text{$\Box$}}

\subsection{A More Formal Version of the Proof} \label{ssec:dull}
Again we start with the hardest, R3:

\begin{proposition} \label{prop:R3} The quantity $\rho_1$ is invariant under R3.
\end{proposition}

\noindent{\em Proof.} We need to know how the Green function $g_{\alpha\beta}$ changes under R3. Here are the two sides of the move, along with the $g$-rules of type~\eqref{eq:CarRules} corresponding to the crossings within, written with the assumption that $\beta$ isn't in $\{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp,\kp\}$, so several of the Kronecker deltas can be ignored. We use $g$ for the Green function of the left side of R3, and $g'$ for the right side:
\[
  \def{$\begin{array}{l{{$\begin{array}{l}
    g_{j,\beta} = \delta_{j\beta} \!+\! Tg_{\jp,\beta} \!+\! (1\!-\!T)g_{\kp,\beta} \\
    g_{k,\beta} = \delta_{k\beta} \!+\! g_{\kp,\beta}
  \end{array}$}}
  \def{$\begin{array}{l{{$\begin{array}{l}
    g_{i,\beta} = \delta_{i\beta} \!+\! Tg_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} \!+\! (1\!-\!T)g_{\kpp,\beta} \\
    g_{\kp,\beta} = g_{\kpp,\beta}
  \end{array}$}}
  \def{$\begin{array}{l{{$\begin{array}{l}
    g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} = Tg_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta} \!+\! (1\!-\!T)g_{\jpp,\beta} \\
    g_{\jp,\beta} = g_{\jpp,\beta}
  \end{array}$}}
  \def{$\begin{array}{l{{$\begin{array}{l}
    g'_{i,\beta} = \delta_{i\beta} \!+\! Tg'_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} \!+\! (1\!-\!T)g'_{\jp,\beta} \\
    g'_{j,\beta} = \delta_{j\beta} \!+\! g'_{\jp,\beta}
  \end{array}$}}
  \def{$\begin{array}{l{{$\begin{array}{l}
    g'_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} = Tg'_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta} \!+\! (1\!-\!T)g'_{\kp,\beta} \\
    g'_{k,\beta} = \delta_{k\beta} \!+\! g'_{\kp,\beta}
  \end{array}$}}
  \def{$\begin{array}{l{{$\begin{array}{l}
    g'_{\jp,\beta} = Tg'_{\jpp,\beta} \!+\! (1\!-\!T)g'_{\kpp,\beta} \\
    g'_{\kp,\beta} = g'_{\kpp,\beta}
  \end{array}$}}
  \input{figs/R3.pdf_t}
\]

A routine computation (eliminating $g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}$, $g_{\jp,\beta}$, and $g_{\kp,\beta}$) shows that the first system of 6 equations is equivalent to the following 3 equations:
\[ g_{i,\beta} = \delta_{i\beta} + T^2g_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta} + T(1-T)g_{\jpp,\beta} + (1-T)g_{\kpp,\beta}, \]
\[
  g_{j,\beta} = \delta_{j\beta} + Tg_{\jpp,\beta} + (1-T)g_{\kpp,\beta},
  \qquad\text{and}\qquad
  g_{k,\beta} = \delta_{k\beta} + g_{\kpp,\beta}.
\]
Similarly eliminating $g'_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}$, $g'_{\jp,\beta}$, and $g'_{\kp,\beta}$ from the second set of equations, we find that it is equivalent to
\[ g'_{i,\beta} = \delta_{i\beta} + T^2g'_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta} + T(1-T)g'_{\jpp,\beta} + (1-T)g'_{\kpp,\beta}, \]
\[
  g'_{j,\beta} = \delta_{j\beta} + Tg'_{\jpp,\beta} + (1-T)g'_{\kpp,\beta},
  \qquad\text{and}\qquad
  g'_{k,\beta} = \delta_{k\beta} + g'_{\kpp,\beta}.
\]
But these two sets of equations are the same, and as stated in $g$-rules lemma (Lemma~\ref{lem:gRules}), along with the $g$-rules corresponding to the other crossings in $D$ (which are also the same between $g$ and $g'$), these equations determine $g_{\alpha\beta}$ and $g'_{\alpha\beta}$, for $\alpha,\beta\notin\{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp,\kp\}$. So with this exclusion on $\alpha$ and $\beta$, we have that $g_{\alpha\beta}=g'_{\alpha\beta}$. But this means that the summations~\eqref{eq:rho1} in the definitions of $\rho_1$ are equal for the two sides of R3, except perhaps for the three summands on each side that come from the crossings that touch $\{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp,\kp\}$.

What remains is completely mechanical. We just need to compute the sum of those three summands for both sides of R3, and apply to it the $g$-rules of types~\eqref{eq:CarRules} and~\eqref{eq:CounterRules} that eliminate the indices $\{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp,\kp\}$. The computation is easy enough to be done by hand, yet why bother? Here's the machine version (it takes less typing to apply all relevant $g$-rules and also eliminate the indices $\{i,j,k\}$):

\input{Invariance-R3.tex} \ {\linebreak[1]\null\hfill\text{$\Box$}}

\begin{proposition} \label{prop:R2c} The quantity $\rho_1$ is invariant under R2c.
\end{proposition}

\noindent{\em Proof.} We follow the exact same steps as in the case of $R3$. First, we write the $g$-rules, assuming $\beta$ isn't in $\{i,j,{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp\}$:
\[
  \def{$\begin{array}{l{{$\begin{array}{l}
    g_{i,\beta} = T^{-1}g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}+(1-T^{-1})g_{\jpp,\beta} \\
    g_{\jp,\beta} = g_{\jpp,\beta}
  \end{array}$}}
  \def{$\begin{array}{l{{$\begin{array}{l}
    g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} = Tg_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta}+(1-T)g_{\jp,\beta} \\
    g_{j,\beta} = g_{\jp,\beta}
  \end{array}$}}
  \def{$\varphi_{j^+}\!=\!1$}{{$\varphi_{j^+}\!=\!1$}}
  \def{$\varphi_{j^{+\!+}}\!=\!1$}{{$\varphi_{j^{+\!+}}\!=\!1$}}
  \input{figs/R2c.pdf_t}
\]

Note that for the right hand side we allowed ourselves to label the edges ${i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}}$ and $\jpp$ as the computation is independent of the labelling and the labelling need not be by contiguous integers (outside of the move area, we assume that the left hand side and the right hand side are labelled in the same way). Note also that for the right hand side, there are no relevant $g$-rules. Now as in the case of R3, for the left hand side we eliminate $g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}$ and $g_{\jp,\beta}$ and we are left with the relations
\[ g_{i,\beta}=g_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta} \qquad\text{and}\qquad g_{j,\beta}=g_{\jpp,\beta}. \]
Otherwise the $g$-rules for the left and for the right are the same, and so their Green functions are the same except if the indices are in $\{i,j,{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp\}$ (these indices do not even appear in the right hand side). Thus the contribution to $\rho_1$ from outside the area of the move is the same for both sides.

