text:
\section{Introduction} In a series of papers by Hatcher and Thurston \cite{HT:1985}, Floyd and Hatcher \cite{FH:1988}, and Hatcher and Oertel~\cite{HO:1989}, the set of incompressible, boundary incompressible surfaces in the complement of a 2-bridge knot or link, or  a Montesinos knot, are completely described and classified. In the case of knots, these papers also describe the possible boundary slopes that occur and in \cite{HO:1989} a table of all boundary slopes of Montesinos knots with 10 or fewer crossings is given. Unfortunately, this table contains several errors. However a  corrected table has been published by Dunfield~\cite{D:2001}. Moreover, the computer program written and used by Dunfield is available at {\tt http://www.CompuTop.org}.

While Floyd and Hatcher explicitly describe all incompressible surfaces in the complement of a 2-bridge link,  they do not compute the boundary slopes of these surfaces,  saying only that it should be possible in principle.  In his Ph.D. thesis~\cite{L:1993}, Lash starts with their construction and shows how to compute the associated boundary slopes. His ultimate goal was to compare the set of boundary slopes of   the Whitehead link, ${\cal L}_{3/8}$, to those predicted by the algebraic-geometric machinery of Culler-Shalen \cite{CS:1983}. Applying his algorithm to ${\cal L}_{3/8}$, Lash was able to show that in this case, every boundary slope arises from a degenerating sequence of representations of the link group into $\mbox{SL}_2 \Bbb C$. Ohtsuki \cite{O:1994} has shown this to be true for all 2-bridge knots (excluding slopes that correspond to a fiber in a fibration), but the question remains open for all 2-bridge links in general. To investigate this question it would obviously be helpful to have boundary slope data for all 2-bridge links. Unfortunately, Lash's thesis has never been published and  tables of boundary slopes of links have not been available.

In this paper we describe Lash's algorithm and develop an improved algorithm that is easier to use. This allows us to describe the types of boundary slopes that can occur. As an illustration of our techniques, we compute the boundary slopes of the $(4k-1)/(8k)$ 2-bridge links. These links may also be described as  $1/k$ surgery on one component of the Borromean rings. For this class of links, we have found  an explicit description of their eigenvalue varieties and in  a forthcoming paper will investigate the relationship between the actual boundary slopes and those detected by the eigenvalue variety.

We have written a computer program to implement our algorithm and include here a table of boundary slopes of all 2-bridge links up to 10 crossings.  In our table, links through 9 crossings are also identified by their index in Rolfsen's table \cite{Ro:1990}. Our program, as well as  a much larger table to 16 crossings, will eventually be available at {\tt http://www.CompuTop.org} and as part of {\sl Knotscape}~\cite{HT:1998}.

In Section~\ref{section:floydandhatcher} we briefly describe Floyd and Hatcher's construction for 2-bridge links. The reader is referred to their original paper for more detail. Then in Section~\ref{section:lash} we describe Lash's algorithm for finding the boundary slopes of a given 2-bridge link using Floyd and Hatcher's construction. In Section~\ref{section:improvedalgorithm} we improve the algorithm and discuss some of its theoretical consequences. The next section includes a nice example for the infinite family of 2-bridge links already mentioned above. Finally, in Section~\ref{section:table} we tabulate boundary slope data for all 2-bridge links with 10 or less crossings.

\section{Floyd and Hatcher's Construction}
\label{section:floydandhatcher}

Let $p$ and $q$ be relatively prime positive integers such that $0<p<q$, $p$ is odd, and $q$ is even. We assume that the reader is familiar with the standard 2-bridge diagram of the 2-bridge link ${\cal L}_{p/q}$. For example, ${\cal L}_{3/4}$ is shown in Figure~\ref{figure:2-bridge link diagram}. It is important to note that our definition of ${\cal L}_{p/q}$ agrees with that of  \cite{HO:1989} and \cite{L:1993}, but is the mirror image of the more conventional depiction of ${\cal L}_{p/q}$ with the ``(straight) bridges on top.''  See for example, \cite{BZ:2003} or \cite{HT:1985}.

\begin{figure}
    \begin{center}
    \leavevmode
    \scalebox{.3}{\includegraphics{3_42bridgeLink.eps}}
    \end{center}
\caption{The 2-bridge link ${\cal L}_{p/q}$, for $p=3$ and $q=4$.}
\label{figure:2-bridge link diagram}
\end{figure}

Viewing $S^3$ as the 2-point compactification of $S^2 \times \Bbb R$, we may place ${\cal L}_{p/q}$ in $S^2 \times I$ so that it meets $S^2 \times \{0\}$ and $S^2 \times \{1\}$ each in two arcs, and each intermediate level in four points.
We may think of each level, $S^2 \times \{z\}$, as the quotient $\Bbb R^2/\Gamma$, where $\Gamma$ is the group generated by $180^\circ$ rotations of $\Bbb R^2$ about the integer lattice points $\Bbb Z^2$. The four points of the link at each intermediate level are precisely the four points of $\Bbb Z^2/\Gamma$.
The arcs at level $z=1$ are the image of  the lines in $\Bbb R^2$ which pass through integer lattice points with slope $p/q$. Similarly,
the two arcs at level $z=0$ are the image of vertical lines through integer lattice points. Finally, PSL$_2\Bbb Z$ acts linearly on each level, leaving $\Bbb Z^2/\Gamma$ invariant.

Floyd and Hatcher next describe four basic branched surfaces, $\Sigma_A, \Sigma_B, \Sigma_C$ and $\Sigma_D$, copies of which will be stacked, one on top of the other, to build a branched surface spanning the 2-bridge link. Actually it is not these surfaces exactly, but rather homeomorphic images of them, that will be compressed vertically and stacked together. To understand the Floyd and Hatcher construction, and Lash's computation of the boundary slopes, we need to first understand these four basic building blocks.

\begin{figure}
\psfrag{a}{$\Sigma_A$}
\psfrag{b}{$\Sigma_B$}
\psfrag{c}{$\Sigma_C$}
\psfrag{d}{$\Sigma_D$}
\psfrag{x}{$\alpha$}
\psfrag{y}{$\beta$}
\psfrag{z}{$\alpha-\beta$}
\psfrag{n}{$n$}
\psfrag{q}{$\alpha-\beta-n$}
\psfrag{bn}{$\beta - n$}
\psfrag{abt}{$\frac{\alpha-\beta}{2}$}
\psfrag{r}{$0$-level}
\psfrag{s}{$\frac{1}{2}$-level}
\psfrag{t}{$1$-level}
    \begin{center}
    \leavevmode
 \includegraphics[width=5.15in]{branchedsurfaces.eps}
\end{center}
\caption{The four basic branched surfaces.}
\label{figure:branchedsurfaces}
\end{figure}

Beautiful illustrations of the four surfaces are given in \cite{FH:1988} which we will not attempt to reproduce here. Instead, we describe the surfaces in a different fashion. Each  is contained in $S^2 \times I$.
 In both of the half-intervals $[0,\frac{1}{2})$ and $(\frac{1}{2}, 1]$ the surface is a product of the half-interval with a finite number of disjoint embedded arcs in the 2-sphere with endpoints at the four points of $\Bbb Z^2/\Gamma$.  The transversality of the surface with the horizontal levels completely degenerates at the $\frac{1}{2}$-level, where branching occurs which allows the arc system at the $0$-level to transition to the arc system at the $1$-level. Figure~\ref{figure:branchedsurfaces} shows cross-sections of each branched surface at heights $0, \frac{1}{2}$, and $1$. At the $\frac{1}{2}$-level the shaded areas indicate horizontal parts of the surface. Each of the four branched surfaces can carry a variety of embedded surfaces in the usual way indicated by the number of sheets, or weights, on each piece of the branched surface. The weights are also indicated in Figure~\ref{figure:branchedsurfaces} as well as arrows at the $\frac{1}{2}$-level which indicate the direction of branching as one moves up from the $0$-level to the $1$-level. Finally, notice that all the surfaces shown in Figure~\ref{figure:branchedsurfaces} carry the implicit assumption that $\alpha > \beta$ (and sometimes that $\alpha$ and $\beta$ have the same parity).

If $g=\left (\begin{array}{cc} a &b \\ c & d \end{array}\right)$ is any element of  PSL$_2\Bbb Z$,  let $\hat g=\left (\begin{array}{cc} d&c \\ b&a \end{array}\right)$. We may take $S^2 \times I$ to itself with the homeomorphism $\hat g \times$id  and carry any one of the four basic surfaces to a new branched surface that begins and ends at arc systems  with slopes depending on $g$. Images like this of the four basic surfaces can then be joined together vertically provided they have matching arc systems where they are attached. In this way we can piece together a branched surface that begins with the two arcs of ${\cal L}_{p/q} $ at level 0 of slope $\frac{1}{0}$ and ends with the two arcs of ${\cal L}_{p/q} $ at level 1 of slope $\frac{p}{q}$.  For example, suppose we begin with the Hopf link, ${\cal L}_{1/2}$. We wish to piece together a branched surface that starts at slope $\frac{1}{0}$ and ends at slope  $\frac{1}{2}$. Starting with a copy of $\Sigma_A$ we may move from two arcs at slope $\frac{1}{0}$ to an arc system consisting of three arcs: one at slope $\frac{1}{0}$ and two at slope $\frac{0}{1}$. We may then attach to this an upside-down copy of $\Sigma_D$ which then takes us to an arc system of three arcs: two at slope $\frac{0}{1}$ and one at slope $\frac{1}{2}$. Finally, we will end with an upside-down copy of $\Sigma_A$ transformed by  $\hat g \times$id where $g=\left (\begin{array}{cc} 1&0 \\ 2&1 \end{array}\right)$. Notice that the linear transformation $\hat g$ takes lines of slope $\frac{1}{0}$ to lines of slope $\frac{1}{2}$ and  lines of slope $\frac{0}{1}$ to lines of slope $\frac{0}{1}$, because
$$  \left (\begin{array}{cc} 1 &2 \\ 0&1 \end{array}\right) \left ( \begin{array}{c} 0\\1\end{array} \right )= \left ( \begin{array}{c} 2\\1 \end{array} \right )
\quad \mbox{and} \quad
\left (\begin{array}{cc} 1 &2 \\ 0&1 \end{array}\right) \left ( \begin{array}{c} 1\\0 \end{array} \right )= \left ( \begin{array}{c}1\\ 0 \end{array} \right ).
$$
Therefore, the upside-down copy of $\Sigma_A$ transformed by $\hat g \times$id ends with the desired arc system of two arcs of slope $\frac{1}{2}$.

