\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}

\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{tikz, tikz-cd}
\usetikzlibrary{decorations.markings}
\usepackage{xcolor}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=blue,
    pdftitle={Knot Theory},
    pdfpagemode=FullScreen,
    }
% \usepackage{ulem}

% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real nubers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
\newcommand{\st}{ \ : \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\img}{\mathrm{im}} % image
\newcommand{\into}{\hookrightarrow} % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map


% knot theory
\newcommand{\lk}{\mathrm{lk}} % linking number


% SOME ENVIRONMENTS

\newcommand{\exheading}[1]{\section*{Exercise #1}}
\newcounter{exnum}
% \setcounter{exnum}{1} % default 0 start
\newcommand{\exercise}{
    \stepcounter{exnum} 
    \exheading{\theexnum}
    }

\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
\newcommand{\ee}{\end{enumerate}}




\title{MAT180 HW06}
\author{(ADD NAME)}
\date{Due 5/12/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle


\noindent {\footnotesize
\textbf{Reminder.} 
Your homework submission must be typed up in full sentences, with proper mathematical formatting.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

Below is a depiction of the Stevedore knot $6_1$, drawn as the boundary of an oriented Seifert surface built out of one disk and two twisted bands:
\begin{center}
\begin{tikzpicture}
    % \def\r{.2}; % gap
    % diagram
    \draw[thick] (0,2) -- (0,0) -- (8,0) -- (8,2);
    \draw[thick] (0,4) -- (2,6) -- (6,2) -- (7,2);
    \draw[thick] (1,4) -- (2,5) -- (5,2) -- (3,2);
    \filldraw[white] (4,3.5) circle (.7cm);
    \draw[thick] (3,2) -- (7,6);
    \draw[thick] (1,2) -- (2,2) -- (7,7) -- (8,6);
    % crossings
    \foreach \y in {2,3}{
        \begin{scope}[yshift= \y cm]
            \draw[thick] (0,0) -- (1,1);
            \filldraw[white] (.5,.5) circle (.2);
            \draw[thick] (1,0) -- (0,1);
        \end{scope}
    }
    \foreach \y in {2,3,4,5}{
        \begin{scope}[yshift= \y cm, xshift=7cm]
            \draw[thick] (1,0) -- (0,1);
            \filldraw[white] (.5,.5) circle (.2);
            \draw[thick] (0,0) -- (1,1);
        \end{scope}
    }
    % orientation
    \node (plus) at (4,1) {$+$};
    \draw (4,1) circle (.2cm);
\end{tikzpicture}
\end{center}

\bea
    \item Let $\gamma_1$ be the counter-clockwise loop that goes through the left band (only, exactly once), and let $\gamma_2$ be the counter-clockwise loop going through the right band (only, exactly once). Sketch a diagram of the surface $F$, the curves $\gamma_1, \gamma_2$, and their pushoffs $\gamma_1^+, \gamma_2^+$. 
    \item Construct the Seifert matrix $V$ for this surface. 
    \item Compute $\det(V - tV^\top)$; then, if needed, multiply it by $t^n$ for some $n \in \Z$ (possibly negative) so that the powers of $t$ appearing in the Alexander polynomial of $6_1$ are balanced around $0$. 
\ee


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

% linking matrix of a Seifert surface in two different ways

Consider the usual diagram of the \href{http://katlas.org/wiki/4_1}{Figure 8 in the Rolfsen knot table}.


\bea
\item Draw the Seifert surface $F$ resulting from Seifert's algorithm such that `most' of the surface is facing upward (i.e. you see more of the positive side of $F$ than the negative side).
Since there are four crossings, you should have four bands connecting the Seifert disks.

\item The surface $F$ is \emph{homotopy equivalent} to a graph with four edges. Draw this graph $\Gamma$. Then, contract edges until you have a graph $\Gamma_*$ with only one vertex; draw this process as a sequence of graphs ending with $\Gamma_*$.

\item Using your $\Gamma_*$, choose a basis $\{\gamma_i\}$ for the first homology $H_1(F)$ of $F$. (Remember to orient your curves!) Draw these curves on a copy of your surface $F$; be careful near the crossings! Draw push-offs $\{\gamma_i^+\}$ in a different color.

\item Compute the Seifert matrix $V$ with respect to the data $(F, \{\gamma_i\})$ that you've chosen.

\item Compute $\det(V-tV^\top)$ and compare this with the Alexander polynomial of the Figure 8 knot.
    
\ee





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% \emph{This problem will be graded for completion.}
% Stabilization and Alexander polynomial.

Let $V$ be a matrix corresponding to some data $(F,\{\gamma_i\}_{i=1}^{2g})$, where $g = g_3(F)$. Let $\tilde F$ be a stabilization of $F$, and let $\widetilde{V}$ be the Seifert matrix computed from the data $(\tilde F, \{\gamma_i\} \cup \{\tilde \gamma_1, \tilde \gamma_2\})$, where 
\[
    \lk(\tilde \gamma_1,\tilde \gamma_1^+)
    = \lk(\tilde \gamma_2,\tilde \gamma_1^+)
    = \lk(\tilde \gamma_2,\tilde \gamma_2^+)
    = 0
    \qquad \text{and} \qquad
    \lk(\tilde \gamma_1,\tilde \gamma_2^+)
    = 1.
\]
Therefore $\widetilde{V}$ must have the following form:
\[
    \widetilde{V} = 
    \left(\begin{array}{@{}c|c@{}}
    V & U \\ \hline
    L & 
        \begin{matrix}
            0 & 1 \\
            0 & 0
        \end{matrix}
    \\ 
    \end{array}\right),
\]
where $U \in \R^{2g \times 2}$ and $L \in \R^{2 \times 2g}$.  

We want to understand how $\det(\widetilde{V} - t \widetilde{V})$ is related to $\det(V - tV)$. This is somewhat difficult in general. However, if we consider the special case where $U = L^\top = 0 \in \R^{2g \times 2}$ (as in the example drawn in class), then 
\[
    \widetilde{V} = 
    \left(\begin{array}{@{}c|c@{}}
    V & 0 \\ \hline
    0 & 
        \begin{matrix}
            0 & 1 \\
            0 & 0
        \end{matrix}
    \\ 
    \end{array}\right).
\]
In this case, it is not difficult to compute $\det(\widetilde{V} - t \widetilde{V}^\top)$, by following the steps below.


\bea

\item Write down the matrix $\widetilde{V} - t \widetilde{V}^\top$ in terms of $V$.

\item Let $D$ denote the bottom right $2\times2$ block of the matrix $\widetilde{V} - t \widetilde{V}^\top$. Prove that $D$ is invertible, i.e.\ its determinant is a nonzero Laurent polynomial (a polynomial in negative, 0, and positive powers of the variable $t$).
Then, find the matrix inverse of $D$. 

\item Compute the Alexander polynomial $\det(\widetilde{V} - t \widetilde{V}^\top)$ by using the following fact from linear algebra, in terms of the polynomial $\det(V - t V^\top)$. 

\paragraph{Fact.} 
If $\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ is a block matrix where $D$ is invertible, then 
\[
    \det \left ( \begin{bmatrix} A & B \\ C & D \end{bmatrix} \right )
    = \det(D)\det(A - BD\inv C).
\]



\ee












\end{document}