Next we write the contribution to $\rho_1$ coming from the two crossings and one rotation that appear on the left, and use the $g$-rules to push all the indices in $\{i,j,{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\jp\}$ up to ${i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}}$ and $\jpp$. This can be done by hand, but seeing that we have tools, we use them:

\input{Invariance-R2c.tex}

This result is clearly equal to the single rotation contribution to $\rho_1$ that comes from the right hand side. {\linebreak[1]\null\hfill\text{$\Box$}}

\begin{proposition} \label{prop:R1l} The quantity $\rho_1$ is invariant under R1l.
\end{proposition}

\noindent{\em Proof.} We start with the relevant $g$-rules:
\[
  \def{$\begin{array}{l{{$\begin{array}{l}
    g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} = \delta_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} + Tg_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta} + (1-T)g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} \\
    g_{i,\beta}=\delta_{i,\beta}+g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}
  \end{array}$}}
  \input{figs/R1l.pdf_t}
\]
The first of these rules is equivalent to $g_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta}=T^{-1}\delta_{{i^+}} \def\jp{{j^+}} \def\kp{{k^+},\beta} + g_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta}$. For $\beta\neq i,{i^+}} \def\jp{{j^+}} \def\kp{{k^+}$ we find as before that $g_{i,\beta}=g_{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}},\beta}$ and we can ignore the contributions to $\rho_1$ coming from outside the area of the move. The contribution to $\rho_1$ coming from the single crossing and single rotation on the left hand side is computed below, and is equal to the empty contribution coming from the right hand side:

\input{Invariance-R1l.tex} \ {\linebreak[1]\null\hfill\text{$\Box$}}

\vskip 1mm
\noindent{\em Second Proof of Theorem~\ref{thm:Main}, ``Invariance''.} After the Upright Reidemeister Theorem (Theorem~\ref{thm:UprightRMoves}) which sets out what we need to do, and Propositions~\ref{prop:R3}, \ref{prop:R2c}, and~\ref{prop:R1l} which prove invariance under R3, R2c, and R1l, it remains to show the invariance of $\rho_1$ under R1r, R2b, and Sw$^+$. This is done exactly as in the examples already shown, so in each case we show only the punch line:

\input{Invariance-Rest.tex} \ {\linebreak[1]\null\hfill\text{$\Box$}}

\section{Some Context and Some Morals} \label{sec:Context} We would
like to emphasize again that $\rho_1$ seems very close to the Alexander
polynomial, yet we have no topological interpretation for it. Until that
changes, where is $\rho_1$ coming from?

It comes via a lengthy path, which we will only sketch here. For
a while now \cite{PP1, PG, Talk:NCSU-1604, Talk:Indiana-1611,
Talk:LesDiablerets-1708, Talk:Matemale-1804, Talk:Toronto-1811,
Talk:DaNang-1905, Talk:UCLA-191101} we've been studying quantum invariants
related to the Lie algebra ${sl_{2+}^\epsilon}$, the 4-dimensional Lie algebra with generators $y,b,a,x$ and
brackets
\[ [b,x]=\epsilon x, \quad [b,y]=-\epsilon y, \quad [b,a]=0, \quad [a,x]=x, \quad [a,y]=-y, \quad [x,y]=b+\epsilon a, \]
where $\epsilon$ is a scalar. The beauty of this algebra stems from the following:
\begin{itemize}
\item It is a ``classical double'' of a two dimensional the Lie bialgebra $\langle a,x\rangle$, with
  \[ [a,x]=x,\qquad \delta(a)=0, \qquad \delta(x)=\epsilon x\wedge a, \]
  and hence quantization tools are available and are used below (e.g.~\cite{EtingofSchiffman:QuantumGroups}).
\item At invertible $\epsilon$ it is isomorphic to $sl_2\oplus\langle
  t\rangle$, where $t=b-\epsilon a$ is a central element. Quantum
  topology tells us that the algebra $sl_2$ is related to the
  Jones polynomial. In fact, the universal quantum invariant
  (see \cite{Lawrence:UniversalUsingQG, Lawrence:Universal,
  Ohtsuki:QuantumInvariants}) for the Lie algebra $sl_2$ is equivalent
  to the coloured Jones polynomial of~\cite{Jones:Hecke}.
\item At $\epsilon=0$ it becomes the diamond Lie algebra $\diamond$, a solvable
  algebra in which computations are easier. The algebra $\diamond$
  is the semi-direct product of the unique non-commutative 2D Lie
  algebra ${\mathfrak a}$ with its dual, and quantum topology tells us that it
  is related to the Alexander polynomial~\cite{Talk:Chicago-1009, WKO1}.
\end{itemize}
The last two facts taken together tell us that the Alexander
polynomial is some limit of the coloured Jones polynomial
(originally conjectured~\cite{MelvinMorton:Coloured, Rozansky:TrivialFlatI} and proven
by other means~\cite{Bar-NatanGaroufalidis:MMR}).

We can make this a bit more explicit.  By using the Drinfel'd quantum
double construction~\cite{Drinfeld:QuantumGroups} we find that the
universal enveloping algebra ${\mathcal U}({sl_{2+}^\epsilon})$ has a quantization $QU$,
which has an $R$-matrix solving the Yang-Baxter equation (meaning,
satisfying the R3 move, in the appropriate sense). These are given by:
\[ QU =
  A\langle y,b,a,x\rangle\left/\begin{pmatrix}
    [b,a]=0, \quad [b,x]=\epsilon x,\quad [b,y]=-\epsilon y, \\
    [a,x]=x, \quad [a,y]=-y, \quad xy-qyx=\frac{1-{\mathbb e}^{-\hbar(b+\epsilon a)}}{\hbar}
  \end{pmatrix}\right.,
\]
where $A\langle\text{\it gens}\rangle$ is the free associative algebra
with generators {\it gens}, and $q={\mathbb e}^{\hbar\epsilon}$, and
\[ R=\sum_{m,n\geq 0}\frac{y^nb^m\otimes (\hbar a)^m(\hbar x)^n}{m![n]_q!}
  \qquad\left(\text{where $[n]_q!=\prod_{k=1}^n\frac{1-q^k}{1-q}$ is a ``quantum factorial''}\right).
\]

Thus there is an associated universal quantum invariant of knots $Z_\epsilon(K)\in QU$
(which, as stated, is equivalent to the coloured Jones polynomial). In
our talks and papers we show that $Z_\epsilon$ can be expanded as a power
series in $\epsilon$, that at $\epsilon=0$ it is equivalent to the Alexander
polynomial, and that in general, the coefficient $Z_{(k)}$ of $\epsilon^k$
in $Z_\epsilon$ can be computed in polynomial time and is homomorphic,
meaning that it leads to an ``algebraic knot theory'' in the sense of
(say)~\cite{Talk:Sydney-190916}. We also know that the excess information in $Z_{(k)}$ (beyond the
information in $\{Z_{(0)},\ldots,Z_{(k-1)}\}$) is contained in a single polynomial, $\rho_k$. The first of
these polynomials is $\rho_1$ of this paper.

But how did we arrive at the specific formulas of this paper? As often seen with quantizations, $QU$ is
isomorphic (though only as an algebra, not as a Hopf algebra) with ${\mathcal U}({sl_{2+}^\epsilon})$, and the latter can be
represented into the Heisenberg algebra ${\mathbb H}=A\langle p,x\rangle/([p,x]=1)$ via
\[ y\to -tp-\epsilon\cdot xp^2,\qquad b\to t+\epsilon\cdot xp,\qquad a\to xp,\qquad x\to x, \]
(abstractly, ${sl_{2+}^\epsilon}$ acts on its Verma module
${\mathcal U}({sl_{2+}^\epsilon})/({\mathcal U}({sl_{2+}^\epsilon})\langle y,a,b-\epsilon a-t\rangle)\cong{\mathbb Q}[x]$
by differential operators, namely via ${\mathbb H}$). So $QU$'s $R$-matrix
can be expanded in powers of $\epsilon$ and pushed to ${\mathcal U}({sl_{2+}^\epsilon})$ and on
to ${\mathbb H}$, resulting in ${\mathcal R}={\mathcal R}_0(1+\epsilon {\mathcal R}_1+\cdots)$, with
${\mathcal R}_0={\mathbb e}^{t(xp\otimes 1-x\otimes p)}$ and ${\mathcal R}_1$ a quartic
polynomial in $p$ and $x$. Now all the computations for $\rho_1$ can be carried out by pushing around a
rather small number of $p$'s and $x$'s (at most 4), and this can be done using the rules
\begin{align*}
  (p\otimes 1){\mathcal R}_0 &= {\mathcal R}_0({\mathbb e}^t(p\otimes 1)+(1-{\mathbb e}^t)(1\otimes p)), \\
  (1\otimes p){\mathcal R}_0 &= {\mathcal R}_0(1\otimes p),
\end{align*}
which, after setting $T={\mathbb e}^t$, must remind the reader of
Equation~\eqref{eq:A}. When all the dust settles, the resulting formulas
are similar to the ones in Equations~\eqref{eq:R1} and~\eqref{eq:rho1}
(but only similar, because we applied some ad hoc cosmetics to make the
formulas appear nicer).