There is a beautiful correspondence between branched surfaces constructed in this way and continued fraction expansions of $p/q$ which in turn may be viewed as paths in the following diagrams.
Consider first the tessellation $D_1$ of $\Bbb H^2$ by ideal triangles shown in Figure~\ref{figure:D0andD1}. The rationals, together with $\frac{1}{0}$, are arranged around the unit circle as shown, and two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are connected by a geodesic if and only if $ad-bc=\pm 1$. (This diagram contains the Stern-Brocot tree generated from $\frac{1}{0}$ and $\frac{0}{1}$ by adding fractions the ``wrong way'' according to the (mis)rule $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{c+d}$. See \cite{GKP:1994}, for example.) The group of orientation preserving symmetries of $D_1$ is PSL$_2\Bbb Z$. Let $G\subset \mbox{PSL}_2\Bbb Z$ be the subgroup of Mobius transformations given by $z\to \frac{az+b}{cz+d}$ where $c$ is even. It follows that the  triangle $\{\frac{1}{0}, \frac{0}{1}, \frac{1}{1}\}$ is a fundamental domain for the action of $G$ and the $G$-images of the ideal quadralateral $Q=\{\frac{1}{0}, \frac{0}{1},\frac{1}{2}, \frac{1}{1}\}$ tessellate $\Bbb H^2$. If we delete the $G$-orbit of the diagonal $\{\frac{0}{1}, \frac{1}{1}\}$ and replace it with the $G$-orbit of the opposite diagonal $\{\frac{1}{0}, \frac{1}{2}\}$, we obtain a new diagram called $D_0$, which is also shown in Figure~\ref{figure:D0andD1}.

\begin{figure}
\psfrag{a}{$1/1$}
\psfrag{b}{$3/4$}
\psfrag{c}{$2/3$}
\psfrag{d}{$1/2$}
\psfrag{e}{$1/3$}
\psfrag{f}{$1/4$}
\psfrag{g}{$0/1$}
\psfrag{h}{$-1/2$}
\psfrag{i}{$-1/1$}
\psfrag{j}{$1/0$}
\psfrag{k}{$2/1$}
\psfrag{l}{$3/2$}

\psfrag{x}{C}
\psfrag{y}{A}
\psfrag{z}{B}
\psfrag{w}{D}

\psfrag{u}{$D_1$}
\psfrag{v}{$D_0=D_\infty$}

    \begin{center}
    \leavevmode
    \scalebox{.8}{\includegraphics{D0.eps}}
    \end{center}
\caption{The diagrams $D_0, D_1$, and $D_\infty$.}
\label{figure:D0andD1}
\end{figure}

Between $D_0$ and $D_1$ there exists a family of oriented diagrams $D_t$, for $0<t<1$, obtained by expanding each diagonal in $D_1$ (labeled C in Figure~\ref{figure:D0andD1}) to a rectangle which may then be collapsed to the opposite diagonal (labeled D in  Figure~\ref{figure:D0andD1}), thus giving $D_0$. The intersection of each of $D_0, D_t$, and $D_1$ with the fundamental quadralateral $Q$ is shown in Figure~\ref{figure:quadInDt}.  The edges of $D_t$ fall into four $G$-orbits which are named $A, B, C$, and $D$ and which have, respectively, representative edges $A_0, B_0, C_0$ and $D_0$ as defined in Figure~\ref{figure:quadInDt}. As $t\to 0$ the edges degenerate into $B$ and $D$-type edges in $D_0$. If instead, $t\to 1$, the edges degenerate into $A$ and $C$-type edges in $D_1$. We may orient the edges of $D_t$ by using the orientations of $A_0, B_0, C_0$ and $D_0$  shown in Figure~\ref{figure:quadInDt}, but there is no coherent way to orient the edges of $D_0$ or $D_1$. Finally, by setting $D_t=D_{1/t}$ for $1\le t \le \infty$, we obtain a diagram for every $t\in[0,\infty]$.

\begin{figure}
\psfrag{a}{$1/1$}
\psfrag{b}{$1/2$}
\psfrag{c}{$0/1$}
\psfrag{d}{$1/0$}

\psfrag{e}{A}
\psfrag{f}{B}
\psfrag{g}{C}
\psfrag{h}{D}
\psfrag{e0}{A$_0$}
\psfrag{f0}{B$_0$}
\psfrag{g0}{C$_0$}
\psfrag{h0}{D$_0$}
\psfrag{r}{D$_1$}
\psfrag{s}{D$_t$}
\psfrag{t}{D$_0$}

\begin{center}
    \leavevmode
    \scalebox{.85}{\includegraphics{QinDt.eps}}
    \end{center}
\caption{Expanding the diagonals of $D_0$ and $D_1$ to obtain $D_t$
\
(pictured here with $t=3/4$).}
\label{figure:quadInDt}
\end{figure}

Floyd and Hatcher show that the diagram $D_t$ provides a beautiful way of describing all the incompressible surfaces in the 2-bridge link exteriors. In particular, {\it minimal edge paths} in $D_t$ from $\frac{1}{0}$ to $\frac{p}{q}$ will correspond to branched surfaces which in turn will carry the incompressible surfaces. An edge  path in $D_t$ is {\it minimal} if it never contains two consecutive edges that lie in the same triangle or rectangle of $D_t$. It is not hard to see that a minimal edge path in $D_t$ (with $t \not\in \{0,1, \infty\}$) will collapse to a minimal edge path in either $D_0$ or $D_1$ as $t$ approaches 0 or 1 respectively. Moreover, these limiting paths in $D_0$ and $D_1$ uniquely determine the path in $D_t$. For a particular fraction $\frac{p}{q}$ there can only be a finite number of minimal edge paths connecting it to $\frac{1}{0}$. This follows from the fact that these  minimal paths are all contained in a unique minimal chain of quadralaterals consisting of  $Q$ and a finite number of its translates under $G$.

Each minimal edge path in $D_t$ (with $t \not\in \{0,1,\infty\}$) from $\frac{1}{0}$ to $\frac{p}{q}$ provides a recipe for piecing together images of the four basic surfaces $\Sigma_A, \Sigma_B, \Sigma_C$, and $\Sigma_D$ as follows. Suppose $\gamma$ is a path consisting of consecutive edges $e_1, e_2, \dots, e_m$. For each edge $e_i$ there exists an element $g_i \in G$ taking $E_i$ to $e_i$, where  $E_i$ is  one of the four representative edges $A_0, B_0, C_0$, or $D_0$ in $Q$.  Let $S_i$ be $\Sigma_A, \Sigma_B, \Sigma_C$, or $\Sigma_D$  depending on whether $E_i$ is  $A_0, B_0, C_0$, or $D_0$ respectively. Now take $S_i$ to $(\hat g_i \times \mbox{id})(S_i)$, rescaling vertically so as to place the image in $S^2 \times [\frac{i-1}{m}, \frac{i}{m}]$. Moreover, if the orientation of $g_i(E_i)$ is opposite that of $e_i$, we first reflect $S_i$ through the 2-sphere $S^2 \times \{\frac{1}{2}\}$ before applying $\hat g_i \times \mbox{id}$.

If $t$ is rational and equal to the reduced fraction $\alpha/\beta$, we can further use this information to assign weights to the branched surface as follows. If $t>1$ the branched surface is weighted as shown in Figure~\ref{figure:branchedsurfaces}. If instead, $0<t<1$  the above construction is altered by first rotating each of the basic surfaces $\Sigma_A, \Sigma_B, \Sigma_C$, and $\Sigma_D$ through $180^\circ$ in the obvious way so as to interchange the two components of ${\cal L}_{p/q}$. We then swap $\alpha$ and $\beta$ and proceed as before. Thus for every minimal edge path in $D_t$ with $t$ a positive rational number different from $1$, we have associated a weighted branched surface and thus an actual surface in the complement of the link.

If $t=0, 1$, or $\infty$ we must modify this recipe slightly. If we let $t$ approach zero or infinity, then the above constructions will limit at surfaces for which $\alpha=0$ or $\beta=0$ respectively.  But the limiting surfaces which arise from the above constructions when $\alpha \to \beta$ do not give all desired surfaces with $\alpha=\beta$. Instead, the process must be modified slightly. As $t \to 1$ the minimal edge paths in $D_t$ approach minimal edge paths in $D_1$ that consist entirely of $A$ or $C$-type edges. Returning to Figure~\ref{figure:branchedsurfaces}, notice that if $\alpha-\beta=0$, then $\Sigma_A$ with $n=\alpha$ and $\Sigma_C$ are isotopic. Thus  if $C$-type edges are involved, then $\Sigma_A$ allows for more general branching than $\Sigma_C$ and for this reason we replace each use of $\Sigma_C$ with $\Sigma_A$.

Finally, the main result of Floyd and Hatcher is the following theorem.

\begin{theorem}[Floyd and Hatcher] The orientable incompressible and meridionally incompressible surfaces in $S^3-{\cal L}_{p/q}$, without peripheral components, are exactly (up to isotopy) the orientable surfaces carried by the collection of branched surfaces associated to minimal edge paths in $D_t$ from   $\frac{1}{0}$ to $\frac{p}{q}$ and with $t \in [0, \infty]$.
\end{theorem}

Recall from \cite{FH:1988} that a surface $S$ in the complement of the link $L$ is {\it meridionally incompressible} if whenever there is a disk $D\subset S^3$ with $D\cap S=\partial D$ and such that $D$ meets $L$ transversely in one interior point, then there exists a disk $D^\prime\subset S \cup L$ with $\partial D= \partial D^\prime$ and such that $D^\prime$ also meets $L$ transversely in a single interior point.

Floyd and Hatcher then go on to explicitly describe when two surfaces constructed in this way are isotopic. However, they do not compute the boundary slopes of these surfaces,  saying only that it should be possible in principle.

\section{Lash's Algorithm}
\label{section:lash}

The boundary of a branched surface derived from the Floyd-Hatcher construction  defines a train track on the boundary of the regular neighborhood of the link. Thus the boundary of any incompressible surface carried by the branched surface is carried by this train track. Lash's first step is to determine the train tracks for each of the four basic surfaces $\Sigma_A, \Sigma_B, \Sigma_C$ and $\Sigma_D$.

Before doing this,  we introduce some notation. Let the four points of $\Bbb Z^2/\Gamma$ be $(0,0), (1,0), (0,1)$ and $(1,1)$. At the 0-level, assume that the two arcs of ${\cal L}_{p/q}$ join the points $(0,0)$ to $(0,1)$ and $(0,1)$ to $(1,1)$. Furthermore, let ${\cal L}_{p/q}=\{K_1, K_2\}$ where $K_1$ contains $(0,0)$ and $(0,1)$. Orient $K_1$ so that it runs vertically  upward from $(0,0)$ and orient $K_2$ so that it runs vertically upward from $(1,1)$. Choose as a fundamental domain of $\Bbb R^2/\Gamma$ the region ${\cal D}=[0,1] \times [-1/2, 3/2]$. Removing small disks of radius $\epsilon$ centered at the four points of $\Bbb Z/\Gamma$ will remove semi-disks from the fundamental domain $\cal D$.  These correspond to meridional cross sections of
the regular neighborhood of the link.

Let $\mu_1$ be the oriented meridian of $K_1$ having linking number $+1$ with $K_1$. Let $\lambda_1$ be the oriented  longitude of $K_1$ defined as follows. Start with  the line segment from $(0,1-\epsilon, \epsilon)$ to  $(0,\epsilon, \epsilon)$. Join to this the vertical segments $\{(0,\epsilon)\} \times [ \epsilon, 1-\epsilon]$ and $\{(0,1-\epsilon)\} \times [ \epsilon, 1-\epsilon]$.  Next, add the curve in $\Bbb R^2 \times \{1-\epsilon\}$ which starts at $(0,\epsilon, 1-\epsilon)$,
ends at $(0,1-\epsilon, 1-\epsilon)$, and is parallel to $K_1$. Finally, orient $\lambda_1$ parallel to $K_1$.

The $180^\circ$ rotation of Figure~\ref{figure:2-bridge link diagram} about the vertical axis $\{(1/2, 1/2)\} \times \Bbb R$ interchanges the components of the link, and  preserves their orientations.
We  define the oriented meridian $\mu_2$ and longitude $\lambda_2$ of $K_2$ as the images of  $\mu_1$ and $\lambda_1$ respectively under this rotation.

