\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{\sgn}{\mathrm{sgn}} % sign of permutation

% \newcommand{\bpmat}{\begin{pmatrix}}
% \newcommand{\epmat}{\end{pmatrix}}


% 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{MAT 150A HW01}
\author{[ADD YOUR NAME HERE]}
\date{Due Tuesday, 10/10/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

\begin{reminder} 
Your homework submission must be typed up in full sentences, with proper mathematical formatting. The following resources may be useful as you learn to use TeX and Overleaf:
\begin{itemize}
    \item Overleaf's introduction to LaTeX: \\ \url{https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes}
    \item Detexify: \\ \url{https://detexify.kirelabs.org/classify.html}
\end{itemize}
\end{reminder}

\paragraph{Covered in this HW}
Parts of Chp.\ 1, esp.\ \S1.5, 2.1--2.4. Matrices, rotations, definition of a group, symmetric groups and permutations, etc.


\paragraph{Grading} Some of the (parts of) problems will be graded in detail out of several points, and necessary feedback will be given. The rest will be graded out of 2 points. I will reveal which problems are fully graded in the solutions, which will be posted on the Friday following the due date.

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

In Lecture 1, we described a group $G$ generated by rotations of $\theta = \frac{\pi}{4}$ about the $x$-, $y$-, and $z$- axes. To be more precise, we set a convention for the coordinate axes 

\begin{center}
\begin{tikzpicture}
    \node (x) at (1,0) {$x$};
    \node (y) at (0,1) {$y$};
    \node (z) at (-.3,-.5) {$z$};
    \draw[->] (0,0) to (x);
    \draw[->] (0,0) to (y);
    \draw[->, thick] (0,0) to (z);
    \node (zlabel) at (0,-1) {($e_3$ points out of the board)};
\end{tikzpicture}
\end{center}
%
and wrote down the associated matrices for our chosen generators:
%
\begin{center}
$p = 
\begin{pmatrix} 
    1 & 0 & 0 \\
    0 & \cos \theta & \sin \theta  \\
    0 & -\sin \theta & \cos \theta
\end{pmatrix}$
\hspace{.3in}
$y = 
\begin{pmatrix} 
    \cos \theta  & 0 & -\sin \theta \\
    0 & 1 &  0 \\
    \sin \theta & 0 & \cos \theta
\end{pmatrix}$
\hspace{.3in}
$r = 
\begin{pmatrix} 
    \cos \theta & -\sin \theta  & 0 \\
    \sin \theta  & \cos \theta  & 0 \\
    0 & 0 & 1
\end{pmatrix}
$
\end{center}
Since $\theta = \frac{\pi}{4}$, $\cos \theta = \sin \theta = \frac{1}{\sqrt{2}} := t$, so we can rewrite our matrices as follows:

\begin{center}
$p = 
\begin{pmatrix} 
    1 & 0 & 0 \\
    0 & t & t  \\
    0 & -t & t
\end{pmatrix}$
\hspace{.3in}
$y = 
\begin{pmatrix} 
    t & 0 & -t \\
    0 & 1 &  0 \\
    t & 0 & t
\end{pmatrix}$
\hspace{.3in}
$r = 
\begin{pmatrix} 
    t & -t & 0 \\
    t  & t  & 0 \\
    0 & 0 & 1
\end{pmatrix}
$
\end{center}

We used spatial reasoning to try to find a sequence (composition) of the actions (elements of $G$) $p, p\inv, y, y\inv$ that is equal to $r$; this would prove that only $G$ is generated by $\{p,y\}$, and in particular, $r$ is not needed as a generator. 
Some guesses included $p^2y\inv p^{-2}$ and $p y\inv p\inv$. 

\textit{Typesetting tip: To typeset matrices, copy and paste my code for typesetting matrices above, then modify the entries.}

\bea
    \item Find matrices representing $p\inv$ and $y\inv$; prove that these are indeed inverses to $p$ and $y$, respectively.
    \item Use matrix multiplication and trigonometric identities to compute $p^2y\inv p^{-2}$ and $p y\inv p\inv$. Which of these, if any, is equal to $r$?
\ee

\solution

% ADD YOUR SOLUTION HERE.


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

In this exercise, you will explore the symmetric group on 4 indices, $S_4$. 
%
First, read page 42 of the text to see how $S_3$ is described. Lecture 2 and the accompanying note may also be useful.

\paragraph{Warning} The text defines a \emph{transposition} to be any 2-cycle $(i~j)$. In class, we restricted this definition to 2-cycles involving only adjacent indices, i.e.\ $\tau_i = (i~~~ i+1)$. 


\bea

\item Write the permutations
\[
    p = (1~2~3~4)
    \qquad
    q = (1~3~2~4)
    \qquad
    r = (1~4~2)
\]
as products of (adjacent) transpositions $\tau_i$. 

\item In class, we discussed why the set of transpositions $\{\tau_1,\tau_2,\tau_3\}$ generate $S_4$ \emph{intuitively}, but we did not prove it:
\begin{proposition}[A]
The symmetric group $S_n$ is generated by the set of (adjacent) transpositions $\{\tau_1, \tau_2, \ldots, \tau_{n-1}\}$. 
\end{proposition}

Assuming Proposition A is true, prove that $S_4$ is generated by $\{(1~2), (1~2~3~4)\}$. 

\textit{Hint: Can you show that $\tau_2$ and $\tau_3$ are generated by these two elements?}

\ee


\solution





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

This exercise focuses on permutation matrices and determinants. For a refresher on determinants, see \S1.4 in the text. In particular, we will need the following facts, labeled here as Lemmas A and B:

\begin{lemma}[A]
The determinant of the $n\times n$ identity matrix $I_n \in M_{n\times n}(\R)$ is 1.
\end{lemma}

\begin{lemma}[B]
If $M'$ is obtained from $M$ by interchanging two different rows, then $\det A' = - \det A$. 
\end{lemma}

We will also need the following definition:

\begin{definition}[Sign of a permutation] 
Let $p \in S_n$ be a permutation. The \textbf{sign} of $p$ is equal to the determinant of the permutation matrix $P$ representing $p$:
\[
    \sgn(p) := \det(P).
\]
\end{definition}


\bea

\item Prove that the transpose of a permutation matrix is its inverse.


\item Prove that the determinant of a permutation matrix is always $\pm 1$. 

\textit{Therefore the sign of a permutation is always either $+1$ or $-1$. If $\sgn(p) = +1$, we say that $p$ is \textbf{even}; otherwise, if $\sgn(p) = -1$, we say that $p$ is \textbf{odd}.}



\item Let $p\in S_n$, and write $p$ as a composition (or equivalently, product) of $k$ transpositions:
\[
    p = \tau_{i_1} \circ \tau_{i_2} \circ \ldots \circ \tau_{i_k}
\]
Prove that $p$ is even if and only if $k$ is even, and that $p$ is odd if and only if $k$ is odd.

\textit{In other words, we could define $\sgn(p)$ to be $(-1)^k$, where $k$ is the number of transpositions in any composition of transpositions equal to $p$.}
\ee


\solution





\end{document}
