# Tying a portion of a knot diagram

I am having some trouble with a paper by Livingston (Infinite Order Amphicheiral Knots, Algebraic and Geometric Topology 1, 2001, 231-241). He starts with a knot $$K$$, and constructs a new knot "replacing the neighborhood of an unknotted circle $$L$$ in the complement of a Seifert surface for $$K$$ with the complement of a knot $$J$$. The identification map switches longitude and meridian so that the resulting manifold is still $$S^3$$. The effect of this construction is to tie that portion of $$K$$ that passes through $$L$$ into a knot".

I see what happens to $$K$$, but I cannot understand how the construction goes. For example, I think that the neighborhood of $$L$$ is a tubular neighborhood, but then why replacing it affects the original knot $$K$$?

In the background of this construction is JSJ theory, which deals with decomposing $$3$$-manifolds along embedded tori. Let's analyze what happens with tori in knot complements so we understand the particular synthesis.

In $$S^3$$, a key fact about embedded tori is that one side is a solid torus and the other is a knot exterior (possibly also a solid torus if it's an unknot exterior), which is a nice consequence of the loop theorem. Suppose $$T$$ is an embedded torus in a knot exterior $$S^3-\nu(K)$$, where $$\nu(K)$$ is a tubular neighborhood of a knot $$K$$. By the above consideration, $$K$$ is either (1) on the solid torus side or (2) on the knot exterior side of $$T$$. In case (2), $$T$$ bounds a solid torus disjoint from $$K$$, so it is simply the boundary of a regular neighborhood of a knot in the exterior of $$K$$; we won't say anything more about this case.

In case (1), then the solid torus has a loop $$L\subset T$$ that bounds a disk in it. The solid torus also has a curve $$M\subset T$$ so that when gluing in another solid torus with its meridian glued to $$M$$ and its longitude glued to $$L$$, one obtains an $$S^3$$. Let's refer to this $$S^3$$ by $$\Sigma^3$$ since we want to consider $$K$$ both in the original $$S^3$$ and in this $$\Sigma^3$$. In $$\Sigma^3$$, $$L$$ is an unknot in the complement of $$K$$. It is best to think of $$L$$ as being core of the solid torus that was glued in, and we can think of that solid torus as being a tubular neighborhood $$\nu(L)$$ of $$L$$. Hence, $$\Sigma^3-\nu(L)$$ is the solid torus on the $$K$$ side of $$T$$.

Going in reverse, if we take $$\Sigma^3-\nu(L)$$ and glue in the knot complement that was on the other side of $$T$$ from $$K$$, we return to $$K\subset S^3$$. In the following, the left-hand side is $$K$$ and $$L$$ in $$\Sigma^3$$, where $$T$$ is the boundary of the closure of $$\nu(L)$$, and the right-hand side is $$K$$ where the $$L$$ side of $$T$$ has been replaced by a knot exterior of a knot $$J$$. The longitude of this knot exterior corresponds to a meridian loop of $$L$$ (i.e., $$M$$ from before), and the meridian of this knot exterior corresponds to $$L$$ back when we thought of it as being on $$T$$ itself -- or, we may think of the meridian as being $$L$$ pushed slightly into the disk it bounds in the solid torus on the $$K$$ side of $$T$$. This is essentially the definition of a satellite knot, with $$J$$ being the companion and $$(K,L)$$ being the pattern.

Notice we made no use of the fact that $$L$$ is disjoint from a Seifert surface for $$K$$ in $$\Sigma^3$$. This is a separate condition, and it is equivalent to saying that $$L$$ has linking number $$0$$ with $$K$$ in $$\Sigma^3$$, or that $$K$$ is nullhomotopic in the complement of $$L$$.

Roughly speaking, $$K$$ in $$S^3$$ has a Seifert surface that is "knotted up" by $$J$$. $$K$$ always has a Seifert surface where only thin bands pass through $$L$$. Then this "bundle" of bands is knotted up by $$J$$.

One example of this construction is a Whitehead double. It's illuminating to think about a Whitehead double's genus-$$1$$ Seifert surface and how it relates to all of this.

• Thank you so much for your help. My knowledge of 3-manifolds is still very low, but I am working on it! Thanks again. May 5, 2020 at 21:02