There are some morals to this story:

\begin{enumerate}

\item The definition of $\rho_1$ in Section~\ref{sec:Formulas} and the
proofs of its invariance in Section~\ref{sec:Proofs} are clearly much
simpler than the origin story, as outlined above. So quite clearly, we
still don't understand $\rho_1$. There ought to be a room for it directly
within topology, that does not require that one would know anything about
quantum algebra. (And better if that room is large enough to accommodate
morals~(\ref{mor:rhok}) and~(\ref{mor:frakg}) below).

\item \label{mor:rhok}
Like there is $\rho_1$, there are $\rho_k$. The origin story tells us that
$\rho_k$ should have a formula as a summation over choices of $k$-tuples
of features of the knot (crossings and rotations), just as the formula
for $\rho_1$ is a single summation over these features. The summand
for $\rho_k$ will be a degree $2k$ polynomial in the Green function
$g_{\alpha\beta}$ (compare with~\eqref{eq:R1}, which is quadratic). As a
$k$-fold summation, after inverting $A$, $\rho_k$ should be computable in
$O(n^k)$ additions and multiplications of polynomials in $T$, where $n$
is the crossing number.

\item These $\rho_k$ should be equivalent to the invariants in our earlier
works \cite{PP1, PG, Talk:NCSU-1604, Talk:Indiana-1611,
Talk:LesDiablerets-1708, Talk:Matemale-1804, Talk:Toronto-1811,
Talk:DaNang-1905, Talk:UCLA-191101}.

\item These $\rho_k$ should be equivalent to the invariants studied
earlier by Rozansky and Overbay \cite{Rozansky:TrivialFlatI,
Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis}, as their quantum
origin is essentially the same (though strictly speaking, we have not
written proofs of that, and normalizations may differ). Our formulas are
significantly simpler and faster to compute than the Rozansky-Overbay
formulas, and in our language it is easier to see the behaviour of
$\rho_1$ under mirror reflection (Section~\ref{ssec:Demo} here, unproven
in Rozansky-Overbay).

\item Like the Rozansky-Overbay invariants, $\rho_k$ should be equivalent to the ``higher diagonals'' for the
Melvin-Morton expansion (e.g.~\cite{Rozansky:U1RCC}) and should be dominated by the ``loop expansion'' of
the Kontsevich integral~\cite{Kricker:Lines, GaroufalidisRozansky:LoopExpansion}.

\item \label{mor:frakg} The quantum algebra story extends to other Lie
algebras, beyond $sl_2$. So there should be variants $\rho_k^{\mathfrak g}$
of $\rho_k$ at least for every semisimple Lie algebra ${\mathfrak g}$, given by
more or less similar formulas. Quantum algebra suggests that
$\rho_k^{\mathfrak g}$ should be a polynomial in as many variables as the
rank of ${\mathfrak g}$, and should in general be stronger than the ``base''
$\rho_k$. We have not seriously explored $\rho_k^{\mathfrak g}$ yet, though
some preliminary work was done by Schaveling~\cite{Schaveling:Thesis}.

\item It appears that $QU$ has interesting traces and therefore that there should be a link version of
$\rho_1$. We have not pursued this formally.

\item $QU$ has a co-product and an antipode, and so the universal
tangle invariant associated with $QU$ has formulas for strand reversal
and strand doubling (e.g.~\cite{PG, Talk:DaNang-1905}). This implies
(e.g., by following the ideas of~\cite{Talk:Sydney-190916}) that there
should be formulas for $\rho_1$ that start with a Seifert surface for
the knot. We are pursuing such formulas now; we already know that the
degree of $\rho_1$ is bounded by $2g$, where $g$ is the genus of a knot~\cite{PG}.

\item For the same reasons, for ribbon knots $\rho_1$ should
have a formula computable from a ribbon presentation, and its
values might be restricted in a manner similar to the Fox-Milnor
condition~\cite{FoxMilnor:CobordismOfKnots}. We are pursuing this now.

\item The coloured Jones polynomial is invariant under mutation so we
expect $\rho_1$ to likewise be invariant under mutation (and indeed, also
$\rho_k$), yet we do not have a direct proof of that yet. Note that we can expect $\rho_k^{\mathfrak g}$ for
higher-rank ${\mathfrak g}$ to no longer be invariant under mutation.

\end{enumerate}

\input refs.tex

\end{document}
\endinput

\subsection{The Program} This done, here is the full $\rho_1$ program:

\noindent\nbpdfInput{Snips/APAI/4.pdf}

The program uses mostly the same symbols as the text, so even without any knowledge of Mathematica, the reader should be able to recognize at least formulas~\eqref{eq:A}, \eqref{eq:Delta}, and~\eqref{eq:rho1} within it. As a further hint we add that the variables \verb$Cs$ ends up storing the list of crossings in a knot $K$, where each crossing is stored as a triple $(s,i,j)$, where $s$, $i$, and $j$ have the same meaning as in~\eqref{eq:A}. The conversion routine \verb$Rot$ automatically produces \verb$Cs$, as well as a list $\varphi$ of rotation numbers, given any other knot presentation known to the package \verb$KnotTheory`$.

Note that the program outputs the ordered pair $(\Delta,\rho_1)$. The Alexander polynomial $\Delta$ is anyway computed internally, and we consider the aggregate $(\Delta,\rho_1)$ as more interesting than any of its pieces by itself.

\subsection{A Step-by-Step Run-Through}  We start by setting $K$ to be the knot diagram on page~1 using the \verb$PD$ notation of \verb$KnotTheory`$~\cite{Bar-NatanMorrison:KnotTheory}. We then print \verb$Rot[K]$, which is a list of crossings followed by a list of rotation numbers:

\noindent\nbpdfInput{Snips/APAI/5.pdf}

\noindent\nbpdfOutput{Snips/APAI/6.pdf}

Next we set \verb$Cs$ and $\varphi$ to be the list of crossings, and the list of rotation numbers, respectively.
\needspace{20mm}

\noindent\nbpdfInput{Snips/APAI/7.pdf}

\noindent\nbpdfOutput{Snips/APAI/8.pdf}

We set \verb$n$ to be the number of crossings, \verb$A$ to be the $(2n+1)$-dimensional identity matrix, and then we iterate over \verb$c$ in \verb$Cs$,  adding a block as in~\eqref{eq:A} for each crossing.

\noindent\nbpdfInput{Snips/APAI/9.pdf}

\needspace{30mm}
Here's what \verb$A$ comes out to be:

\noindent\nbpdfInput{Snips/APAI/10.pdf}

\noindent\nbpdfOutput{Snips/APAI/11.pdf}

We set $\Delta$ to be the determinant of \verb$A$, with a correction as in~\eqref{eq:Delta}. So $\Delta$ is the Alexander polynomial of $K$.

\noindent\nbpdfInput{Snips/APAI/12.pdf}

\noindent\nbpdfOutput{Snips/APAI/13.pdf}

\needspace{30mm}
\verb$G$ is now the \verb$Inverse$ of \verb$A$:

\noindent\nbpdfInput{Snips/APAI/14.pdf}

\noindent\nbpdfOutput{Snips/APAI/15.pdf}

\needspace{30mm}
It remains to blindly follow the two parts of Equation~\eqref{eq:rho1}:

\noindent\nbpdfInput{Snips/APAI/16.pdf}

\noindent\nbpdfOutput{Snips/APAI/17.pdf}

We replace each ${\mathtt g}_{\alpha\beta}$ with the appropriate entry of \verb$G$:

\noindent\nbpdfInput{Snips/APAI/18.pdf}

\noindent\nbpdfOutput{Snips/APAI/19.pdf}

Finally, we output both $\Delta$ and $\rho_1$. We factor them just to put them in a nicer form:

\noindent\nbpdfInput{Snips/APAI/20.pdf}

\noindent\nbpdfOutput{Snips/APAI/21.pdf}

\subsection{A Demo Run}  \label{ssec:Demo} Here are $\Delta$ and $\rho_1$ of all the knots with up to 6 crossings (a table up to 10 crossings is printed at~\cite{PG}:

