Creating Seifert surface for a given Seifert matrix

Question

I am considering the basic case: $2x2$ Seifert matrix $S$ and trying to create 2 circle $l$ and $k$, such that their linking numbers satisfy $lk(l,l^+) = S_{11}, lk(l,k^+) = S_{12}, ...$ $lk(l,k^+) = S_{12}$ is not difficult, but how about the others? Could you please give me some hints? Thank you.

Answer 1

This is not exactly an algorithm you would want to do, but it might be able to be turned into one (and at least it is computable!). You don't say this, but I will assume that the link is given.

Given a Seifert surface $\Sigma$, embedded arc surgery (as seen in Lickorish's book) is the result of taking a tame arc $f:I\to S^3$ that intersects the surface transversely in the interior at exactly the two endpoints of $f$, taking a regular neighborhood $N\cong I\times D^2$ of $f$ such that $\partial I\times D^2\subset \Sigma$, then constructing the surface $\Sigma'=(\Sigma-\partial I\times D^2)\cup I\times \partial D^2$.

Define surgery equivalence of Seifert surfaces as the equivalence relation generated by saying that two ambient isotopic Seifert surfaces are surgery equivalent, and that two Seifert surfaces related by embedded arc surgery are surgery equivalent. Theorem 8.2 of Lickorish says that this is a trivial relation: every pair of Seifert surfaces for a given link are surgery equivalent.

This leads to Theorem 8.4, which is that there is a corresponding relation on Seifert matrices, called S-equivalence. The relation is the equivalence relation generated by the following: $A$ and $B$ are equivalent if there is a unimodular matrix $P$ (i.e., $\det P=1$ and $P$ has integral entries) such that $B=P^tAP$ or if there is some column vector $\xi$ such that $$B=\begin{pmatrix}A&\xi&0\\0&0&1\\0&0&0\end{pmatrix}\text{ or }B=\begin{pmatrix}A&0&0\\\xi^t&0&0\\0&1&0\end{pmatrix}$$ which is called an elementary enlargement. The first corresponds to a basis change for $H_1(\Sigma)$, and the second corresponds to embedded arc surgery.

So: the problem is a linear algebra problem. Find a sequence of basis changes and elementary enlargements (or their opposites) that carries a Seifert matrix for an arbitrary Seifert surface (created through, say, Seifert's algorithm) to the given Seifert matrix. Then, perform the corresponding embedded arc surgeries or their inverses, and the result will be a Seifert surface with the given Seifert matrix.

One detail I'm not mentioning is how to use $\xi$ to construct the arc for the surgery. It might be that it is easier to find an arc where the resulting matrix has the form $$B=\begin{pmatrix}A&\xi&0\\{*}&{*}&1\\0&0&0\end{pmatrix}$$ or its transpose.

Something I am not sure of myself is whether two Seifert surfaces with the same Seifert matrices are ambient isotopic. I have no reason to believe this is the case, but I also have no counterexample on hand.