We will initially compute all boundary slopes with respect to the basis $\{\mu_i, \lambda_i \}$. However, $\lambda_i$ is not necessarily a preferred longitude of $K_i$ so it will be necessary later to know its linking number with $K_i$. It is a straightforward exercise to compute this. We obtain
\begin{equation}
\mbox{lk}(K_i, \lambda_i)=-\sum_{j=1}^{\frac{q-2}{2}}(-1)^{[\![2 jp/q]\!]}
\label{eqn:linkingnumber}
\end{equation}
where $[\![x]\!]$ is the greatest integer less than or equal to $x$. Note, for example,  that $\lambda_i$ is never the preferred longitude if $q$ is a multiple of 4 since the linking number must be odd in this case.

Figure~\ref{fig:traintracks} depicts the train track boundaries of each of the four basic branched surfaces. Each row of the figure shows the train tracks on the boundary of each of the four ``columns'' which are the regular neighborhoods of the vertical segments of the link. These are depicted in the corresponding regions of ${\cal D} \times [0,1]$, that is, the product of the semicircular arcs surrounding each integer lattice point with the unit interval $[0,1]$. Notice that we have used slopes ranging from $-\infty$ to $\infty$ to parameterize the semicircular arcs.  Thus we see the train tracks for $\Sigma_A$, on each of the four columns, begin at slopes of $\pm \infty$ and end at slopes of $\pm \infty$ and $0$. Similarly, the train tracks for $\Sigma_C$, on all four columns, have curves that begin at slopes of  $1$ and end at slopes of $0$.

\begin{figure}
\psfrag{00}{$(0,0)$}
\psfrag{01}{$(0,1)$}
\psfrag{10}{$(1,0)$}
\psfrag{11}{$(1,1)$}
\psfrag{a}{\small$\alpha$}
\psfrag{b}{\small$\beta$}
\psfrag{n}{\scriptsize$n$}
\psfrag{bn}{\scriptsize$\beta-n$}
\psfrag{i}{$\infty$}
\psfrag{ni}{$-\infty$}
\psfrag{ab}{\small$\alpha-\beta$}
\psfrag{ab2}{\scriptsize$\displaystyle \frac{\alpha-\beta}{2}$}
\psfrag{0}{$0$}
\psfrag{.5}{$.5$}
\psfrag{1}{$1$}
\psfrag{SA}{$\Sigma_A$}
\psfrag{SB}{$\Sigma_B$}
\psfrag{SC}{$\Sigma_C$}
\psfrag{SD}{$\Sigma_D$}
\psfrag{abn}{\scriptsize$\alpha-\beta-n$}
    \begin{center}
    \leavevmode
    \scalebox{.8}{\includegraphics{traintracks.eps}}
    \end{center}
\caption{Train track boundaries of each of the four branched surfaces.}
\label{fig:traintracks}
\end{figure}

Suppose $\Sigma$ is a branched surface obtained by piecing together homeomorphic copies of the four basic branched surfaces and that $S$ is a surface carried by $\Sigma$. Because $\Sigma$ corresponds to a path in $D_t$ from $\frac{1}{0}$ to $\frac{p}{q}$, $\Sigma$ will always begin with a copy of $\Sigma_A$ and end with an upside-down copy of $\Sigma_A$. Let the initial weights on $\Sigma$ at the 0-level be $\alpha$ and $\beta$ as shown in Figure~\ref{figure:branchedsurfaces}. Thus, at the 0-level on the neighborhood of $K_1$, the boundary of $S$ consists of $\alpha $ arcs. If we orient one of these arcs in the direction of $K_1$ and follow it upward, it will follow the train track up  the $(0,0)$ column until it reaches the top, traverse a curve of slope $\frac{p}{q}$ at the 1-level, and then follow the train track down the $(0,1)$ column. In Figure~\ref{fig:traintracks} we have chosen to orient the train tracks parallel to $K_1$ and $K_2$ for this reason. When we return to our starting point at the bottom we may end there, or perhaps continue to travel around again in the longitudinal direction. If the boundary of $S$ consists of several components, we may orient them all parallel to $K_1$ and $K_2$  (even if these orientations are not compatible with an orientation of $S$).

Suppose $\partial S$ has $k$ components  on the regular neighborhood of $K_1$ and that each has algebraic (and geometric) intersection of $l_i$ with $\mu_1$. Thus $$l_1+l_2+\dots+ l_k=\alpha.$$ Furthermore, suppose each has algebraic intersection $m_i$ with $\lambda_1$. Then $\gcd(m_i, l_i)=1$ for all $i$ and moreover, because the components of $\partial S$ are all disjoint and nontrivial, each boundary slope $m_i/l_i$ is the same. From this it follows that $l_i$ and $m_i$ are constant, say $l_i=l$ and   $m_i=m$. Thus to determine the boundary slope $m/l$ we need only compute the total algebraic intersection $M_1=m_1+m_2 + \dots + m_k$ of $\partial S$ with $\lambda_1$ and divide by $\alpha$. Similarly, the boundary slope of $S$ on the regular neighborhood of $K_2$ is $M_2/\beta$ where $M_2$ is the total algebraic intersection of $\partial S$ with $\lambda_2$. Again, these boundary slopes are with respect to the basis $\{\mu_i, \lambda_i\}$.

To compute $M_1$ we must sum the algebraic intersection of the oriented train tracks with $\lambda_1$ on columns $(0,0)$ and $(0,1)$. We may do this one section at a time, with each section coming from one of the four basic surfaces. We may simplify matters by pulling back $\lambda_1$ under the inverse of $\hat g \times$id and counting its intersection with the standard train tracks shown in Figure~\ref{fig:traintracks}, rather than examining the images of the standard train tracks under $\hat g \times$id. If  $g=\left (\begin{array}{cc} a &b \\ c & d \end{array}\right)$ then  $\hat g^{-1}=\left (\begin{array}{cc} a & -c \\ -b & d \end{array}\right)$ and the vector
$\left( \begin{array}{c} 0 \\ 1 \end{array} \right)$,
which represents a slope of $\infty$ (and the longitude $\lambda_1$), pulls back to the vector $\left( \begin{array}{c} -c \\ d \end{array} \right)$, which represents a slope of $-d/c$. Thus we must examine how the standard train tracks intersect the vertical line located at slope $-d/c$.

For example, suppose $\gamma$ is a minimal edge path in $D_t$ beginning at $\frac{1}{0}$ and ending at $\frac{p}{q}$. Let $e_i$ be an edge of $\gamma$ that is the image of $E_i=A_0$ under the element $g_i$ of $G$. Furthermore, suppose that the orientations of $e_i$ and $g_i(A_0)$ agree.  If $0<-d/c<\infty$ then the algebraic intersection on column $(0,0)$ is $-n$ and on column $(0,1)$ is $-(\beta-n)$. Note that on column $(0,0)$ the meridian runs from $0$ towards $\infty$ while on column $(0,1)$ the meridian runs from $0$ to $-\infty$. This explains the minus signs in the above calculations. The total contribution to $M_1$ at this level, in this case, is thus $-\beta$. If instead, $-\infty<-d/c<0$, we obtain $\beta-n$ on column $(0,0)$ and $n$ on column $(0,1)$ for a total contribution to $M_1$ of $\beta$. Note that $-d/c$ cannot be zero. This is because $c$ is even and therefore both $a$ and $d$ must be odd since $\det(g)=1$. However, $-d/c$ may equal $\pm \infty$ if $c=0$. In this case the train track is not transverse to the (pullback of the) longitude. When this occurs, we may istotope the longitude slightly by pushing it in the positive direction of $\mu_1$. So in this case, if $-d/c=\pm \infty$ we obtain an intersection on column $(0,0)$ of $\beta -n$ and on column $(0,1)$ of $-(\beta-n)$ for a net contribution to $M_1$ of zero. Table~\ref{table:m1andm2} lists these results together with the contributions to $M_1$ for the other three types of surfaces, all in the case where the orientations of $e_i$ and $g_i(E_i)$ agree. If these orientations are opposite, then we must flip each surface upside-down and it is easy to see that the effect is to negate the entries in the table.

A similar examination of the standard train tracks allows us to compute $M_2$. However in this case an additional consideration seems necessary. For any $g \in G$, $\hat g$ takes $(0,0)$ to itself and $(0,1)$ to itself since $c$ is even and both $a$ and $d$ are odd. But if $b$ is odd, then  $(1,0)$ and $(1,1)$ will be traded while if $b$ is even these lattice points will each be fixed. Thus it seems necessary to consider these cases separately. However, a close examination of the case when $b$ is odd reveals that both the orientations of the train tracks as well as the orientation of $\mu_2$ are reversed and the total contribution to $M_2$ is unaffected. Thus separate formulae for $b$ even or odd are not needed.  The results for each of the four basic surfaces are listed in Table~\ref{table:m1andm2}.

\begin{table}[htdp]
\renewcommand{\arraystretch}{1.2}
\caption{Contributions to $M_1$ and $M_2$ according to surface type.}
\label{table:m1andm2}
\begin{center}
\begin{tabular}{|c|c|c|c|}

\hline
Surface Type & $M_1$ &  $M_2$&$-d/c$\\
\hline
\hline
$A$&
\begin{tabular}{c}
$\beta$\\
$-\beta$\\
$0$\\
\end{tabular} &
\begin{tabular}{c}
$\beta$\\
$-\beta$\\
$0$\\
\end{tabular} &
\begin{tabular}{c}
$-\infty<-d/c<0$\\
$0<-d/c<\infty$\\
$-d/c=\pm\infty$\\
\end{tabular} \\

\hline
$B$&
\begin{tabular}{c}
$\beta-\alpha$\\
$\alpha-\beta$\\
$0$\\
\end{tabular} &
\begin{tabular}{c}
$0$\\
$0$\\
$0$\\
\end{tabular} &
\begin{tabular}{c}
$-\infty<-d/c<0$\\
$0<-d/c<\infty$\\
$-d/c=\pm\infty$\\
\end{tabular}  \\
\hline

$C$&

\begin{tabular}{c}
$-2\beta$\\
$0$\\
\end{tabular} &
\begin{tabular}{c}
$0$\\
$2 \beta$\\
\end{tabular} &
\begin{tabular}{c}
$0<-d/c<1$\\
otherwise\\
\end{tabular}\\
\hline

$D$&
\begin{tabular}{c}
$\beta-\alpha$\\
$0$\\
$\alpha-\beta$\\
\end{tabular} &
\begin{tabular}{c}
$\alpha-\beta$\\
$\alpha-\beta$\\
$\alpha-\beta$\\
\end{tabular} &
\begin{tabular}{c}
$-\infty<-d/c<1/2$\\
$-d/c=1/2, \pm\infty$\\
$1/2<-d/c<\infty$\\
\end{tabular} \\
\hline

$A$ for $C$&
\begin{tabular}{c}
$2(\beta-n)$\\
$-2n$\\
\end{tabular}
&
\begin{tabular}{c}
$2n$\\
$2(n-\beta)$\\
\end{tabular}
&
\begin{tabular}{c}
$-\infty<-d/c<0$\\
$0<-d/c<\infty$\\
\end{tabular}\\
\hline

\end{tabular}
\end{center}
\end{table}

The data in the first four rows of Table~\ref{table:m1andm2} are sufficient to compute the boundary slopes when $t=\alpha/\beta \not\in \{0,1,\infty\}$ and even in the limiting cases $t\to 0$ or $t \to \infty$. But as mentioned already, when $t \to 1$,  if any  $C$-type edges appear in the limiting minimal path in $D_1$, then  we  replace the corresponding $C$-type surfaces with the more general $A$-type surfaces. Suppose $e$ is such a $C$-type edge in $D_1$. Then $e$ joins $a/c$ and $b/d$ in $D_1$ where both $c$ and $d$ are odd, $ad-bc=1$, and thus, exactly one of $a$ or $b$ is even. Therefore we may choose $g=\left (\begin{array}{cc} a &b \\ c & d \end{array}\right)$ and assume that $b$ is even, while $a,c$, and $d$ are all odd. Now $\hat g$ fixes $(0,0)$ and $(1,0)$ while trading $(0,1)$ and $(1,1)$. To properly use  Figure~\ref{fig:traintracks} now, we must orient columns $(0,0)$ and $(0,1)$ up and the other two down. Furthermore, the meridian $\mu_1$ points from $0$ to $\infty$ in the $(0,0)$ figure and from $0$ to $-\infty$ in the $(1,1)$ figure. Similarly, the meridian $\mu_2$ points from $0$ to $\infty$ in the $(0,1)$ figure and oppositely in the $(1,0)$ figure. It is now a simple matter to compute the contributions to $M_1$ and $M_2$ given in the last row of Table~\ref{table:m1andm2}. Here we see that the number $n$ of horizontal sheets does not cancel from the calculations. Notice also that because both $c$ and $d$ are odd, $-d/c$ can not equal zero or infinity.

