Suppose that $K$ is an orientable knot, $X=\mathbb{R}^3\setminus K$, $x_0\in X$ and $G=\pi_1(X,x_0)$. Suppose $\phi:G\rightarrow G_{ab}=G/G'\cong\mathbb{Z}$.
Use the Wirtinger presentation of $G$ to prove that $\phi([L])=lk(L,K)$ for each loop in $X$ based in $x_0$.
Can someone help me with this question?! I have no idea how to begin -.-
Thanks :)
The Wirtinger presentation gives you one generator for each arc and a relation for each crossing in the diagram. When you abelianize, notice that the relations turn into the statement that the generators corresponding to two neighboring arcs are actually the same! So for a knot, the abelianization just has one generator, which is given by going under any arc of the knot in the positive direction. Therefore, when you calculate $\phi([L])$, you just need to count how many times it passes under $K$, with sign. This is one definition of linking number. A more common definition of linking number also counts the places where $L$ passes over $K$ with sign, and then multiplies by a $1/2$. It is an interesting exercise to show these two definitions of linking number are equivalent.