\noindent\nbpdfInput{Snips/APAI/22.pdf}

\noindent\nbpdfMessage{Snips/APAI/23.pdf}

\noindent\nbpdfOutput{Snips/APAI/24.pdf}

Some comments are in order:
\begin{itemize}
\item $\rho_1$ flips its sign when switching to the mirror of a knot. Indeed in~\eqref{eq:rho1} both $R_1(c)$ and $\varphi_k$ flip sign under reflection in a plane perpendicular to the plane of the knot diagram. Hence $\rho_1$ vanishes on amphicheiral knots, such as $4_1$ and $6_3$ above.
\item $\rho_1$ seems to always be divisible by $(T-1)^2$ and seems to always be palindromic ($\rho_1(T)=\rho_1(T^{-1})$). We are not sure why this is so.
\end{itemize}

\begin{figure}
\[ \resizebox{6in}{!}{\input{figs/GST48-Marked.pdf_t}} \]
\caption{A 48-crossing knot from~\cite{GompfScharlemannThompson:Counterexample}.} \label{fig:GST48}
\end{figure}
Next is one of our favourites, a knot from~\cite{GompfScharlemannThompson:Counterexample} (see Figure~\ref{fig:GST48}), which is a potential counterexample to the ribbon$=$slice conjecture~\cite{Fox:Problems}. It takes about two minutes to compute $\rho_1$ for this 48 crossing knot (note that Mathematica prints \verb$Timing$ information is seconds, and that this information is highly dependent on the CPU used, how loaded it is, and even on its temperature at the time of the computation):

\noindent\nbpdfInput{Snips/APAI/25.pdf}

\noindent\nbpdfOutput{Snips/APAI/26.pdf}

\subsection{The Separation Power of $\rho_1$} Let us check how powerful $\rho_1$ is on knots with up to 12 crossings:

\noindent\nbpdfInput{Snips/APAI/27.pdf}

\noindent\nbpdfOutput{Snips/APAI/28.pdf}

So the pair $(\Delta,\rho_1)$ attains 2,882 distinct values on the 2,977 prime knots with up to 12 crossings (a deficit of 95), whereas the pair $(H,Kh)=$ (HOMFLYPT polynomial, Khovanov Homology) attains only 2,785 distinct values on the same knots (a deficit of 192).

In our spare time we computed all of these invariants on all the prime knots with up to 14 crossings. On these 59,937 knots the pair $(\Delta,\rho_1)$ attains 53,684 distinct values (a deficit of 6,253) whereas the pair $(H,Kh)$ attains only 49,149 distinct values on the same knots (a deficit of 10,788).

Hence the pair $(\Delta,\rho_1)$, computable in polynomial time by simple programs, seems stronger than the pair $(H,Kh)$, which is more difficult to program and (for all we know) cannot be computed in polynomial time. We are not aware of another poly-time invariant as strong as the pair $(\Delta,\rho_1)$.

redpajama_set_name: RedPajamaArXiv


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    (
      data: void | { [key: string]: unknown },
      options: {
        lang?: string;
        notUndoable?: boolean;
      } = {}
    ) => {
      dispatch(
        updateCellData(id)(data, {
          notUndoable: false,
          lang: currentLang,
          ...options,
        })
      );
    },
    [dispatch, id, currentLang]
  );
};

/**
 * @returns a function to remove a cell by id
 */
export const useRemoveCellById = () => {
  const dispatch = useDispatch();
  return useCallback((id: string) => dispatch(removeCells([id])), [dispatch]);
};
/**
 * @param id a cell id
 * @returns a function to remove the given cell
 */
export const useRemoveCell = (id: string) => {
  const removeById = useRemoveCellById();
  return useCallback(() => removeById(id), [removeById, id]);
};

/**
 *
 * @returns a function to remove muliple nodeids
 */
export const useRemoveMultipleNodeIds = () => {
  const dispatch = useDispatch();
  return useCallback(
    (nodeIds: string[]) => {
      dispatch(removeCells(nodeIds));
    },
    [dispatch]
  );
};

/**
 * @returns a function that duplicates a cell
 */
export const useDuplicateCellById = () => {
  const dispatch = useDispatch();
  const editor = useEditorStore();

  return useCallback(
    (id: string) => dispatch(duplicateCell(editor.getNode(id))),
    [dispatch]
  );
};

export const useInsertAfter = () => {
  const dispatch = useDispatch();
  const insertNew = useInsertNew();

  return useCallback(
    (node: Node, insertAfterNodeId: string) => {
      if (insertAfterNodeId) {
        dispatch(
          duplicateNode(node, {
            insertAfterNodeId,
          })
        );
      } else {
        // insert at the end
        insertNew(cloneWithNewIds(node));
      }
    },
    [dispatch, insertNew]
  );
};

/**
 * @returns a function that duplicates multiple cell
 */
export const useDuplicateMultipleCells = () => {
  const editor = useEditorStore();
  const insertAfter = useInsertAfter();

  return useCallback(
    (cellIds: string[]) => {
      const node = getCommonAncestorTree(editor, cellIds);

      const insertAfterNodeId = isRow(node)
        ? node.id
        : node.rows[node.rows.length - 1].id;

      insertAfter(node, insertAfterNodeId);
    },
    [editor, insertAfter]
  );
};

/**
 * @param a cell id
 * @returns a function that duplicates the given cell
 */
export const useDuplicateCell = (id: string) => {
  const duplicate = useDuplicateCellById();

  return useCallback(() => duplicate(id), [duplicate, id]);
};

/**
 * experimental
 * @returns function to set the reference node id. used internally
 */
export const useSetDisplayReferenceNodeId = () => {
  const dispatch = useDispatch();
  const referenceId = useDisplayModeReferenceNodeId();

  return useCallback(
    (nodeId: string) => {
      if (nodeId !== referenceId) dispatch(setDisplayReferenceNodeId(nodeId));
    },
    [dispatch, referenceId]
  );
};

/**
 * @returns a function to focus a cell by id
 */
export const useFocusCellById = () => {
  const dispatch = useDispatch();
  const editor = useEditorStore();

  return useCallback(
    (id: string, scrollToCell?: boolean, mode?: FocusMode) => {
      const parentCellId = editor
        .getNodeWithAncestors(id)
        ?.ancestors?.find((node) => !isRow(node))?.id;

      dispatch(setDisplayReferenceNodeId(parentCellId));
      dispatch(focusCell(id, scrollToCell, mode));
    },
    [dispatch, editor]
  );
};

/**
 * @returns a function to focus a cell by id
 */
export const useFocusCell = (id: string) => {
  const focusCellById = useFocusCellById();

  return useCallback(
    (scrollToCell?: boolean, mode?: FocusMode) =>
      focusCellById(id, scrollToCell, mode),
    [focusCellById]
  );
};
/**
 * @returns function to blur a cell by id
 */
export const useBlurCell = () => {
  const dispatch = useDispatch();

  return useCallback(
    (id: string) => {
      dispatch(blurCell(id));
    },
    [dispatch]
  );
};

/**
 * @returns function to blur all cells
 */
export const useBlurAllCells = () => {
  const dispatch = useDispatch();

  return useCallback(() => {
    dispatch(blurAllCells());
  }, [dispatch]);
};

/**
 * @returns function to insert a cell at the end of the document or the end of the parent cell
 *
 * if the id already exists, it will move that cell
 */
export const useInsertNew = (parentCellId?: string) => {
  const dispatch = useDispatch();
  const cellPlugins = useAllCellPluginsForNode(parentCellId);
  const editor = useEditorStore();
  const lang = useLang();
  return useCallback(
    (partialCell: PartialCell) => {
      const action = parentCellId ? insertCellNewAsNewRow : insertCellAtTheEnd;

      dispatch(
        action({
          cellPlugins,
          lang,
        })(partialCell, { id: parentCellId }, { focusAfter: true })
      );
    },
    [dispatch, editor, cellPlugins, parentCellId]
  );
};