Everything is now in place to compute the boundary slopes for a given 2-bridge link, ${\cal L}_{p/q}$. Beginning with $0<\beta<\alpha$, and hence $1<t<\infty$, we first find all minimal paths in $D_t$ from $1/0$ to $p/q$. For each of these paths, each edge $e$ must be identified as the image of $E \in \{A_0, B_0, C_0, D_0\}$ by some element $g=\left (\begin{array}{cc} a &b \\ c & d \end{array}\right) \in G$. Next, using $-d/c$ and the data in Table~\ref{table:m1andm2}, the contributions to $M_1$ and $M_2$ are computed and these are added over all edges in the path to obtain $M_1$ and $M_2$. Note that the entries in the table must be negated if the orientation of $e$ does not match that of  $g(E)$.  Notice also that $M_1$ and $M_2$ are always integer linear combinations of $\alpha$ and $\beta$. The boundary slope of this surface is then
$$\frac{M_1}{\alpha}=\frac{x \alpha +y \beta}{\alpha}=x+\frac{y}{t}$$
on the boundary associated to $K_1$ and
$$\frac{M_2}{\beta}=\frac{z \alpha +w \beta}{\beta}=z t+w$$
 on the boundary associated to $K_2$. This gives a 1-parameter family of boundary slopes for each rational number $t$ greater than 1. As $t$ approaches $\infty$ we may simply take the limits of the slopes obtained so far provided $z\ne0$. If $z=0$, then $M_2$ and $\beta$ both approach zero as $t$ approaches $\infty$. This means that the surface has no intersection with the regular neighborhood of $K_2$ and thus has no boundary slope associated with $K_2$.  At the other extreme, letting $t$ approach 1 will produce legitimate boundary slopes, but not all possible slopes with $\alpha=\beta$. Instead we must consider the limiting minimal edge path in $D_1$, swap $A$ for $C$-type edges (if there are any) and now use the data from the last row of Table~\ref{table:m1andm2}. Finally,  to allow for $t<1$, we must rotate all our surfaces 180$^\circ$ around the axis $\{(1/2, 1/2)\} \times \Bbb R$, thus trading $\alpha$ with $\beta$. The 4-tuple $(M_1(\alpha, \beta), \alpha, M_2(\alpha, \beta), \beta)$ of the algebraic intersections with $\lambda_1, \mu_1, \lambda_2$, and $\mu_2$ respectively is then transformed to  $(M_2(\beta, \alpha), \alpha, M_1(\beta, \alpha), \beta)$ with it now the case that $\alpha < \beta$.  Letting $t$ approach zero now corresponds to letting $\alpha$ approach zero. Finally, all of these computations are with respect to the bases  $\{\mu_1, \lambda_1\}$ and $\{\mu_2, \lambda_2\}$. To convert to a preferred bases, we must consider the linking number $l=$lk$(\lambda_1, K_1)=$lk$(\lambda_2, K_2)$ given in Equation~\ref{eqn:linkingnumber}. Converting to the preferred basis sends the 4-tuple $(M_1, \alpha, M_2, \beta)$ to $(M_1+l \alpha, \alpha, M_2+l \beta, \beta)$.

\section{An Improved Algorithm}
\label{section:improvedalgorithm}

As mentioned already, $M_1$ and $M_2$ are always integer linear combinations of $\alpha$ and $\beta$. After implementing Lash's algorithm on a computer and looking at sample data, the conclusions of the following theorem were apparent. In searching for a proof, we were led to a revision of Lash's algorithm that is much simpler to apply by hand and implement on a computer. The revised approach will be described in the proof and illustrated in the next section with an interesting example.

Before stating some results we define $M(\gamma)$ to be the pair of intersection numbers $(M_1, M_2)$ associated to the path $\gamma$ in $D_t$.

\begin{theorem}
\label{thm:xyz}
Given any path $\gamma$ (not necessarily minimal) in $D_t$ with $1<t<\infty$,   which begins and ends at vertices  of $D_1$,  $M(\gamma)$ is of the form
$$M(\gamma)=(x \alpha+y\beta, y\alpha+z\beta)$$
where  $x \equiv z \mod 2$. If additionally, $\gamma$ begins at $\frac{1}{0}$ and ends at $\frac{p}{q}$, then $x+y \equiv 1+ q \mod 2$.

\end{theorem}

The proof of this theorem is an easy consequence of the following two lemmas.

\begin{lemma}
\label{lemma:AandBonly}
Suppose $\gamma_t$ is a path in $D_t$ starting at $\frac{p_0}{q_0}$, ending at $\frac{p_n}{q_n}$ and consisting solely of $A$ and $B$-type edges. Letting $t \to 1$, $\gamma_t$ will collapse to the path
$$\gamma_1=\left \{\frac{p_0}{q_0}, \frac{p_1}{q_1}, \dots, \frac{p_n}{q_n} \right \}$$
in $D_1$ containing only $A$-type edges. Then
$$\begin{array}{rl}
M(\gamma_t)&=\left ( \sum_{i=0}^{n-1}\delta_i \right)(\alpha, \beta) \mbox{ and }\\
M(\gamma_1)&=\left ( \sum_{i=0}^{n-1}\delta_i \right)(\beta, \beta)
\end{array}
$$
where
$$\delta_i=\left \{ \begin{array}{ll} 0, & \mbox{if  } q_iq_{i+1}= 0;\\
 p_i q_{i+1}-p_{i+1}q_i, & \mbox{otherwise.}  \end{array}\right.$$
\end{lemma}
{\bf Proof:} Let $e$ be an edge of $\gamma_1$ oriented from $\frac{a}{c}$ to $\frac{b}{d}$ where $a, b, c$ and $d$ are all positive, $c$ is even and $ad-bc=\pm1$. The corresponding matrix $g \in G$ is either  $g=\left (\begin{array}{cc}  a&b\\c&d\\ \end{array}\right  ) $ or $g=\left (\begin{array}{cc}  a&-b\\c&-d\\ \end{array}\right  ) $, whichever one has determinant one. In $\gamma_t$, $e$ begins with an $A$-type edge, oriented forward, and ends with a $B$-type edge, oriented backwards. Referring to Table~\ref{table:m1andm2}, we see that the contribution to $(M_1, M_2)$ is either $(0,0)$ if $c=0$, or $(ad-bc)(\alpha, \beta)$. It is easy to check  the remaining case where $c$ is odd and $d$ is even.
\hfill $\Box$

\begin{lemma}
\label{lemma:pathmovest}
 Let  $g=\left (\begin{array}{cc}  a&b\\c&d\\ \end{array}\right  ) $ be any element of $G$ and label the regions of $g(Q)$ as shown in Figure~\ref{fig:gQ}. If $\gamma_i=\partial R_i$, oriented counterclockwise, then $M(\gamma_i)$  are given by:
$$\begin{array}{l|l}
\gamma_i&M(\gamma_i)\\
\hline
\gamma_0 \mbox{ or } \gamma_2& (0, -2\beta)\\
\gamma_1 \mbox{ or } \gamma_3& (-\alpha+\beta, \alpha-\beta)\\
\gamma_4 & (-2\beta, -2\alpha+4\beta)\\
\end{array}
$$
\end{lemma}

\begin{figure}
\psfrag{a}{$\frac{a+b}{c+d}$}
\psfrag{b}{$\frac{a+2 b}{c+2 d}$}
\psfrag{c}{$\frac{b}{d}$}
\psfrag{d}{$\frac{a}{c}$}

\psfrag{e}{A}
\psfrag{f}{B}
\psfrag{g}{C}
\psfrag{h}{D}
\psfrag{e0}{A}
\psfrag{f0}{B}
\psfrag{g0}{C}
\psfrag{h0}{D}
\psfrag{s}{D$_t$}

\psfrag{r0}{$R_0$}
\psfrag{r1}{$R_1$}
\psfrag{r2}{$R_2$}
\psfrag{r3}{$R_3$}
\psfrag{r4}{$R_4$}

\begin{center}
    \leavevmode
    \scalebox{.85}{\includegraphics{loopsinQ.eps}}
    \end{center}
\caption{Lemma~\ref{lemma:pathmovest} describes the result of travelling around each region.}
\label{fig:gQ}
\end{figure}

{\bf Proof:} Suppose $g=\left (\begin{array}{cc}  a&b\\c&d\\ \end{array}\right  ) $ and consider the path $\gamma_0$. The first edge in the path is of type $A$ and is oriented forward, the second is type $C$ oriented backwards, and the final edge is type $A$ oriented backwards. The matrix $g$ is used to determine the contribution of the first two edges, while $\left (\begin{array}{cc}  a&a+b\\c&c+d\\ \end{array}\right  )$ is used for the third edge. Thus to determine the contribution of the third edge we must consider $-d^\prime/c^\prime=-(c+d)/c=-1-d/c$. Suppose first that $-d/c=\pm \infty$ and hence $-d^\prime/c^\prime=\pm \infty$. Using Table~\ref{table:m1andm2}, we see that the three edges contribute $(0,0), (0,-2\beta)$ and $(0,0)$ respectively, for a sum of $(0,-2\beta)$.
Next, suppose $-\infty<-d/c<0$ and thus  $-\infty<-d^\prime/c^\prime<0$. Now the edges contribute $(\beta, \beta), (0,-2\beta)$ and $(-\beta, -\beta)$, giving the same total as before. As our next case, suppose $0<-d/c<1$ and hence $-\infty<-d^\prime/c^\prime<0$.  The edges now contribute $(-\beta,-\beta), (2\beta,0)$ and $(-\beta, -\beta)$, again giving a sum of $(0,-2\beta)$. Finally, if $1<-d/c<\infty$, then $0<-d'/c'<\infty$ and we obtain $(-\beta,-\beta)+(0,-2\beta)+(\beta, \beta)=(0,-2\beta)$.

The computations for the other regions are similar and are left to the reader.
\hfill $\Box$

{\bf Proof of Theorem~\ref{thm:xyz}:} Let $\gamma$ be a path in $D_t$ with endpoints in $D_1$ and let $\gamma'$ be obtained from $\gamma$ by pushing each ``diagonal'' edge across a region of type $R_0$ through $R_3$ (as in Lemma~\ref{lemma:pathmovest}) to eliminate all edges of type $C$ and $D$. Thus $M(\gamma')=k(\alpha, \beta)$ for some integer $k$. Notice that any path in $D_1$ must pass through fractions whose denominators alternate in parity. Thus if $\gamma'$ starts at $\frac{1}{0}$ and ends at $\frac{p}{q}$ it must have a number of edges equivalent to $q \mod 2$. Since the first edge contributes zero to $k$ and every other edge contributes $\pm 1$, we have that $k$ and $q$ have opposite parity.

