\documentclass[11pt, oneside]{article}   	% use "amsart" instead of "article" for AMSLaTeX format
\usepackage{geometry}                		% See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   		% ... or a4paper or a5paper or ... 
%\geometry{landscape}                		% Activate for rotated page geometry
%\usepackage[parfill]{parskip}    		% Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}				% Use pdf, png, jpg, or eps§ with pdflatex; use eps in DVI mode
								% TeX will automatically convert eps --> pdf in pdflatex		
\usepackage{amssymb}

%SetFonts

%SetFonts


\title{Brief Article}
\author{The Author}
%\date{}							% Activate to display a given date or no date

\begin{document}
{\bf Math 177 HW 9 Due May 7}
\bigskip

Oriented knots can be depicted with {\it oriented knot diagrams} as shown in Figure 1. Here five knots are depicted. Their names are $0_1 \mbox{(unknot)}, 3_1 \mbox{(trefoil)}, 4_1 \mbox{(figure eight)}, 5_1 \mbox{(pentafoil)}$, and $5_2 \mbox{(twist knot)}$. Associated to every knot is the fundamental group of its complement. If two knots are really the same knot, then their associated groups will be isomorphic. The goal of this exercise set is to study the group of a knot and to use these groups to show that these five knots are all different.

Given a diagram of an oriented knot, we can write down a presentation of its group, called the {\it Wirtinger presentation}, as follows. Place one generator on each {\it arc} of the diagram. These are called the {\it Wirtinger generators}. (This has already been done in Figure 1.) Next, write down one relation for each crossing as shown in Figure 2. The orientation of the overcrossing strand is need to write down the relation! (But the orientation of the undercrossing strand is not needed.) The group of the unknot is thus $\langle x\,|\hspace{.1 in}\rangle$ which is $\mathbb Z$, the free group on one generator, also known as the infinite cyclic group. This group is Abelian.
\begin{enumerate}

\setcounter{enumi}{57}
\item\label{presentations} Write down the Wirtinger presentation for the knot group for each of the knots in Figure 1. Using Tietze transformations, change the presentation of each of $3_1, 4_1, 5_1$ and $5_2$ to one having two generators and one relation.

\item Start with the Wirtinger presentation for {\it any} knot and show that if we Abelianize the group by adding the relations that say that every generator commutes with every other generator, then we always get $\mathbb Z$, the free group on one generator. Explain why this means that every knot group is infinite.

\item If $G$ is a group and $x\in G$, we say that any element of the form $g^{-1}xg$, where $g$ is any element of $G$,  is a {\it conjugate} of $x$. We say that $x, y \in G$ are {\it conjugate} if there exists $g\in G$ such that $y=g^{-1}xg$. 
\begin{enumerate}
\item Show that conjugacy is an equivalence relation on the elements of $G$.
\item Given any knot diagram, show that all the Wirtinger generators are in the same conjugacy class (that is, they are all conjugate to each other).
\item Let $g \in G$ be fixed. Define $\Phi_g:G\to G$ by $\Phi_g(x)=g^{-1}xg$. Show that $\Phi$ is an isomorphism. (An isomorphism from a group to itself is called an {\it automorphism}.)
%\item Find all the conjugacy classes in $S_3=\{(1), (1\,2),(1\,3),(2\,3),(1\,2\,3),(1\,3\,2)\}$.
\item Let $p$ be an odd prime. Find all the conjugacy classes in the dihedral group
$$D_p=\langle f, r\,|\, f^2=1, r^p=1, frfr=1\rangle.$$
\end{enumerate}

\item Suppose that $\sigma:G \to H$ is a group homomorphism. 
\begin{enumerate}
\item Show that if $x$ and $y$ are conjugate elements in $G$, then $\sigma(x)$ and $\sigma(y)$ are conjugate elements in $H$.
\item Show that if $\sigma$ is an epimorphism and $G$ is Abelian, then $H$ is Abelian. 
\item Let $X$ be a set of generators for $G$ and let $Y=\{\sigma(x)\,|\,x\in X\}\subset H$. Show that $\sigma$ is an epimorphism if and only if $Y$ generates $H$.
\end{enumerate}

\item\label{map onto dihedral} The goal of this exercise is to study when a knot group has an epimorphism onto a dihedral group.
\begin{enumerate}
\item Suppose that $G$ is the group of a knot with generating set $X$ and furthermore that all the elements of $X$ are in the same conjugacy class. Suppose there is an epimorphism from $G$ onto $D_p$ for some odd prime. Use the previous exercises to conclude that all the generators of $G$ must be sent into the same conjugacy class and determine what conjugacy class that is.
\item Let $G$ be the group of the trefoil $3_1$ and consider the presentation with two generators and one relation that was found in Exercise~\ref{presentations}. We want to find all possible odd primes $p$ such that there exists an epimorphism from $G$ onto $D_p$. Explain why the previous exercises imply that the two generators of the knot group must be sent to reflections, that is, to elements of the form $fr^k$ for $0\le k<p$. Thus there are potentially $p^2$ choices for the images of the two generators. A given choice will give a homomorphism if and only if the single relator in the knot group is sent to the identity. Hence, for a given $p$, we have a finite number of things to check to see if there is an epimorphism onto $D_p$. However, we can reduce the amount of checking that must be done!
\begin{enumerate}
\item Suppose there is an epimorphism $\sigma:G \to D_p$. Let $g$ be any element in $D_p$. Show that $\Phi_g\circ \sigma:G\to D_p$ is an epimorphism.
\item Show that if there is an epimorphism $\sigma:G \to D_p$, then  there is an epimorphism $\tau:G \to D_p$  such that $\tau$ takes one of the two generators of $G$ to the reflection $f$. (Hint: We will have $\tau=\Phi_g\circ \sigma$ for some $g$.)
\item Thus to see if there is an epimorphism from $G$ to $D_p$, we may assume that one generator is taken to $f$ and the other to $fr^k$ for some $k$. Show that the only odd prime $p$ for which there is an epimorphism of $G$ onto $D_p$ is $p=3$.
\end{enumerate}
\item Repeat the above work for the knot $4_1$ in order to find all odd primes $p$ for which the knot group maps onto $D_p$.
 \item Repeat the above work for the knot $5_1$ in order to find all odd primes $p$ for which the knot group maps onto $D_p$.
\item Repeat the above work for the knot $5_2$ in order to find all odd primes $p$ for which the knot group maps onto $D_p$.

\end{enumerate}

\item Do these exercises show that the knots $0_1, 3_1, 4_1, 5_1$, and $5_2$ all have different groups and hence are all different knots?

\end{enumerate}
\begin{center}
\includegraphics[scale=.7]{HW9figure.pdf}
\end{center}
\end{document}  