A classification of spanning surfaces for alternating links

Abstract

A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus, orientability, and aggregate slope that a surface spanning that link can have. To this end, we describe a straightforward algorithm, much like Seifert’s algorithm, through which to construct certain spanning surfaces called state surfaces, obtained by splitting each crossing one of the two ways, filling in the resulting circles with disks and connecting these disks with half twisted bands at the crossings. A particularly important subset of these will be what we call basic state surfaces. We can alter these surfaces by performing the entirely local operations of adding handles and/or crosscaps, each of which increases genus.

The main result then shows that if we are given an alternating projection $P\left(L\right)$ and a surface $S$ spanning $L$, we can construct a surface $T$ spanning $L$ with the same genus, orientability, and aggregate slope as $S$ that is a basic state surface with respect to $P$, except perhaps at a collection of added crosscaps and/or handles. Furthermore, $S$ must be connected if $L$ is nonsplittable.

This result has several useful corollaries. In particular, it allows for the determination of nonorientable genus for alternating links. It also can be used to show that mutancy of alternating links preserves nonorientable genus. And it allows one to prove that there are knots that have a pair of minimal nonorientable genus spanning surfaces, one boundary-incompressible and one boundary-compressible.