/**
 * used for the trash target
 */
export const useTrashDrop = () => {
  const removeCell = useRemoveCellById();
  return useDrop<
    CellDrag,
    void,
    {
      isHovering: boolean;
    }
  >({
    accept: 'cell',
    collect: (monitor) => ({
      isHovering: monitor.isOver({ shallow: true }),
    }),
    drop: (item, monitor) => removeCell(item.cell.id),
  });
};

redpajama_set_name: RedPajamaGithub


text:
Q: Android: How to record video from required portion of the screen? i need to capture video from required portion of the screen like -

I have tried
...
try {
        camera.unlock();
        recorder.setCamera(camera);
        recorder.setOrientationHint(cameraRotationDegree);
        recorder.setAudioSource(MediaRecorder.AudioSource.MIC);
        recorder.setVideoSource(MediaRecorder.VideoSource.CAMERA);
        recorder.setOutputFormat(MediaRecorder.OutputFormat.MPEG_4);
        **recorder.setVideoSize(sz.width, sz.height);**
        recorder.setAudioEncoder(MediaRecorder.AudioEncoder.DEFAULT);
        recorder.setVideoEncoder(MediaRecorder.VideoEncoder.MPEG_4_SP);
        recorder.setOutputFile(fileName);
        recorder.prepare();
        recorder.start();
        isRecording = true;
    } catch (IllegalStateException e) {
        e.printStackTrace();
    } catch (IOException e) {
        e.printStackTrace();
    }

but it records while screen.
is it possible to capture video from required portion of the screen?

A: The video is being cut off on the display by being obscured by static content.  You can't record from the display, and even if you could, you'd just end up with the static content in the video as well.
One approach is to direct the video to a SurfaceTexture, which converts each incoming frame to an OpenGL ES "external" texture.  You would then render the desired portion of the frame twice, once to the display, once to a video encoder.  This is fairly involved, but you can find examples of the various bits in Grafika.
A less efficient but perhaps more straightforward approach would be to post-process the video file with something like ffmpeg.  I haven't worked with it, but it seems fairly sophisticated, and may provide an easy way to re-encode a video with cropping.
The simplest option is to simply fit the camera preview into your layout, and work with the full frame.

redpajama_set_name: RedPajamaStackExchange


Lunch & Snacks provided; Fee of $30 payable to University of Delaware covers cost of refreshments and some program materials. Toolkit is free.
For more information: Contact Maria Pippidis, pippidis@udel.edu or call 302-831-1239.


Praveen Kailas is a longtime enthusiast of audio/visual work and modern electronics. He primarily has experience on the user side of audio/visual implementation, and this experience comes not only from personal endeavors but from professional work in home construction, energy efficiency projects, and home automation.
Praveen's first experience in the audio/visual field was back in the late 1990s, when he began wiring systems and gaining some experience in home automation (including the control of security measures or lighting). Although he primarily studied psychology and business in school, he quickly realized that he enjoyed these technical projects as well. Now Praveen continues to configure these options through the use of apps, wiring of commercial buildings, and more.
In addition to home automation, Praveen loves to improve his own understanding of audio and visual tools. He has experience with various types of speakers, receivers, and subwoofers. He explores AVS forums to learn more about the topic and some of his favorite brands are Marantz, Sony, and Pioneer. Praveen also likes to support smaller internet companies which are making some of the most exciting headway in the industry.
The quality of audio and visual content that can be put on the Internet is improving while requiring less and less bandwidth. Praveen follows the evolution of audio quality to ensure that he is utilizing and installing the best options possible. Back in the 1990s CDs were the pinnacle of audio quality, and then the trend of ripping CDs to MP3s unfortunately ended up sacrificing a lot of this quality. Now, as more high-quality streaming services become available online and we have more platforms to choose from, audio quality is becoming better than it has been in years.
In addition to electronics, home automation, and audio improvements, Praveen loves visual forms of media– and the technological progress that has improved their overall quality. He enjoys watching movies with family and friends, especially now that screening services are getting more advanced. One of his favorite movies of all time is Kingdom of Heaven, but he's also a big fan of movies that are inspired by comic books.
Praveen's favorite superhero movie is The Avengers and he is excited to see the revitalization of superhero stories in pop culture nowadays. From the many Marvel and DC Comics movies that have recently been released, to the increasing popularity of industry entertainment events like Comic-Con, superhero entertainment is clearly in a second golden age.


The Menzingers have premiered a new song 'After The Party', which is the title track of their new album, out February 3rd via Epitaph Records.
'After the Party' is the follow-up to the Philadelphia-based band's widely acclaimed 'Rented World'. Pre-orders for 'After The Party' are available here. Produced by Will Yip (Title Fight, Balance & Composure, Pianos Become the Teeth), 'After the Party' taps into the Menzingers' everyman romanticism to reflect on getting older but not quite growing up. Throughout the album, singer/guitarists Greg Barnett and Tom May, bassist Eric Keen, and drummer Joe Godino offset that deeply nuanced songwriting with anthemic harmonies, furious power chords, and larger-than-life melodies.
Next Month, the Menzingers will embark on a US headline tour with Jeff Rosenstock and Rozwell Kid as support before heading to the UK and Europe in April for a headline tour with special guests The Flatliners and support just added from The Dirty Nil. They'll also be playing Groezrock Festival in Belgium.


on Microsoft's trademarks. "The action, though, is not about dragging the hackers into court. The lawsuit is a tool for Microsoft to target what it calls 'the most vulnerable point' in Fancy Bear's espionage operations: the command-and-control servers the hackers use to covertly direct malware on victim computers," the report added. So far, Microsoft has used the lawsuit to wrest control of 70 different command-and-control points from Fancy Bear.
Microsoft has "identified over 120 new targets of the Kremlin's cyber spying and control-alt-deleting segments of Russian President Vladimir Putin's hacking apparatus, the report added. Microsoft's approach is indirect, but effective, it said.
"Rather than getting physical custody of the servers, which Fancy Bear rents from data centres around the world, Microsoft has been taking over the Internet domain names that route to them," the report noted.
These are addresses like "livemicrosoft[.]net" or "rsshotmail[.]com" that Fancy Bear registers for about $10 each.
"Once under Microsoft's control, the domains get redirected from Russia's servers to the company's, cutting off the hackers from their victims, and giving Microsoft a omniscient view of that servers' network of automated spies, the report said.
A judge in Alexandria, Virginia is scheduled to rule whether to grant Microsoft a permanent injunction against Fancy Bear. Fancy Bear, also known as 'APT28', 'Sofacy', 'Pawn Storm' and 'Strontium' has been conducting cyber attacks since 2007. According to the US intelligence findings, Fancy Bear targeted the Democratic National Committee (DNC) and the Clinton campaign as part of Moscow's efforts to help Donald Trump win the 2016 election.


text:
Q: PHP / SQL: searching against html entities stored in database I work a lot with languages that use accented characters, e.g. é. I store content in "utf_8_bin" encoded tables AND I convert accented characters to HTML entities too.
So, for example, "Términator" would be stored as "T & eacute ; rminator" (had to spaces in that to stop it rendering online) in the database.
When a user searches for "términator" a match is found because the query is also converted to HTML entities and the SQL query "lowercases" both sides of the argument with "lcase".
The problem I am having now, is that the client wants to be able to search for "Terminator" (no accent on the "e") to get results matching "Términator".
I would prefer not to change the way I store my data, particularly because storing HTML entities solves a number of other problems. So I'm asking in case there's a simpler solution. Thanks!

