What is $X_\infty$ for a topological space $X$?

Question

Lickorish gives in his 'An Introduction to Knot Theory' a discription about how to construct a space $X$-cut-along-$F$ and uses this to construct a space called $X_{\infty}$. This you can see [here]. In this situation $F$ is a Seifert surface for an oriented link $L$, i.e. $\partial F=L$ and $X$ is the closure of the complement of a tubular neighbourhood for $L$ in $\mathbb{S}^3$.

Question 1. Can someone explain by using an easy example, e.g. the unknot, what $X_{\infty}$ actually is?

We have $L=\mathbb{S}^1$ and $F=\mathbb{D}^2$, so $X$ is the complement of a torus. The result should be $X_{\infty}=\mathbb{D}^2\times\mathbb{R}$, but I don't arrive there.

Question 2. Besides I do not understand Lickorish's drawing, the first one. What is $\mu$ and how do I get $Y$ out of this?

Answer 1

Question 1. For the unknot, the Seifert surface is the disk, as you say. I find it easier to imagine the space $Y$ by rotating $S^3$ so that infinity occurs within the disk; the space is homeomorphic to a cylinder with the top and bottom faces being copies of the Seifert surface and the side being the boundary of the torus. Then, $X_\infty$ is just countably many cylinders $\mathbb{D}^2\times [0,1]$ stacked upon each other, which gives $\mathbb{D}^2\times\mathbb{R}$.

Question 2. $\mu$ is the meridian of the torus, cut where it crosses the Seifert surface. The diagram is showing what is happening locally at the torus (since it is much too difficult to really see $X_\infty$ with all its twistiness). Right near the meridian, $Y$ is the result of unfurling the space so that it is locally a wall of the cylinder in the previous question.

The way I imagine this construction is that there is a space $X_\infty$, which for all intents and purposes feels exactly like $X$ when you are inside it, but while you wander around, the space keeps track of the net number of times you've crossed the Seifert surface ($+1$ for going back-to-front, $-1$ for going front-to-back). If you put a jelly bean down and go around the knot, you'll find the jelly bean missing, but if you go back around the other way, the jelly bean will reappear.

If you are familiar with covering spaces, all Lickorish is doing is constructing a covering space with fiber $\mathbb{Z}$ using a Seifert surface. In particular, it is the connected covering space corresponding to the commutator subgroup of $\pi_1(X)$. It's the one corresponding to the abelianization map $\pi_1(X)\to H_1(X)$, with $\mu$ being a loop whose lift goes from sheet to sheet.