Longitude of knot

Question

I am trying to understand the peripheral system of knot. In that direction, I don't see why the description of longitude of knot in Remark 3.13 does actually lie second commutator as shown in proposition 3.12?

Answer 1

Let me first just explain why it is that this word $\ell$ corresponds to the longitude. A longitude is any peripheral curve (i.e, a curve lying on the torus boundary of the knot exterior) that is the boundary of a Seifert surface. What characterizes these curves is that they are parallel to the knot and have linking number $0$ with it.

In the following, I drew a blue curve that is the longitude. Every time the curve goes under an arc of the knot, I labeled the event with which Wirtinger presentation generator this corresponds to. Collecting these generators, you get the word $\ell$.

The linking number of a Wirtinger presentation word with a knot is the sum of the exponents. The sum here is $0$, so the linking number is $0$.

A way to explain why longitudes are in the second commutator subgroup is thinking in terms of homology (this is some of the essence of 3.12). Let $K$ be the knot and $G=\pi_1(S^3-K)$ its knot group. The first homology group $H_1(S^3-K)$ is isomorphic to the abelianization of $G$, hence it is isomorphic to $G/G^{(1)}$ where $G^{(1)}$ is the first commutator subgroup. One can check that this is isomorphic to $\mathbb{Z}$ (for instance, via the Wirtinger presentation). The function of a Seifert surface $S$ of $K$ is that given an element of $G$ represented as a smooth curve in $S^3-K$, its value in $G/G^{(1)}$ can be evaluated through calculating the algebraic intersection number of the curve with $S$.

Given an element of $G$ whose curve can be put purely on $S$, by pushing it off the surface in one direction, we see it intersects with $S$ zero times. Hence, such elements lie in $G^{(1)}$. Similarly, the longitude is a word that lies on $S$, so it lies in $G^{(1)}$. However, something special happens. Since the boundary of $S$ is a word that is a composition of commutators of the canonical system of curves of $S$, it lies in the second commutator subgroup $G^{(2)}$.

This is saying that the longitude is not only zero in $H_1(S^3-K)$, but zero in $H_1(\overline{S^3-K})$, where $\overline{S^3-K}$ denotes the covering space associated to $G^{(1)}$ (the "infinite cyclic cover"). $H_1(\overline{S^3-K})$ is isomorphic to the abelianization of the commutator subgroup, i.e., $G^{(1)}/G^{(2)}$.