A: You should use the correct collation in your query in your case utf8_unicode_ci (This is without the html entities)
https://dev.mysql.com/doc/refman/5.5/en/charset-unicode-sets.html
This collation you use determines which results you get back from you database. And how that characters are compared.
SELECT * FROM some_table WHERE title LIKE "Terminator" COLLATE utf8_unicode_ci

This query will return records with the title términator, Terminator etc, note that it does a case insensitive comparison (the _ci part in the collation).
The utf8_unicode_ci is a bit slower but that's really minimal and you probably wouldn't even notice the difference.
There are more collations that can fit your needs, not sure if there is one which can be used for html entities. You could add your own collation to mysql database to create the htmlentities support yourself something like utf8_htmlentities_ci. https://dev.mysql.com/doc/refman/5.7/en/adding-collation.html
Here a nice example with phone numbers https://dev.mysql.com/doc/refman/5.7/en/ldml-collation-example.html


CTQ's current efforts to engage teachers and administrators as co-leaders of learning, teaching, and leading is a new way of thinking about school improvement. But our collective leadership work today grows out of decades of focused research, innovative practice, networked learning, and thought leadership.
Southeast Center for Teaching Quality staff at the Hillsborough, NC office (2003).
The Center for Teaching Quality (CTQ) got its start in 1998, when Barnett Berry founded the organization with a grant from the BellSouth Foundation. The young think tank focused on how to improve teaching and close the achievement gap in schools.
Over the next seven years, staff tackled a wide range of projects with research partners: assessing the impact of No Child Left Behind, documenting the influence of high-stakes accountability on teachers' professional development, analyzing how to recruit and retain accomplished teachers, and identifying innovative models of teacher preparation.
Bob Hughes, now serving as Director of K-12 Education for the Bill & Melinda Gates Foundation, and NYC teachers Ariel Sacks, José Vilson, and Stephen Lazar (L to R) discuss Teacherpreneurs at Ford Foundation convening (2014).
In the early 2000s, we launched the Teacher Leaders Network (TLN). We learned how this first virtual community of practice for teacher leaders could elevate their voices of expertise on education policy and practice. This work grounded the our skill in building and sustaining virtual communities among the busiest of educators.
As practitioners became more integrated into our ongoing work, their ideas and expertise increasingly informed CTQ's research and and thought leadership through various publications, reports, and books, including TEACHING 2030 and Teacherpreneurs.
We continued to elevate teachers' voices and build communities through our TeacherSolutions teams, TLN networks, New Millennium Initiative, and finally, in 2013, the CTQCollab, an incubator for teachers' bold ideas and innovative solutions. In recent years, we've brought educators together for storytelling retreats and collective leadership workshops to drive change in public schools, as well as championed the creation and growth of Teacher-Powered Schools.
Teacher Julie Hiltz (L) and staffer Jennifer Barnett collaborate during a CTQ-hosted storytelling retreat (2016).
Over the past two decades, we've invested in research to create learning experiences that prepare educators to further support and develop teacher leadership. Cultivating Communities for Impact (CCI), a learning tool that supports educators to use their skills and knowledge to build trust and generate results, provides a solid foundation for sustaining online learning and leadership that focuses on improvement and innovation.
We've also worked with districts and states to advance hybrid roles for teachers ("teacherpreneurs"). We promote the adoption of micro-credentials as a means to modernize professional learning and use new impact assessment and storytelling tools to help educators share success stories and spread new approaches and ideas for better teaching.
Teacher-Powered Schools National Conference attendees learn from students during a school tour (2017).
Over the years, CTQ has been dedicated and committed to teachers and their roles in improving education. But we've also learned that even the most exceptional teachers can't make big changes alone. Meaningful change needs teachers, administrators, and system leaders collaborating to find and implement meaningful changes.
In 2017, we rededicated our work to help bring these groups together to create better schools for every student. As CTQ enters into our twentieth year, we remain focused and guided in our vision for an education system that works better for every student.


We have all kinds of tours from one day, multi-day, theater, and quilting tours. See the tours page for all the details.
Coaches can be contracted for all your needs, one day as well as multi-day events.
Contact us today to schedule your special event.


Although you might have experienced much success in your career, have you ever had that nagging feeling that you can do better? You might want to take your business performance to a higher level, but you aren't quite sure how to do it.
As a CEO, you can't confide in just anyone about your concerns, especially others within your organization. A CEO coach can be your sounding board, by listening to your concerns, ideas and the various situations that always need your attention. You benefit from the years of incredibly valuable business experience and knowledge your CEO coach brings to you.
Business coaching is a recent development. Businesses and executives have been managing without them for years, so why start getting coaching now? Unfortunately, a lot of people think that you don't need coaching unless you're failing—there's a mentality of "if it ain't broke, don't fix it," and that getting coaching is a sign of weakness. The reality is that having somebody with the experience and wisdom that Mark has to help you make the best possible decisions is your greatest strength.
Join Mark for a workshop and choose from either the Vistage Version or Advantage Version. Both will let your business harness and direct positive sales energy to grow your customer base like you haven't done before.
The right coach can help you grow as a business leader and help you ward off problems before they ever have a chance to come up. Professional athletes use coaches to get and stay at the top of their game—why shouldn't successful business owners? Like athletes, if you're not actively improving, you're probably stagnating, which neither you or your business can afford to let happen.
Markets, technologies, and workforces are all shifting in ways that can be hard to keep track of. There's a lot of legal and financial scrutiny and pressure on businesses. Board members can be focused more on concrete financial results rather than improving a CEO's performance. You employees are naturally hesitant to criticize you and you can't really talk about problems with staff members with other staff members. Trying to handle all of this on your own is a perfect way to burn out and lose a lot of opportunities.
A CEO coach, like Mark T. Moore, can help. Mark is the founder of Sales KnowHow and author of Superstar Business Secrets: The Top Five Keys to Big Success and Bigger Profits for Your Business. He has over 30 years of international business experience that he's invested not only on building his own companies, but also speaking, presenting workshops, consulting, and writing. Now, he can give you the personal CEO coaching you need to grow as a business leader to dramatically improve your performance.
You win by having a trusted confidante and a neutral party. Mark has the experience to know where to look and how to fix your blind spots in a safe, focused, one-on-one environment. He is passionate about helping people just like you reach your highest potential and is only concerned with your success as a leader. He looks at your needs and then helps improve your talents so you can develop solutions to whatever comes your way.
What's more is that he's a truly engaging leader who can help you with those "soft skills"—the ones that are more important to running a business than people think. As a leader you don't just tell people what to do—you need to inspire the people around you to be successful, which you do by modeling the very qualities that attract success. Things like motivation, compassion, commitment, authenticity, and reliability. You may be the leader, but you're also part of the team, and those skills can help your team work together effectively, leading to far more success for everyone.
A good leader knows what works. A great leader does everything in their power to do better. A Superstar Business leader contacts Mark Moore for business coaching. Fill out our online contact form for more information. You are going to have so much fun with Mark as your trusted advisor!


text:
Q: how to find span tag iside iframe? I am using TinyMCE editor that resides inside an iframe. There is a span having class AMedit inside this iframe. The id of this iframe is elm1_ifr. How can I find the span tag having class AMedit in this Iframe in jquery ? I have to do some stuff in this span once I find it.
Thanks

A: Try this:
$("#elm1_ifr").contents().find("span.AMedit ");

To attach a focus event listener on that contenteditable span you can use this:
var span = $("#elm1_ifr").contents().find("span.AMedit ");
span.on('focus',function(){
   //your code here
});

FIDDLE (check the console)

A: if ( $("#elm1_ifr span").hasClass("AMedit") )
{
  // your code
};
else
{
  //your code
}


1 Planning – the first stage of waste management reviews.
2 Organization – the second stage of waste management reviews.
3 Motivation – the third stage of the waste management reviews.
4 Control – the fourth stage of waste management reviews.
Management, like the waste management reviews, always includes several stages. To get the result, you need to go through these stages.
Planning – the first stage of waste management reviews.
Organization – the second stage of waste management reviews.
Motivation – the third stage of the waste management reviews.
Control – the fourth stage of waste management reviews.
Know that management is a whole science. You have received basic knowledge of waste management reviews. Now, if you do not feel yourself strong, you just need to turn to professionals.