To go back to $\gamma$ from $\gamma'$ suppose that we must move across $n_0^+$ regions of type $R_0$ (or $R_2$) in the positive sense and $n_0^-$ in the negative sense. Similarly, let $n_1^+$ and $n_1^-$ be the number of regions of type $R_1$ (or $R_3$) that we must push across in the positive or negative sense respectively. Then
$$\begin{array}{rl}
M(\gamma)&=k(\alpha, \beta)+(n_0^+-n_0^-)(0, -2\beta)+(n_1^+-n_1^-)(-\alpha+\beta, \alpha-\beta)\\
&=((k-n_1^++n_1^-)\alpha+(n_1^+-n_1^-)\beta, (n_1^+-n_1^-)\alpha+(k-n_1^++n_1^--2n_0^++2n_0^-)\beta).
\end{array}
$$
Thus
$$\begin{array}{rl}
x&=k-n_1^++n_1^-\\
y&=n_1^+-n_1^-\\
z&=k-n_1^++n_1^--2n_0^++2n_0^-
\end{array}
$$
and $x \equiv z \mod 2$. Furthermore, $x+y=k$. Thus if the path begins at $\frac{1}{0}$ and ends at $\frac{p}{q}$ we see that $x+y \equiv 1+q \mod 2$.
\hfill $\Box$

The following lemma, which is analogous to Lemma~\ref{lemma:pathmovest}, allows us to handle the  case where $C$-type edges are replaced with $A$-type edges in $D_1$.
The proof is similar to the proof of Lemma~\ref{lemma:pathmovest} and is left to the reader.

\begin{lemma}
\label{lemma:pathmoves1}
 Let  $g=\left (\begin{array}{cc}  a&b\\c&d\\ \end{array}\right  )$  be any element of $G$ where $b$ is even and label the regions of $g(Q)$ as shown in Figure~\ref{fig:gQ1}. If $\gamma_i=\partial S_i$, oriented counterclockwise, then
$$
\begin{array}{rl}
M(\gamma_0)&=( -2\beta+2n, -2n) \mbox{ and}\\
M(\gamma_1)&=(-2n,  -2\beta+2n)
\end{array}
$$
where $0 \le n \le \beta$.

\end{lemma}

\begin{figure}
\psfrag{a}{$\frac{a+b}{c+d}$}
\psfrag{b}{$\frac{a+2 b}{c+2 d}$}
\psfrag{c}{$\frac{b}{d}$}
\psfrag{d}{$\frac{a}{c}$}

\psfrag{e}{A}
\psfrag{f}{B}
\psfrag{g}{C}
\psfrag{h}{D}
\psfrag{e0}{A}
\psfrag{f0}{B}
\psfrag{g0}{C}
\psfrag{h0}{D}
\psfrag{r}{D$_1$}

\psfrag{r0}{$S_0$}
\psfrag{r1}{$S_1$}

\begin{center}
    \leavevmode
    \scalebox{.85}{\includegraphics{loopsinD1.eps}}
    \end{center}
\caption{Lemma~\ref{lemma:pathmoves1} describes the result of travelling around each region.}
\label{fig:gQ1}
\end{figure}

Using this lemma we may determine the form of $M(\gamma)$ for any path in $D_1$.

\begin{theorem}
\label{thm:xys}Let $\gamma$ be any path in $D_1$. Then $M(\gamma)$ has the form
$$M(\gamma)=((x+y s)\beta, (x-y s)\beta)$$
where $y$ is the number of $C$-type edges in $\gamma$ and $s$ is a rational parameter with $-1\le s \le 1$. If additionally, $\gamma$ begins at $\frac{1}{0}$ and ends at $\frac{p}{q}$, then $x+y \equiv 1+q \mod 2$.
\end{theorem}

{\bf Proof:}  If $\gamma$ has no $C$-type edges the result is the same as Lemma~\ref{lemma:AandBonly}. If $C$-type edges are present, push each such edge across a region of type $S_0$ (as in Lemma~\ref{lemma:pathmoves1}) to obtain a path with only $A$ and $B$-type edges. Applying Lemma~\ref{lemma:pathmoves1}, we now have that
$$M(\gamma)=k(\beta, \beta)+\sum_{i=1}^P(-2\beta+2n_i, -2n_i)-\sum_{i=P+1}^{P+N}(-2\beta+2n_i, -2n_i)$$
for some integer $k$, nonnegative integers $P$ and $N$, and integers $n_i$ where $0\le n_i \le \beta$. Let
$$X=2\sum_{i=1}^Pn_i-2\sum_{i=P+1}^{P+N}n_i.$$
 If we let
$$s=\frac{X-(P-N)\beta}{(P+N)\beta}$$
then it is not hard to show that $-1 \le s \le 1$. Substituting for $X$ in $M(\gamma)$ we obtain the desired result with $y=P+N$ and $x=k-P+N$.
\hfill $\Box$

Theorems~\ref{thm:xyz} and \ref{thm:xys} place restrictions on the form of $M(\gamma)$ where $\gamma$ is a path in either $D_t$ or $D_1$. The following theorem shows that these are in fact the only restrictions that apply and moreover that minimal paths may be used to realize any desired value of $M(\gamma)$.

\begin{theorem}
There exist minimal paths in either $D_t$ or $D_1$ which begin at $\frac{1}{0}$ and end at some $\frac{p}{q}$ that realize all possible values of $M$ subject only to the constraints of Theorems~\ref{thm:xyz} and \ref{thm:xys}.
\end{theorem}

{\bf Proof:} Focussing on Theorem~\ref{thm:xyz}, let $x, y$, and $z$ be any three integers such that $x \equiv z \mod 2$. Suppose also that $x+y \equiv 1 \mod 2$. We seek a fraction $\frac{p}{q}$, with $q$ even, and a minimal path $\gamma$ from $\frac{1}{0}$ to $\frac{p}{q}$ such that $M(\gamma)=(x\alpha+y\beta, y\alpha+z\beta)$. Referring to the proof of Theorem~\ref{thm:xyz}, we will show how to build a minimal path with compete control over $k, n_0^+, n_0^-, n_1^+$, and $n_1^-$.

\begin{figure}
\psfrag{a}{$1$}
\psfrag{b}{$0$}
\psfrag{c}{$\frac{1}{0}$}
\psfrag{d}{$-1$}
\psfrag{e}{$\begin{array}{l}\Delta n_1^+=1\\ \Delta k=2 \end{array}$}
\psfrag{g}{$\begin{array}{l}\Delta n_1^-=1\\ \Delta k=0 \end{array}$}
\psfrag{i}{$\begin{array}{l}\Delta n_0^+=1\\ \Delta k=4 \end{array}$}
\psfrag{l}{$\begin{array}{l}\Delta n_0^-=1\\ \Delta k=0 \end{array}$}

\psfrag{n}{$\Delta k=1, 3, 5, \dots$}
\psfrag{p}{$\Delta k=-1,-3,-5, \dots$}

    \begin{center}
    \leavevmode
    \scalebox{.5}{\includegraphics{pathmodules.eps}}
    \end{center}
\caption{Building blocks for a minimal path $\gamma$ with desired $M(\gamma)$.}
\label{fig:pathblocks}
\end{figure}

Figure~\ref{fig:pathblocks} shows six blocks of quadrilaterals, each containing a minimal path. Heavy dots are placed at vertices with even denominators. Starting with either of the last two blocks in the second row,  we may then paste on any number of the first four blocks (in any order), always gluing blocks together as indicated by the arrows.  No matter how  the blocks are joined  together, the path will remain minimal.  The first four blocks involve $C$ and $D$ type edges and can be used to create any desired values for  $n_0^+, n_0^-, n_1^+$, and $n_1^-$. Once these parameters are fixed, a sufficiently long starting block of one of the two types may be prepended to create any desired value of $k$.

It is not hard to adapt this argument to the case where $x+y \equiv 0 \mod 2$ or to the case of Theorem~\ref{thm:xys}. Moreover, more efficient sets of building blocks than these can easily be designed.
\hfill $\Box$

\section{An Example}
\label{section:example}

In this section we will apply our improved algorithm to compute the boundary slopes of the links
 ${\cal L}_\frac{4 k-1}{8 k}$. Since $\frac{4 k-1}{8 k}$ has the continued fraction expansion $[0,2,-2k,-2]$, these links can be pictured as shown in Figure~\ref{figure:the links}. Here we have replaced the $-2k$ right-handed crossings with $\frac{1}{k}$-surgery on an unknot surrounding the parallel strands. Viewed this way we see that ${\cal L}_\frac{4 k-1}{8 k}$, which we will henceforth simply denote  as ${\cal L}_k$,  is $\frac{1}{k}$ surgery on one component of the Borromean rings.

\begin{figure}[t]
\psfrag{a}{\large $\frac{1}{k}$}
    \begin{center}
    \leavevmode
        \scalebox{.85}{\includegraphics{rotatedLinkDiagram.eps}}
    \end{center}
\caption{The links ${\cal L}_k$}
\label{figure:the links}
\end{figure}

The minimal chain of quadralaterals in the diagram $D_t$ connecting $\frac{1}{0}$ to $\frac{4k-1}{8k}$ is abstractly depicted in Figure~\ref{figure:quadchain}. It is not hard to show by induction that for $k>1$ there are precisely six minimal paths, $\{\gamma_1, \dots, \gamma_6 \}$, in $D_t$ from $\frac{1}{0}$ to $\frac{4k-1}{8k}$. These are listed in Table~\ref{table:minimalpaths} by listing the consecutive vertices in each path. The vertices $R_i, S_i$, and $T_i$ each lie between two vertices of $D_1$ (or $D_0$) as indicated in the figure. If $k=1$, path $\gamma_4$ is no longer minimal and should be deleted from the list.

