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

\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{xcolor}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{hyperref}

% 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


% 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 HW02}
\author{(ADD NAME)}
\date{Due 4/14/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle


{\footnotesize
\paragraph{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}
}

\paragraph{How will this be graded?}  You will be graded for accuracy, but also effort. As such, you should attempt every problem, and even if you don't have the full solution, you should write down your thoughts (coherently, please). You are highly encouraged to collaborate with classmates! You must submit your own solution in your own words, though. 

If you get stuck for a long time on a problem, talk to people! Discuss with classmates, go to office hours, etc. While it's an important skill to be able to keep at a problem, your time is also valuable; you know what's a reasonable amount of time for yourself to spend on a problem.





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

(Ex.~1.2 in the book)
Recall that the \emph{crossing number} of a knot $K$ is the minimal number of crossings needed in a diagram representing $K$. If the crossing number of $K$ is $c$, then we say $K$ is a \emph{$c$-crossing knot}. 

Prove that there are no two-crossing nontrivial knots.

\solution

% ADD YOUR SOLUTION HERE.


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

This is a multi-part exercise about the writhe of diagrams.

\bea

\item Compute the writhe of the knot diagram shown in Figure \ref{fig:10_36}.
\begin{figure}
    \centering
    \includegraphics{180-HWs/images/10_36.png}
    \caption{A diagram of the knot $10_{36}$, taken from Knot Atlas.}
    \label{fig:10_36}
\end{figure}

\item Describe how the writhe of a diagram changes under all the Reidemeister moves. (Note that there are two different Reidemeister 1 and 3 moves.)

\ee

\solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
(1.17 in the book) Compute the absolute values of the linking numbers of the two links shown in Figure 1.39 of the book in order to show that they must be distinct links. 

\solution


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
(1.25, 1.27 in the book) 
\bea
\item Show that the composition of any knot with a tricolorable knot yields a new tricolorable knot.
\item Prove that the figure-eight knot ($4_1$ in the Rolfsen knot table) is \emph{not} tricolorable. Conclude that the figure-eight knot and the trefoil knot are distinct knots. 
\ee

\solution


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

(1.6, 1.7 in the book)

\bea
\item Show that by changing the crossings from over to under or vice versa, any projection of a knot can be made into the projection of an alternating knot. 
(This isn't as easy as it might seem. How do you know your procedure will always work?) 
\item In a projection with $n$ crossings, what is the maximum number of crossings that would have to be changed in order to make the knot alternating?
\item Show that by changing some of the crossings from over to under or vice versa, any projection of a knot can be made into a projection of the unknot.
\ee

\solution

% ADD YOUR SOLUTION HERE.






\end{document}