Add some teardrop glamour to your ensembles with these radiant earrings. These earrings are from the Sparkle Collection and as the name suggests, these earrings shimmer with brilliance through the perfectly encrusted diamonds. Set in 14kt and 18kt yellow gold, these earrings are sure to catch all the attention from your onlookers.


text:
Kalei Mau digs UA volleyball's stout defense
Published Sep 11, 2015 12:00pm
By Ivan Leonard
Rebecca Noble | The Daily Wildcat
Arizona's Kalei Mau (10) spikes against California State Northridge on Saturday, Sept. 5. Mau has been one of the team's top defensive players.
From becoming a situational player after transferring from Minnesota last year to becoming one of the top two-way players in the Pac-12 Conference, Arizona volleyball outside hitter Kalei Mau is learning that the sky is the limit.
"Last year was a transition year and a great learning year for her, and now she is taking what she went through in the spring and starting to apply to this fall season," said Arizona volleyball coach Dave Rubio.
"I like what she is doing this season. She still has a long way to go to be an efficient player for us, but she's got some real athleticism and she is shouldering a big load for us."
The hitter from Kahaluu, Hawaii, won Pac-12 Defensive Player of the Week to start the season on a high note.
Led by Mau, the Wildcats boast one of the Pac-12's stingiest defenses, allowing a hitting percentage of only .076.
After tallying 28 digs all of last season, Mau accumulated 31 digs against SMU and tied for the fifth-highest single game total in school history.
She also became the third Arizona player with over 30 digs in a four-set match.
Mau is becoming a better offensive player for the Wildcats, too, as she currently leads the team in kills, while ranking eighth in the conference in points per set and digs per set.
The Wildcats are on the right track as a team, but Mau believes she has room to improve. She noted that she loves the position she is in with Rubio and her teammates.
"There [are] so [many] more things I can improve with my game, but probably scoring from the back row," Mau said about what she needs to improve on most.
"This is my first time playing all rotation and I have never had a coach trust me to play back row, so this is a big deal for me to make plays and score points. The main thing for me is to be smart about the attacks I choose when I am hitting in the back row."
Mau was a key player for the Wildcats last weekend at the Arizona Invitational, when they swept South Dakota, Florida A&M and Cal State Northridge to extend their winning streak to six. The winning weekend also improved Arizona's national ranking to No. 13.
The junior led the team in kills in all three of those games, including a career-high 15 against Cal State Northridge.
Mau will look to once again lead the Wildcats when they travel to Brooklyn, New York, this weekend for the Blackbird Invitational. The tournament includes a date with Ohio State, which is currently ranked No. 15 in the nation.
For Mau, it will be another opportunity to display her already impressive game, and to continue to improve.
Follow Ivan Leonard on Twitter.

redpajama_set_name: RedPajamaCommonCrawl


Cricket Green – the next 50 years – get involved!
We're celebrating the 50th anniversary of Mitcham Cricket Green Conservation Area this year!
The things that we all know make Cricket Green a special place – the ancient greens, variety of historic buildings, wonderful views and rich story – were officially recognised by Merton Council in 1969 and now by law every effort has to be made by developers and others to "preserve and enhance" its character.
We have a programme of celebrations and events planned, including a 50th anniversary Community on the Green on 18th July and a special Mitcham Heritage Day on 14th September.
Keep an eye out on Twitter (@MitchamCrktGrn) for our 50 things initiative telling the story of the Conservation Area through photographs of things that people have told us are special to them. We are also looking to plant something golden!
The 50th anniversary is also a time to look forward and think about how Cricket Green is going to change in the future.
We all know there are lots of pressures from development and some recent changes have not been for the better. We are also keen to see the open spaces and wildlife better cared for, trees planted and ponds protected. There are too few places for people to meet, we lack community facilities and we are losing shops and businesses. We want to see beautiful homes built and less pollution.
Over the last few years our work has been guided by the Cricket Green Charter and its principles to inform policies and planning decisions about the future of the area.
Among other successes the Charter is the starting point for the £5m now being invested in The Canons and its grounds.
It is time for a refresh and we want your views.
How would you like to see Cricket Green change and develop?
What is special about Cricket Green that should be protected?
What has happened in the last few years you would rather not see repeated?
What kind of new development would be welcome?
What community facilities are we lacking?
You can see our early thoughts on a new Cricket Green Charter here.
We know Cricket Green is special.
In this 50th anniversary year please play your part in keeping it that way.
This entry was posted in Cricket Green Conservation Area on February 3, 2019 by mitchamcricketgreen.


t + detox is a cleansing blend of herbs, green tea, fruit and 50% RDA in 9 daily essential vitamins per clever little tea bag.
We blend together functional herbs milk thistle, dandelion root and ginger root used for centuries for internal cleansing, along with antioxidant packed green tea and 50% RDA in vitamins B2 and vitamins C ( vitamins that can help protect cells from oxidative stress ). We then finish this blend off with a delicious natural apple & blackcurrant flavours to create the perfect cleansing super tea!
The Detox Pack includes 4 boxes of t + detox wellness tea. Each box contains 15 foil wrapped dual chamber tea bags.


New in Terminal B; passengers can now discover the Aelia Duty Free boutique dedicated to Fashion. Superdry, Calvin Klein and Guess accessories are just a sample of all the brands you will find in this new store, with specific temporary animations. You will also be able to enjoy last minute shopping in perfumes, alcohol, tobacco and electronic devices.


With the school expanding our staffing structure is evolving too. Here is a guide to the Staffing Structure at SPS.
My name is Cassandra Malone and I am the Head Teacher of Sevenoaks Primary School. I joined the school in January 2019 after over a decade of working in London Boroughs.
I have been in the teaching profession for 16 years and have worked across four schools, in a range of areas and contexts.
I began my teaching career in Kent, where I also studied for my degree. Since qualifying, I have had the privilege of teaching across the Primary range. Alongside my teaching experience, I have held a vast array of responsibilities; these range from various English and curriculum specialisms to being a local authority moderator. My leadership roles have been just as varied. In addition to being a class teacher, I have also had roles as an Assistant Head, Deputy Head and Head Teacher.
My specialist studies at university allowed me to publish an article on gender stereotypes, which still remains a firm area of interest to me. I have since studied to become MBTi (Myers Briggs Type Indicator) qualified, and have undertaken various coaching roles and training programmes in order to support my work on coaching cultures and leadership development in teams.
In my previous headship, my school was a National Support school that offered various support to local schools and schools within the MAT (Multi Academy Trust) that it founded. This support included leadership development, the development of outstanding teaching, and provision for the most vulnerable pupils.
During my eight years at my previous school, we gained various accreditations, including the IQM (Inclusion Quality Mark), to many sporting and basic skills awards. I moved on from the school after leading it through an Ofsted that achieved Outstanding judgements in every inspection category.
Today, in my role as Head Teacher, I regularly provide training and support to leaders and other schools and really enjoy the mutual learning that takes place through this support. I have joined the Sevenoaks partnership group and look forward to being part of the exciting opportunities that this work can bring to the children of Sevenoaks.
Hi, I am one of the Assistant Heads, Jane Durkin. I have been teaching for over twenty years and at Sevenoaks Primary since 2005.
Having taught in three other Primary Schools in Sevenoaks and two British International Schools, I have gained experience teaching in both key stages and with a number of managerial roles.
Here, at Sevenoaks Primary, I have been able to build on that experience with my role as Director of Studies which involved, among other things, overseeing the school improvement plan and the curriculum development. I am proud to be a part of the Sevenoaks team as its values and visions are akin with mine. That is, firstly, a drive for improvement, always challenging ourselves to be the best that we can, to meet the needs of all of our children. Secondly, a curriculum that reaches beyond the classroom to develop the whole child. I particularly enjoy the sport and SPSA events at our school but love the wealth of creative opportunities we also provide.
This school buzzes with activity and I am enjoying being a part of that 'buzz' in my new role!
Hello, I am Sarah White and very pleased to have joined the Sevenoaks Primary School community. My roles include being one of the Assistant Head Teachers and the class teacher of 3B.
I have spent the last six years developing my skills as a senior leader. I enjoy working with and supporting others, sharing my expertise and ensuring all children have the opportunity to reach their potential. I am excited by continuing to develop a growth mindset culture within Sevenoaks Primary and supporting children on their learning journey.
If you see me around, please come and say hello! I look forward meeting you all and being part of the school community.
I have been the English manager at three primary schools and in 2004 I completed my Master's degree in Literacy and Learning. I have also lectured at Canterbury University on the teacher training course. Gifted and Talented leader has also been a role I have enjoyed and for several years I chaired the Sevenoaks Partnership's Gifted and Talented group . However, my great passion within education is to raise children's self-esteem, help them to recognise their own potential and provide experiences and challenges which enable them to achieve their goals. The greatest moments in teaching are watching a child move from, "I can't" to "I can" and to play any part in that is a great privilege.
Hello, my name is Matthew Walker and I am very proud to have joined the team at Sevenoaks Primary School as the Phase 2 leader and class teacher in Year 2. I have a degree in education from London's South Bank University and have spent the past 10 years teaching across both KS1 and KS2 at a community primary school in West London. During this time I have gained a huge amount of experience in the differing approaches needed to ensure that every child is making progress in class, as well as the pastoral care that is vital to make sure children are happy and enjoy coming to school. I love being a teacher and am passionate about children becoming lifelong learners; this process needs to begin as early as possible in a child's life and I want to ensure that children transition into Key Stage 2 with not only all the skills they need to thrive but a desire to keep on constantly improving themselves as they get older.
I completed a BA (Hons) in Primary Education at Canterbury Christ Church University and gained my teacher training experience in three Kent schools across Early Years and Key Stage One classes. My teaching career was spent in three more schools in Medway and South London with the majority of those years spent as a Reception teacher and year group leader. In 2010 I decided to undertake the first part of a Masters Degree in Early Years. This gave me the opportunity to study the work of many different early childhood theorists and also undertake some action research looking at high quality play experiences in my own setting. I have, most recently, worked for Kent County Council as an Early Years Improvement Adviser, supporting teachers, teaching assistants and headteachers in schools across Dartford, Gravesham, Sevenoaks and occasionally down by the seaside! It's lovely to now be back in the classroom with my terrific Tiger class.