Associated to each of these paths, in the case where $t>1$, is a weighted branched surface with $\alpha>\beta$. To find the boundary slopes of surfaces carried by such a branched surface we must first compute the intersection numbers $(M_1, M_2)$ and then adjust for our choice of basis. We will derive  $(M_1, M_2)$ from each path by using the Lemmas given in the last section.

\begin{figure}[b]
\psfrag{a}{$\frac{1}{0}$}
\psfrag{b}{$S_0$}
\psfrag{c}{$\frac{1}{1}$}
\psfrag{d}{$\frac{2k}{4k+1}$}
\psfrag{e}{$T_{2k}$}
\psfrag{f}{$\frac{4k-1}{8k}$}
\psfrag{h}{$T_0$}
\psfrag{i}{$R_0$}
\psfrag{j}{$\frac{1}{2}$}
\psfrag{k}{$R_{2k+1}$}
\psfrag{l}{$S_{2k}$}
\psfrag{m}{$\frac{0}{1}$}
\psfrag{n}{$R_1$}
\psfrag{o}{$R_{2k}$}
\psfrag{p}{$\frac{2k-1}{4k-1}$}
\psfrag{q}{$R_i$}
\psfrag{r}{$R_{i+1}$}
\psfrag{s}{$\frac{i-1}{2 i-1}$}
\psfrag{t}{$\frac{i}{2i+1}$}
\psfrag{u}{$S_i$}
\psfrag{v}{$T_i$}
\psfrag{w}{$\frac{2i-1}{4i}$}

    \begin{center}
   \leavevmode
      \scalebox{.7}{\includegraphics{quadchain.eps}}
    \end{center}
\caption{The minimal chain of quadralaterals from $\frac{1}{0}$ to $\frac{4k-1}{8k}$.}
\label{figure:quadchain}
\end{figure}

\begin{table}
\renewcommand{\arraystretch}{1.5}
\caption{Minimal $D_t$ paths from $\frac{1}{0}$ to $\frac{4 k-1}{8k}$.}
\label{table:minimalpaths}
\begin{center}
\begin{tabular}{rl}
$\gamma_1$: & $\frac{1}{0}, S_0, R_0, \frac{1}{2}, R_{2 k+1}, T_{2k}, \frac{4k-1}{8k}$\\
$\gamma_2$: & $\frac{1}{0}, S_0, R_0, \frac{1}{2}, R_{2 k},  S_{2k}, \frac{4k-1}{8k}$\\
$\gamma_3$: &$\frac{1}{0}, T_0, R_1, \frac{1}{2}, R_{2 k+1},  T_{2k}, \frac{4k-1}{8k}$\\
$\gamma_4$: & $\frac{1}{0}, T_0, R_1, \frac{1}{2}, R_{2 k},  S_{2k}, \frac{4k-1}{8k}$\\ $\gamma_5$: & $\frac{1}{0}, T_0, R_1, R_2, \dots, R_{2k}, S_{2k}, \frac{4k-1}{8k}$\\
$\gamma_6$:  & $\frac{1}{0}, T_0, \frac{0}{1}, S_1, T_1, \frac{1}{3}, S_2, T_2, \dots, \frac{2k-1}{4k-1}, S_{2k},  \frac{4k-1}{8k}$

\end{tabular}
\end{center}
\end{table}

Let $\sigma_1$ be the path in $D_1$ given by $\sigma_1=\{\frac{1}{0},\frac{1}{1},\frac{1}{2},\frac{2k}{4k+1},\frac{4k-1}{8k}\}$. It follows easily from Lemma~\ref{lemma:AandBonly} that $M(\sigma_1)=(3\alpha, 3\beta)$. We may now deform $\sigma_1$ to $\gamma_1$ by moving the path over two regions, each of type $R_1$ (or $R_3$) as shown in Figure~\ref{fig:gQ}. This results in adding $2(-\alpha+\beta, \alpha-\beta)$. We therefore obtain $(2 \alpha+\beta, \alpha+2 \beta)$.
To pass to either $\gamma_2$ or $\gamma_3$, we deform $\gamma_1$ across two regions of type $R_0$ and one of type $R_4$. In either case then, we must add
$(-2\beta, -2\alpha)$ and obtain a total of $(\alpha, \beta)$.
To move from $\gamma_2$ (or $\gamma_3$) to $\gamma_4$, we again move across two regions of type $R_0$ and one of type $R_4$ and again add  $(-2\beta, -2\alpha)$. This gives $(\alpha-2\beta, -2\alpha+\beta)$ for $M(\gamma_4)$.
We may now obtain $\gamma_5$ from $\gamma_4$ by moving across $2k-1$ regions of type $R_0$. Thus the value of $M(\gamma_5)$ is $(\alpha-2\beta, -2\alpha+\beta)+(2k-1)(0, -2\beta)=(\alpha-2\beta, -2\alpha+(3-4k)\beta)$.
Now let $\sigma_2=\{\frac{1}{0},\frac{0}{1},\frac{1}{2},\frac{2k-1}{4k-1},\frac{4k-1}{8k}\}$. We may either think of moving $\gamma_4$ to $\sigma_2$, or start over again with Lemma~\ref{lemma:pathmovest}. Either way we obtain $(-\alpha, -\beta)$ for $M(\sigma_2)$.
Finally, to obtain $\gamma_6$ from $\sigma_2$, we must move across $2k-1$ regions, each made up of one region of type $R_0$, two of type $R_1$ and one of type $R_4$. This results in adding $(2k-1)(-2\alpha, 0)$ to  $M(\sigma_2)$ giving a total of $((1-4k)\alpha, -\beta)$.

We have now accounted for all the possibilities when $\alpha>\beta$. We may determine all cases where $\beta=0$, and thus $t=\infty$, from these by substitution, but letting $\alpha$ equal $\beta$  does not give all possibilities for $t=1$. Because both $\gamma_5$ and $\gamma_6$ involve $C$-type edges, so do their limiting minimal edge paths in $D_1$.  In this example, both $\gamma_5$ and $\gamma_6$  limit to the same minimal edge path $\gamma_5^1$ in $D_1$. To recover all possible boundary slopes when $\alpha=\beta$ we must now consider the branched surface corresponding to this path where $C$-type surfaces are replaced with $A$-type surfaces.

Starting from $\sigma_2$ we may move to $\gamma_5^1$ by moving across $k$ regions of type $S_0$ and $k-1$ regions of type $S_1$. Thus we must add $\sum_{i=1}^k(-2\beta+2n_{2i-1}, -2n_{2i-1})$ and $\sum_{i=1}^{k-1}(-2n_{2i}, -2 \beta+2n_{2i})$ to $M(\sigma_2)=(-\alpha, -\beta)$. This gives $(-(1+2k)\beta+X, (1-2k)\beta-X)$, where $X=2\sum_{i=1}^{2k-1}(-1)^{i+1}n_i$. Here $0\le n_i\le \beta$.

The final 4-tuples $(M_1, \alpha, M_2, \beta)$ for ${\cal L}_{\frac{4k-1}{8k}}$ in all cases where $\alpha \ge \beta \ge 0$ are given in  Table~\ref{table:boundaryslopes}. All of these data  are still with respect to the basis $\{\mu_i, \lambda_i\}$.

From Table~\ref{table:boundaryslopes} we may easily derive the corresponding 4-tuples when $0 \le \alpha \le \beta$. Namely, $(M_1(\alpha, \beta),\alpha, M_2(\alpha, \beta), \beta)$ is changed to $(M_2(\beta, \alpha), \alpha, M_1(\beta, \alpha), \beta)$. Notice that if  a minimal path in $D_t$ does not involve any $C$-type edges, then the contributions to $M_1$ and $M_2$ given in Table~\ref{table:m1andm2} are equal when $\alpha=\beta$. Thus the same 4-tuple will result as we approach $\frac{\alpha}{\beta}=1$ from either above or below.
Using Equation~\ref{eqn:linkingnumber} it is not difficult to show that $\mbox{lk}(\lambda_1, K_1)=\mbox{lk}(\lambda_2, K_2)=-1$ for all $k$.  The final boundary slopes $M_1/\alpha$ and $M_2/\beta$  for all rational values of $t$, and with respect to the preferred basis are now given in Table~\ref{table:finalslopes}. Furthermore, the slopes in  this table are described by the rational parameters $t=\alpha/\beta$ in the case where $\alpha\ne \beta$, and
 $$s=\frac{X-\beta}{(2k-1)\beta}$$
  in the case where $\alpha=\beta$. Note that a pair of boundary slopes of the form $(0,\phi)$ means that the corresponding surface has no boundary components on $K_2$ (and a slope of zero on $K_1$).

\begin{table}
\caption{Boundary slope data for ${\cal L}_{\frac{4k-1}{8k}}$, $k>0$,  with respect to the basis $\{\mu_1, \lambda_1\}$ and for $\alpha \ge \beta$.
The path $\gamma_4$ is not counted if $k=1$. Here $X=n_1-n_2+n_3- \dots +n_{2k-1}$.}
\begin{center}
\begin{tabular}{lcc}
path&algebraic intersection with & restrictions\\
&$(\lambda_1, \mu_1, \lambda_2, \mu_2)$&\\
\hline
$\gamma_1$ &$(\alpha+2 \beta,\alpha,2 \alpha+\beta, \beta)$&$\alpha>\beta \ge 0$\\
$\gamma_2$ &$(\alpha,\alpha,\beta, \beta)$&$\alpha>\beta\ge 0$\\
$\gamma_3$ &$(\alpha,\alpha,\beta, \beta)$&$\alpha>\beta\ge 0$\\
$\gamma_4$ &$(\alpha-2 \beta,\alpha,-2\alpha+\beta, \beta)$&$\alpha>\beta\ge 0$\\
$\gamma_5$ &$(\alpha-2 \beta,\alpha,-2 \alpha+(3-4k)\beta, \beta)$&$\alpha>\beta\ge 0$\\
$\gamma_6$ &$((1-4k)\alpha,\alpha, -\beta, \beta)$&$\alpha>\beta\ge 0$\\
$\gamma_5^1$ &$(-(1+2k)\beta+2X,\beta, (1-2k)\beta-2X, \beta)$& $0\le n_i\le \beta=\alpha$\\

\end{tabular}
\end{center}
\label{table:boundaryslopes}
\end{table}

\begin{table}
\caption{Boundary slope pairs for  ${\cal L}_{\frac{4k-1}{8k}}$, $k>0$,  with respect to the preferred basis $\{\mu_1, \lambda_1^0\}$. Both $t$ and $s$ are rational parameters.}
\begin{center}
\begin{tabular}{|l|c|}
\hline
$\partial$-slopes&restrictions\\
\hline
\hline
$(0,0)$&\\
$(0, \phi),(\phi, 0)$&\\
$(-4k, \phi),(\phi, -4k)$&\\
$(-4k, -2),(-2, -4k)$&\\
$(2 t^{-1}, 2t)$&$0\le t \le \infty$\\
$(-2 t^{-1}, -2t)$&$0\le t \le \infty, k>1$\\
$(-2 t^{-1}+2-4k, -2t)$&$0\le t \le 1$\\
$(-2 t^{-1},2-4k- 2t)$&$1\le t \le \infty$\\
$(-1-2k+(2k-1)s, -1-2k-(2k-1)s)$&$-1 \le s\le 1$\\

\hline
\end{tabular}
\end{center}
\label{table:finalslopes}
\end{table}

\section{Boundary Slopes of 2-bridge Links up to 10 Crossings}
\label{section:table}

We have written a computer program to implement the algorithm described in Section~\ref{section:improvedalgorithm}. One begins by finding all minimal paths in $D_0$ and $D_1$ from $\frac{1}{0}$ to $\frac{p}{q}$. These determine the minimal paths in $D_t$. Each minimal path $\gamma$ in $D_t$ is then deformed into a path $\gamma'$ of only $A$ and $B$-type edges. Lemma~\ref{lemma:AandBonly} is used to compute $M(\gamma')$ and $M(\gamma)$ is derived from this using Lemma~\ref{lemma:pathmovest}.  In order to minimize the chance of producing errors, each of us coded a program independent of the other with debugging proceeding until our results agreed.

The second Tait conjecture states that any reduced alternating diagram of a link is minimal. Thus is it easy to determine the crossing number of ${\cal L}_{p/q}$. If we express $p/q$ as the continued fraction $[0, a_2, \dots, a_n]$ where each $a_i$ is positive, then ${\cal L}_{p/q}$ has a reduced alternating 4-plat diagram with $a_2+a_3+\dots a_n$ crossings. Thus the crossing number of ${\cal L}_{p/q}$ is $a_2+a_3+\dots a_n$. It is now a simple matter to determine all 2-bridge links with 10 or fewer crossings by finding all such continued fractions. Furthermore, from the classification of 2-bridge links, we know precisely when two fractions represent the same link.

The boundary slopes of all 2-bridge links having ten or less crossings are presented in Tables~\ref{data2to8}-\ref{data10c}. All slopes are  with respect to a preferred longitude and meridian basis. For those links with 9 or less crossings, the index of the link in Rolfsen's table \cite{Ro:1990} is also given. Two types of data are listed. First, all boundary slopes in the case where $1<t<\infty$  are given. From these the slopes with $0<t<1$ can then be determined as described earlier. Furthermore, one may then derive from these all boundary slopes in the cases $t=0$ and $t=\infty$ and some of the boundary slopes in the case $t=1$. Finally, all additional slopes in the case $t=1$ are given in terms of the rational parameter $s$ with $-1 \le s \le 1$.

