\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
\newcommand{\ind}{\mathrm{index}} % index of crit pt in Morse theory
\newcommand{\ord}{\mathrm{ord}} % order of element in group
\newcommand{\CC}{\mathcal{C}} % category, or chain complex
\newcommand{\Ob}{\mathrm{Ob}} % objects
\newcommand{\Mor}{\mathrm{Mor}} % morphisms
\newcommand{\Mat}{\mathrm{Mat}} % additive closure


% 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 HW07}
\author{(ADD NAME)}
\date{Due 5/19/23 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
% morse theory
In this exercise, you'll compute Euler characteristic using ideas from Morse theory. 
%
Consider the following picture of $\Sigma_2$, a closed, orientable surface of genus 2:

\begin{center}
\begin{tikzpicture}[scale=1]
    \draw (1,3) arc (0:180:1cm);
    \draw (1,1) arc (0:-180:1cm);
    \draw (-1,3) -- (-1,1);
    \draw (1,3) -- (1,1);
    % genus holes
    \begin{scope}[xscale=.5, yshift=2.8cm]
        \draw (.2,.5) arc (90:270:.5cm);
        \draw (0, .45) arc (70:-70:.48cm);
    \end{scope}
    \begin{scope}[xscale=.5, yshift=1.2cm]
        \draw (.2,.5) arc (90:270:.5cm);
        \draw (0, .45) arc (70:-70:.48cm);
    \end{scope}
\end{tikzpicture}
\end{center}

\bea

\item Draw picture of $\Sigma_2$ similar to the above, and then imagine it as a two-holed donut with glaze dripping down from the top. Identify the \emph{critical points} on the surface, i.e.\ points where some particles of the glaze do not flow. Label these critical points with variable names $c_1, c_2, \ldots$.

\item For each $c_j$, identify the number of \emph{stable dimensions}, i.e.\ directions where glaze is flowing downward toward $c_j$, and also the number of \emph{unstable dimensions}, i.e.\ directions where glaze is flowing downward away from $c_j$. 

\emph{Note: ``Directions'' here is imprecise, because we really just want to choose two linearly independent directions at each $c_j$. The number if ``directions'' is obviously infinite!} 

\item For a critical point $c_j$, the \emph{index} of $c_j$, denoted $\ind(c_j)$, is the number of unstable dimensions at $c_j$. Write down the index of each critical point. Then, compute the Euler characteristic of $\Sigma_2$ using the formula
\[
    \chi(\Sigma_2) = \sum_i (-1)^i \#\{c_j \st  \ind(c_j) = i\}.
\]
Check for yourself that this agrees with our previous definition of $\chi$ using genus. 




\ee


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% concordance group

In class, we defined the mathematical object known as a \emph{group}. 
Let $g$ be an element in a group $(G, \cdot)$, where $\cdot$ is the group's binary operation.
We think of $\cdot$ like multiplication; for example, we write $g^3$ for $g\cdot g \cdot g$, and $g^{-2}$ for $g\inv \cdot g\inv$. 

\begin{definition}
    The \emph{order} of $g$ is 
    \[
        \ord(g) = \min \{ n \in \N \st g^n = e\},
    \]
    where $e \in G$ is the identity element.
    For example, $\ord(e) = 1$. 
    If $g^n \neq e$ for all $n$, then $\ord(g) = \infty$. 
\end{definition}

\bea

\item Let $K$ be an amphicheiral knot that is \emph{not slice}. Prove that, in the knot concordance group, the equivalence class $[K]$ of $K$ is an order 2 element, i.e.\ $\ord([K]) =2$. 

\item Draw a sequence of sketches (i.e.\ a movie) depicting the \emph{concordance} from the unknot to $4_1 \# 4_1$. (Hint: You get to choose which diagrams of $4_1$ to connect sum.) This shows that $[4_1]\cdot[4_1] = [U]$ in the knot concordance group.

\ee


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% category theory (Matrices)
\emph{This exercise will be graded for effort. The goal is to prime you for a more topological construction that we will discuss soon in class.}

In this exercise, you'll think about $2 \times 2$ matrices as morphisms of a simple category, and then build a bigger category out of it.


