Homework Assignment 7

Question 1. By using van Kampen's theorem a number of times, prove that for any $n$-crossing knot diagram $K$ the knot group $\pi_1(K)$ has a presentation with $n$ generators and $n$ relations, where the generators correspond to the arcs of $K$ and the relations correspond to the crossings of $K$, as shown in class. (Recall that an "arc" is an uninterrupted segment in $K$).

Question 2. In class we got two presentations for the group of the trefoil knot $T_{3,2}$: $\langle x,y\rangle/(x^3=y^2)$ and $\langle a,b,c\rangle/(a^b=c,b^c=a,c^a=b)$. Find an explicit isomorphism $\phi\colon\langle x,y\rangle/(x^3=y^2)\to\langle a,b,c\rangle/(a^b=c,b^c=a,c^a=b)$ between these two presentations. (Hint. Don't search in the dark! $x,y,a,b,c$ all had a geometric meaning, and it should be possible to figure out in a geometric way how to write $x,y$ in terms of $a,b,c$.)

This assignment is due on Crowdmark by the end of Wednesday November 18, 2020.