\bibliographystyle{hplain}

redpajama_set_name: RedPajamaArXiv


text:
Q: regarding Remote Simulator to Windows When I run my Xamarin iOS test via Visual Studio 2017, I got the following message: "iOS tests are not supported on Windows". I have set up Remote Simulator to Windows option so it should open simulator but it didn't happen so please tell me what's wrong?
https://learn.microsoft.com/en-us/xamarin/tools/ios-simulator/index
Here is my test configuration:
I have installed Visual Studio 2017 Enterprise Version 15.8.5 with Xamarin version 4.11.
I have created new Test Project using Xamarin.UITest Cross-Platform Test Project template.
I have also configured Mac Book using Pair to Mac option and it is connected fine.
I have checked Remote Simulator to Windows option in IOS Settings via VS Tools options in order to run my test on remote simulator.
I have following code in my Test.cs file
if (platform == Platform.iOS)
{
return ConfigureApp
.iOS
.InstalledApp("com.example.yourappbundleid")
.EnableLocalScreenshots()
.StartApp();
}

When I run the test, I got the following message: iOS tests are not supported on Windows.
As I have said earlier, I have set up Remote Simulator to Windows option so it should open simulator but it didn't happen so please tell me what's wrong?
I knew we can't run test directly on windows machine using actual device but I should open the remote simulator.
Please suggest me the solution or any updated information you have regarding running iOS test on Remote Simulator.
Many thanks in advance.
SPatel

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Clifford M. Lewis brings to Currenex more than 25 years experience in the international financial markets as well as a unique network of long-standing business relationships in over 30 countries. Prior to joining Currenex, Mr Lewis was the Chief Operating Officer of BRIDGE eMarkets, a unit of Bridge Information Systems responsible for wholesale financial market transactions. Before that, he was an executive with CBOT and a Management Board Member for CBOT-EUREX Joint Venture, Inc., an alliance between two of the world's largest derivatives exchanges. From 1978 to 1991, Mr. Lewis was an executive with the World Bank Group, where he had various high-level strategic planning and operational assignments. His experience also includes managing strategy, restructuring and technology engagements at leading banks, exchanges and clearing houses as a partner in the International Financial Services [Group] at KPMG Peat Marwick. Articles by Mr. Lewis on a range of policy and technical subjects have appeared in the Wall Street Journal, Atlantic Monthly, National Interest and other publications. Mr. Lewis is an active member of various advisory groups on international economics and finance at the Heritage Foundation Cato Institute and American Enterprise Institute and is a Fellow of Harvard University's Center for International Affairs.
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A new Japanese innovation using blockchain technology has made lab-grown diamonds chemically, physically and optically identical to natural diamonds.
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Gene expression change in cyclin D1 -/- retinas in comparison to wildtype.
Ontology highlight
ABSTRACT: Cyclin D1 belongs to the core cell cycle machinery1, and it is frequently overexpressed in human cancers2. The full repertoire of cyclin D1 functions in normal development and in cancer cells is currently unknown. To address this question, here we introduce a novel approach that allows one to determine the set of cyclin D1-interacting proteins (D1 "interactome") and cyclin D1-bound genomic fragments (D1 "cistrome") in essentially any mouse organ, at any point of development or at any stage of cancer progression. Using this approach, we detected several novel tissue-specific interactors of cyclin D1. A significant number of these partners represent proteins involved in transcription. We show, using genome-wide location analysis3, that cyclin D1 occupies promoters of a very large number of genes in the developing mouse, where it binds in close proximity to transcription start sites. Bioinformatics analyses of cyclin D1-bound genomic segments in the developing embryo revealed DNA recognition sequences for several transcription factors. By querying SAGE libraries4, promoter CpG content5 and gene expression profiles of cyclin D1-null organs, we demonstrate that cyclin D1 binds promoters of highly expressed genes, and that it functions to activate or to repress gene expression in vivo. Analyses of cyclin D1 transcriptional targets reveal that cyclin D1 contributes to cell proliferation by upregulating genes required for S-phase entry and progression. Hence, cyclin D1 plays a broad transcriptional regulatory function in vivo during normal mouse development. We wished to determine whether cyclin D1 functioned to positively or negatively regulate transcription of the target genes (identified by chip-on-chip) in vivo. To address this question, we asked how expression of cyclin D1-bound genes changes when cyclin D1 has been knocked-out. We dissected retinas from eyes of cyclin D1-/- or wild-type neonates, isolated RNA and hybridized it onto Affymetrix expression arrays.
ORGANISM(S): Mus musculus
SUBMITTER: Frederic Emile Bienvenu Frederic E Bienvenu
PROVIDER: E-GEOD-13635 | ArrayExpress | 2010-03-09
SECONDARY ACCESSION(S): GSE13635PRJNA113873
REPOSITORIES: GEO, ArrayExpress
Json Xml
Transcriptional role of cyclin D1 in development revealed by a genetic-proteomic screen.
Bienvenu Frédéric F Jirawatnotai Siwanon S Elias Joshua E JE Meyer Clifford A CA Mizeracka Karolina K Marson Alexander A Frampton Garrett M GM Cole Megan F MF Odom Duncan T DT Odajima Junko J Geng Yan Y Zagozdzon Agnieszka A Jecrois Marie M Young Richard A RA Liu X Shirley XS Cepko Constance L CL Gygi Steven P SP Sicinski Piotr P
Nature 20100101 7279
Cyclin D1 belongs to the core cell cycle machinery, and it is frequently overexpressed in human cancers. The full repertoire of cyclin D1 functions in normal development and oncogenesis is unclear at present. Here we developed Flag- and haemagglutinin-tagged cyclin D1 knock-in mouse strains that allowed a high-throughput mass spectrometry approach to search for cyclin D1-binding proteins in different mouse organs. In addition to cell cycle partners, we observed several proteins involved in trans ...[more]
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Flag Chromatin IP in Cyclin D1 Flag-HA tagged E.14.5 embryos versus Flag Chromatin IP in wildtype E.14.5 embryos
Project description:Cyclin D1 belongs to the core cell cycle machinery1, and it is frequently overexpressed in human cancers2. The full repertoire of cyclin D1 functions in normal development and in cancer cells is currently unknown. To address this question, here we introduce a novel approach that allows one to determine the set of cyclin D1-interacting proteins (D1 "interactome") and cyclin D1-bound genomic fragments (D1 "cistrome") in essentially any mouse organ, at any point of development or at any stage of cancer progression. Using this approach, we detected several novel tissue-specific interactors of cyclin D1. A significant number of these partners represent proteins involved in transcription. We show, using genome-wide location analysis3, that cyclin D1 occupies promoters of a very large number of genes in the developing mouse, where it binds in close proximity to transcription start sites. Bioinformatics analyses of cyclin D1-bound genomic segments in the developing embryo revealed DNA recognition sequences for several transcription factors. By querying SAGE libraries4, promoter CpG content5 and gene expression profiles of cyclin D1-null organs, we demonstrate that cyclin D1 binds promoters of highly expressed genes, and that it functions to activate or to repress gene expression in vivo. Analyses of cyclin D1 transcriptional targets reveal that cyclin D1 contributes to cell proliferation by upregulating genes required for S-phase entry and progression. Hence, cyclin D1 plays a broad transcriptional regulatory function in vivo during normal mouse development. Overall design: We hypothesized that cyclin D1 may play role in transcription by interacting with transcriptional machinery at the promoters of target genes. To test this possibility, we employed chromatin immunoprecipitation coupled to DNA microarray analysis (genome-wide location analysis or ChIP-chip) to study association of cyclin D1 with genomic DNA sequences. In this procedure, a protein of interest is crosslinked to DNA, immunoprecipitated, and the bound DNA is hybridized to an array containing probes that span the genome. Since anti-FLAG antibodies have been successfully used for ChIP-chip in several different systems including murine cells, knock-in mice expressing tagged cyclin D1 provided us with a novel tool to query the association of cyclin D1 with the genome in vivo, in a living mouse.
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Project description:The chromosomal translocation t(11;14)(q13;q32) leading to cyclin-D1 over-expression plays an essential role in the development of mantle cell lymphoma (MCL), an aggressive tumor that remains incurable with current therapies. Cyclin-D1 has been postulated as an effective therapeutic target, but its evaluation has been hampered by our incomplete understanding of its oncogenic functions and by the lack of valid MCL murine models. To address these issues, we generated a cyclin-D1-driven mouse model whereby cyclin-D1 expression can be externally regulated. These mice developed lymphomas capable of recapitulating most features of human MCL. We found that cyclin-D1 inactivation was not sufficient to induce lymphoma regression in vivo. However, using a combination of in vitro and in vivo assays, we identified a novel pro-survival cyclin-D1 function in MCL cells. Specifically, we demonstrate that cyclin-D1 sequestrates the pro-apoptotic protein BAX, thereby favoring BCL2 anti-apoptotic function. Accordingly, cyclin-D1 inhibition sensitized the lymphoma cells to apoptosis through BAX release. Thus, genetic or pharmacologic targeting of cyclin-D1 combined with a pro-apoptotic BH3 mimetic synergistically killed murine lymphomas and human MCL cells. Our study identifies a novel role of cyclin-D1 in deregulating apoptosis and highlights the potential benefit of simultaneously targeting cyclin-D1 and survival pathways in patients with MCL. Dose response involving 20 samples. Two replicates for each cell line, 4 cell lines sensitive to ABT-737 and 6 cell lines resistant to ABT-737. Mantel cell lymphoma cell lines sensitive to ABT-737 vs. Mantel cell lymphoma cell lines resistant to ABT-737
Sustained expression of cyclin D1 or cyclin D1/KE in mouse mammary gland
Project description:Analysis of mammary glands from tet-inducible(rtTA) transgenic mice expressing cyclin D1 using Affymetrix Mouse Gene 1.0 ST GeneChip arrays. MMTV-rtTA transgenic mice (MMTV-Mouse Mammary Tumor Virus promoter) were cross-mated to cyclin D1 transgenic mice under control of tet operon. 8-week-old tetracycline-inducible cyclin D1/rtTA bi-transgenic pregnant female mice (12 days postcoitus) were treated with doxycycline through drinking water supplementation at a final concentration of 2 mg/ml. Control mice were rtTA transgenics alone and treated in the same manner. After 7 days of doxycycline treatment, the mice were sacrificed and mammary glands taken for RNA isolation. Results provide insight into the in vivo gene expression pattern regulated by cyclin D1 through acute induction. Analysis of mammary glands from MMTV-cyclin D1/WT and MMTV-cyclin D1/KE using Affymetrix Mouse 430A v2.0 GeneChip arrays. Cyclin D1 point mutant, cyclin D1/KE K112E (K112E) contains a lysine to glutamine substitution at amino acid position 112. cyclin D1. The cyclin D1/KE mutant fails to induce cyclin D1-dependent kinase activity. Female MFD1, MFD1-KE, and WT mice were monitored twice weekly, up to 760 days, for the development of palpable tumors. Those developing palpable tumors were sacrificed within a week of tumor detection. Tumors were dissected and portions snap frozen for RNA isolation. Results provide insight into the in vivo gene expression pattern regulated by cyclin D1 that is kinase independent. Two separate control mice were positive for MMTV-rtTA transgene compared to 3 separate cyclin D1/rtTA bitransgenic female mice and 3 separate cyclin D1 KE mutant/rtTA bitransgenic female mice (Mouse Gene 1.0 ST arrays). Three separate control WT FvBmice were compared to three MMTV-cyclin D1/WT and 3 MMTV-cyclin D1/KE mice (Mouse 430A v2.0 arrays).
Acute expression of cyclin D1 in mouse mammary gland using the tetracycline system
Project description:Analysis of mammary glands from tet-inducible (rtTA) transgenic mice expressing cyclin D1 (Ccnd1). MMTV-rtTA transgenic mice (MMTV-Mouse Mammary Tumor Virus promoter) were cross-mated to cyclin D1 transgenic mice under the control of the tet operon. 8-week-old tetracycline-inducible cyclin D1/rtTA bi-transgenic pregnant female mice (12 days postcoitus) were treated with doxycycline through drinking water supplementation at a final concentration of 2 mg/ml. Control mice were rtTA transgenics alone and were treated in the same manner. After 7 days of doxycycline treatment, the mice were sacrificed and mammary glands taken for RNA isolation. Results provide insight into the in vivo gene expression pattern regulated by cyclin D1 through acute induction. Two separate control mice positive for the MMTV-rtTA transgene were compared to 3 separate cyclin D1/rtTA bi-transgenic female mice.
Comparison of Normal Prostate Fibroblasts Overexpressing Cyclin D1 and Carcinoma Associated Fibroblasts
Project description:The cyclin D1 oncogene encodes the regulatory subunit of a holoenzyme that phosphorylates and inactivates the Rb protein and promotes progression through G1 to S phase of the cell cycle. Several prostate cancer cell lines and a subset of primary prostate cancer samples have increased cyclin D1 protein expression. However, the relationship between cyclin D1 expression and prostate tumor progression has yet to be clearly characterized. This study examined the effects of manipulating cyclin D1 expression in either human prostatic epithelial or stromal cells using a tissue recombination model. The data showed that overexpression of cyclin D1 in the initiated BPH-1 cell line increased cell proliferation rate, but did not elicit tumorigenicity in vivo. However, overexpression of cyclin D1 in Normal Prostate Fibroblasts (NPF) that were subsequently recombined with BPH-1 did induce malignant transformation of the epithelial cells. The present study also showed that recombination of BPH-1 + cyclin D1 overexpressing fibroblasts (NPF cyclin D1) resulted in permanent malignant transformation of epithelial cells (BPH-1 NPF-cyclin D1 cells) similar to that seen with Carcinoma Associated Fibroblasts (CAFs). Microarray analysis showed that the expression profiles between CAFs and NPF cyclin D1 cells were highly concordant including cyclin D1 upregulation. These data indicated that the tumor-promoting activity of cyclin D1 may be tissue-specific. Keywords: cyclin D1; stromal-epithelial interactions; prostate cancer; cDNA microarray NPF cyclin D1 cells were generated from NPFs, which were isolated from two different patient samples; CAFs were isolated from two different patient samples as well. RNA was isolated from NPFs, CAFs, and NPF cyclin D1 cells using total RNA isolation kit (Qiagen). Custom spotted cDNA microarrays were constructed as previously described (True, L., Coleman, I., Hawley, S., Huang, C. Y., Gifford, D., Coleman, R., Beer, T. M., Gelmann, E., Datta, M., Mostaghel, E., Knudsen, B., Lange, P., Vessella, R., Lin, D., Hood, L., and Nelson, P. S. A molecular correlate to the Gleason grading system for prostate adenocarcinoma. Proc Natl Acad Sci U S A, 103: 10991-10996, 2006) using a non-redundant set of 6,700 prostate-derived cDNA clones identified from the Prostate Expression Data Base (PEDB), a public sequence repository of expressed sequence tag data derived from human prostate cDNA libraries. Total RNA was amplified through one round of linear amplification using the MessageAmp aRNA kit (Ambion, Austin, TX). Sample quality and quantification was assessed by agarose gel electrophoresis and absorbance at A260. Cy3 and Cy5 labeled cDNA probes were made from 4 µg of amplified RNA. Two NPF cyclin D1 and two CAF samples (labeled with Cy3) were hybridized head-to-head with an NPF control sample labeled with Cy5. Probes were hybridized competitively to microarrays under a coverslip for 16 h at 63°C. Fluorescent array images were collected for both Cy3 and Cy5 by using a GenePix 4000B fluorescent scanner, and image intensity data were gridded and extracted using GenePix Pro 4.1 software. Differences in gene expression between NPF cyclin D1/NPF and CAF/NPF groups were determined using a two-sample t-test with Significance Analysis of Microarrays (SAM) software (http://www-stat.stanford.edu/_tibs/SAM/) with a False Discovery Rate (FDR) of ≤ 10% considered significant (Tusher, PNAS, 2001). Similarities in gene expression between NPF cyclin D1/NPF and CAF/NPF groups were determined using a one-sample t-test in SAM with an FDR of ≤ 0.1% considered significant. These results were reduced to unique genes by eliminating all but the highest scoring clones for each gene. A Pearson correlation coefficient was calculated in Excel to assess the strength of the linear relationship between NPF cyclin D1/NPF and CAF/NPF average log2 ratios.
2007-08-31 | E-GEOD-6936 | ArrayExpress
The undifferentiated phenotype in neuroblastoma depends on Cyclin D1 and CDK4 activity
Project description:Genomic aberrations of Cyclin D1 (CCND1) and CDK4 in neuroblastoma indicate that dysregulation of the G1 entry checkpoint is an important cell cycle aberration in this pediatric tumor. Here we report that analysis of Affymetrix expression data of primary neuroblastic tumors shows an extensive over-expression of Cyclin D1 and CDK4 which correlates with histological subgroups and prognosis respectively. Immunohistochemical analysis demonstrated an over-expression of Cyclin D1 in neuroblasts and a low Cyclin D1 expression in all cell types in ganglioneuroma. This suggests an involvement of G1 regulating genes in neuronal differentiation processes which we further evaluated using RNA interference against Cyclin D1 and its kinase partner CDK4 in several neuroblastoma cell lines. This resulted in pRb pathway inhibition as shown by an almost complete disappearance of CDK4 specific pRb phosphorylation; reduction of E2F transcriptional activity and a decrease of Cyclin A protein levels. The Cyclin D1 and CDK4 knock-down resulted in a significant reduction in cell proliferation, a G1 specific cell cycle arrest and moreover an extensive neuronal differentiation. Affymetrix microarray profiling of siRNA treated cells revealed a shift in expression profile towards a neuronal phenotype. Several new potential downstream players are identified. We conclude that neuroblastoma functionally depend on over-expression of G1 regulating genes to maintain their undifferentiated phenotype. Keywords: Neuroblastoma, CCND1, Cyclin D1, CDK4 Overall design: The Cell line IMR-32 at time point 0 and transiently transfected with siRNA against GFP, Cyclin D1 and CDK4 at time point 48 hours. All experiments are biological triplicates.
2008-08-20 | GSE8866 | GEO
Analyses of cyclin D1 function using a "genetic-proteomic" approach
Project description:We examined the transcriptional function of cyclin D1 in mouse development using two approaches. First, we queried association of cyclin D1 with the genome of E14.5 mouse embryos using ChIP-on-chip approach. We observed binding of cyclin D1 to several promoter regions. Second, we compared gene expression profiles between wild-type and cyclin D1-null retinas. We observed several transcripts with altered levels in cyclin D1-null organs. Refer to individual Series
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Thank you to all those who joined us for our Parent and Children Craft sessions.
Lots of grown ups came in to join us in making our crafts in all year groups across the school which were very successful! The children had a wonderful time!