First, let's define a category  $\CC$ where 
\begin{itemize}
    \item there is one object called $\R^2 \in \Ob(\CC)$, and 
    \item the only morphism set is $\Mor_\CC(\R^2, \R^2)$, the set of linear transformations from $\R^2$ to itself.
\end{itemize}
In order to get a handle on this category, let's choose the standard basis for $\R^2$, $\{e_1, e_2\}$. Then, in terms of this basis, we can write 
\[
    \Mor_\CC(\R^2, \R^2) =
    \left \{
        \begin{bmatrix}
            a & b  \\
            c & d 
        \end{bmatrix}
        \st 
        a,b,c,d \in \R
    \right \}.
\]

We now define a new category $\Mat(\CC)$, where the objects are \emph{direct sums} of copies of objects of $\CC$, which we now explain.
% 
Recall that we decided $\R^2$ is given as the span of the vectors $e_1$ and $e_2$: we write
\[ \R^2 = \langle e_1, e_2 \rangle, \]
knowing that $e_1 \perp e_2$.
Imagine we now have two copies of $\R^2$, 
\[
    \R^2_{(1)} = \langle e_1^{(1)}, e_2^{(1)} \rangle
    \qquad \text{and}  \qquad
    \R^2_{(2)} 
    =\langle e_1^{(2)}, e_2^{(2)} \rangle.
\]
Then their direct sum is 
\[
    \R^2_{(1)} \oplus \R^2_{(2)} 
    = \langle e_1^{(1)}, e_2^{(1)}, e_1^{(2)}, e_2^{(2)}\rangle,
\]
with the basis in that particular order, and where we imagine all four basis vectors as mutually orthogonal.

Then $\R^2_{(1)} \oplus \R^2_{(2)}  \in \Ob(\Mat(\CC))$. You can then direct sum more copies of $\R^2$'s to get vector spaces of any even dimension. But in this category, we really imagine all the copies of $\R^2$ (and direct sums of them) as different objects. For example, $\R^2_{(1)}$ and $\R^2_{(2)}$ are themselves objects in $\Mat(\CC)$, and they are not the same object. 

Now let's consider morphisms. For any $2\times 2$ matrices $A$ and $B$ (i.e.\ $A,B \in \Mor_{\CC}(\R^2, \R^2)$), we can view 
$A$ as a morphism $\R^2_{(1)} \to \R^2_{(1)}$, and 
$B$ as a morphism $\R^2_{(1)} \to \R^2_{(2)}$. Then the block matrix 
\[
    \begin{bmatrix}
        A & B 
    \end{bmatrix}
\]
is a morphism $\R^2_{(1)} \oplus \R^2_{(2)} \to \R^2_{(1)}$, i.e.\
\[ 
    \begin{bmatrix}
        A & B 
    \end{bmatrix} 
    \in \Mor_{\Mat(\CC)}\left (\R^2_{(1)} \oplus \R^2_{(2)}, \R^2_{(1)} \right ).
\]

\bea

\item Read the whole background to this problem; we will later need this example to talk about a more topological category constructed in this way. Then, write down one question you have about this. 

\item Let $V = \R^2_{(1)} \oplus \R^2_{(2)} $. Describe the set $\Mor_{\Mat(\CC)}(V,V)$. What is the identity morphism?

\item Suppose $U$ is 2-dimensional, $V$ is 6-dimensional, and $W$ is 4-dimensional. 
Let $A$ be a morphism $U \to V$, and let $B$ be a morphism $V \to W$. 
Describe what $A$ and $B$ look like, using $A_1, A_2, \ldots$ and $B_1, B_2,\ldots$ to denote $2 \times 2$ matrices in $\Mor_{\CC}(\R^2, \R^2)$. 

\item Write down the composition $B\circ A \in \Mor_{\Mat(\CC)}(U, W)$ is, in terms the matrices $A_i,B_i$ you used in the previous part.

\ee







\end{document}