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I'm coming late to the new forum, but: just a quick note that my combomarginsplot does provide some functionality to combine the multiple calls to marginsplot.
What I would personally would love to have is a replace option for generate. It's a small thing in the grand scheme of things, but the lack of one keeps annoying me - especially when doing ad-hoc trials with .do-files and the like. Also, looping could be simpler in some cases.
I agree completely with this one. And even more so for -egen-.
2. Somewhat similaris the possibility to save empty data sets - yes, I know there's an addon to do it and also easy ways to work around, but it would make some things much more elegant to be able to do so from scratch.
Not sure what you intend here that -save, emptyok- doesn't address.
Last edited by Jeph Herrin; 07 Aug 2014, 08:42.
1. More and more I find myself working with very large databases - where analytic files are on the order of 100gb, say - and in this environment Stata does not shine. In the long history of Little Data, the memory model employed by Stata had a clear and evident advantage. But no more - I find myself not only using SAS for data management, but (reluctantly) suggesting it to others who are working in the same space.
2. Large data is typically maintained in relational databases, accessed via some flavor of SQL. Stata has some support for SQL, in that one can submit an SQL statement and retrieve the output, but this is not helpful if one wants to, eg, use an index file to match IDs against the SQL database. Instead (and this often leads to problems of type 1 above) one must retrieve every record and do the matching locally in Stata, which means working with even larger datasets than one would need to otherwise.
Last edited by Jeph Herrin; 07 Aug 2014, 08:52.
You are presumably referring to ==, >, <, >=, <=, !=.
I guess he is referring to precision control in the lines of http://docwiki.embarcadero.com/RADSt...s_%28Delphi%29 , meaning in short that the system should know what it is comparing (types of LHS and RHS) and know it's own limitations (what it can and cannot compare).
Combomarginsplot is a great command. I do hope that Stata corps doesn't think that just because somebody has written a command that it doesn't have to worry about the issue anymore. For a user it would be easier to add an option to margins than to learn a whole new command.
Large data is typically maintained in relational databases, accessed via some flavor of SQL. Stata has some support for SQL, in that one can submit an SQL statement and retrieve the output, but this is not helpful if one wants to, eg, use an index file to match IDs against the SQL database. Instead (and this often leads to problems of type 1 above) one must retrieve every record and do the matching locally in Stata, which means working with even larger datasets than one would need to otherwise.
I would even add that my impression is that other tools like monetdb (esp. linked to R) has shocking performance advantages for data retrieval, selection, manipulation and aggregation, let alone "MapReduce" operations on many cores (btw, even Hadoop will be 10 years old when Stata 14 comes out). I understand that Stata is fantastic for -regress- and -ivregress- and -teffects- and clustered standard errors and what not, but it is all the sadder if we waste time with -by- looping over the data all the time, spend time with disk I/O for preserving, rarely transparent to the user, doing needless dances for merges, waiting to have the data resorted again and again because there is no better index for the data etc.
I am obviously no database scientist, but I do see other tools (not only SQL, but also pandas in Python, e.g.) doing "merges, egens, collapses, ifs, bys" much, much more efficiently. I really wonder why StataCorp does not invest in this.
Native JDBC support analogous to ODBC. Although the Java API and/or C plugin capabilities for end user development are great, some end users (myself in particular) have little to no experience in these languages. Depending on the data system, JDBC can also be a bit better performing than ODBC. Support for multiple datasets residing in memory simultaneously. With sufficient ODBC/JDBC support, this could potentially be a matter of interfacing with whichever flavor SQL end users prefer (maybe have a set sql [mysql, sqlite, postgre, oracle, sqlserver] [, permanently]) to make integration of Stata within the "big data" space a bit easier on the development team at Stata as well as retaining flexibility. Using graph names as a method for creating multilayered graphs. For example, we can already use : tw scatter x y || scatter abc z But having a way to use something like: tw scatter x y, name(gr1) tw scatter abc z, name(gr2) tw scatter varx vary, name(gr3) // use gr1 as base, layer gr3 on top of that, and gr2 on top of all of them gr layer gr1 gr3 gr2 It doesn't seem as though alpha transparency is something that will be coming up in the immediate future, but making it easier for the end users to layer things without having to rewrite code could be a good stop gap that would also make it easier for people to create complex graphs more easily. Better documentation of the graphics engine to make it easier for the user/programmer community to add to the graphics capabilities. Better documentation/net courses for programming mata. Latent class/transition analysis (categorical latent variables more generally) modeling. Native support for 3PL IRT (allowing a single fixed and individually constrained/estimated c parameters in the model).
And one simpler thing, much less frustrating for StataCorp than my previous suggestions: a new case(,,…) function, or anything to replace nested cond(,,) functions. And this should replace the everday use of repeated -replace if- and other uses (or the recode command!), which are painfully slow, esp. on bigger data.
I would really like a few simple things.
2. A Do file editor that is divorced from the main package, so when Stata takes a nose dive, it doesnt take your code with it.
+1 for Laszlo's suggestion of a case() function!
One other thing: Could -margins- and (some) other factor-variable magic work with predefined interaction terms? Stata is lovely that it takes care of interactions for us (though numerical derivates are painfully slow sometimes), but repeated use of interaction terms in various models needlessly repeats the generation of those temporary variables, unless I am missing something. Why can't I save/define the interaction term, use it repeatedly, but still not fool -margins- thinking it's a separate variable?
And one more: if StataCorp does not feel like their comparative advantage is in reinventing the wheel and writing up regex in C etc., they could invest in fast, easy, efficient back-and-forth with other environments. Python comes to mind, as it is free, and much of it a complement to Stata, not a substitute (unlike with R or Matlab or SAS etc.). It would be great to manipulate variables with essentially Python, esp. strings (and now with strL…).
which would greatly simplify do files that use local macros with date literals as the parameters.
Last edited by Michael Anbar; 15 Aug 2014, 12:20.
Ricahard Williams wrote "* Also, I understand that some other programs (e.g. MPLUS) let you specify auxiliary variables that help improve the handling of missing data."
the (<- gender minority alienation) line adds three auxiliary variables.
Oh, while I'm at it, it is crazy that -save- cannot just save a subset of observations even in version 14, 30 years after the first release. -preserve-keep-save-restore- is needlessly costly with big data. Preparing clean extracts can be a nightmare.