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text:
Q: Перенос виртуального диска vhdx в физический В Hyper-V под Windows 2012 R2 есть виртуальная машина, используется виртуальный диск vhdx.
Возникла необходимость использовать физический диск вместо виртуального.
Каким образом можно перенести данные с диска vhdx на физический?


A keynote speaker welcomes the crowd at Made by Few web design conference at the Ron Robinson Theater in Downtown Little Rock on Oct. 3, 2015.
Creatives, web developers and designers of all sorts, get your little black book planners, iPads, sketchbooks, or whatever you take notes on out. Little Rock's popular creative conference Made by Few is moving up to Bentonville this year the weekend of Oct. 26-29.
Made by Few started five years ago as a conference to inspire and provide networking for designers, developers, creatives, and entrepreneurs in the startup economy culture. Several workshops, guest speakers and events are planned for attendees to participate in.
The event will be spread out in venues of downtown Bentonville and Northwest Arkansas, but the main event is being held at the brand new venue, The Record, near the Bentonville square, which used to house the newspaper offices of the Bentonville Daily Record, as well as Crystal Bridges and 21c Hotel.
This year's keynote speakers include PayPal's Product Design Lead Victoria Prater, Cassie McDaniels of the Mozilla Foundation, Daniel Burka of Google Ventures, Cards Against Humanity's Design Director Amy Schwartz, Alex Cornell of Facebook and James Victore, just to name a few.
The speeches range in content variety from managing inspiration, working remotely and influencing remote communities, balancing multiple projects at once and discussing design from top level of the system.
Keynote speakers from all over the country from companies such as Spotify, PayPal, Facebook and Google are regulars to the Made by Few conference.
Friday night also features the Designed by Few competition at 21c Hotel, where three designers compete for a cash prize as they do design work benefiting local non-profit ArtFeeds. There will also be a Halloween costume contest, music, food and drinks. Saturday night will be the afterparty, with live music from Adam Faucett and the Tall Grass.
Tickets to conference are available online at madebyfew.com and are $424.22 for a full pass, $186.37 for students and $212.80 for a day pass. Ticket sales will cap off at 300, Lowry said.
There will also be Handmade by Few, which will be open to the public for free, from the folks behind The Little Craft Show. Some of the artisans at the craft fair will be Sally Nixon, Big Bot, Chipper Things, Hello Cocoa, KYYA, Jason Jones, Sara L. Walker, Arsenic and Lace and several others. The show will be at The Record during the main conference, starting at 8:30 a.m.
All of the speeches given at the conference will be uploaded online for free viewing following the conference, as previous years speeches have been, Lowry said.
To no surprise, there is an impressively designed website with more info surrounding the conference at madebyfew.com.


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Q: Getting KeyError: 'project' when try to run apportable load command I am trying to load my coming objective C project using apportable load command on terminal. But I am getting KeyError: 'project' in starting. Could anyone please suggest.
e.g.
Building with TARGET_ARCH_ABI:armeabi ARM_NEON:False
Building to /Users/nimitja/.apportable/SDK/Build/android-armeabi-debug
Traceback (most recent call last):
  File "/Users/nimitja/.apportable/SDK/bin/apportable", line 745, in
    run(env)
File "/Users/nimitja/.apportable/SDK/bin/apportable", line 709, in run
    results = actionsargs.action
File "/Users/nimitja/.apportable/SDK/bin/apportable", line 86, in LoadAction
    return env.LoadApp(site_init.BuildApplication(env, env['BUILD_TARGET']))
File "/Users/nimitja/.apportable/SDK/site_scons/site_init.py", line 359, in BuildApplication
    return build.App(env, app_sconscript)
File "/Users/nimitja/.apportable/SDK/site_scons/build/init.py", line 477, in App
    print "Updating configuration parameters... Building Xcode project " + target["project"]
KeyError: 'project'
Suggestion are welcome.

A: This is likely an issue from installing Xcode 5. Its xcodebuild does sub-projects (even without our tools scanning things).
Do xcode-select --switch /Applications/Xcode.app/Contents/Developer to switch your Xcode version back.