A classification of spanning surfaces for alternating links

TL;DR

This paper classifies spanning surfaces for alternating links based on genus, orientability, and a new invariant called aggregate slope, providing an algorithm to construct and analyze these surfaces.

Contribution

It introduces a classification framework and an algorithm for constructing spanning surfaces, including layered and basic layered surfaces, for alternating links.

Findings

Determines all possible combinations of genus, orientability, and aggregate slope for spanning surfaces.

Provides an algorithm to construct surfaces with specified properties from alternating link projections.

Shows that mutancy preserves nonorientable genus and identifies knots with multiple minimal nonorientable genus surfaces.

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 layered surfaces. A particularly important subset of these will be what we call basic layered surfaces. We can alter these surface 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(L) 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 layered surface with respect to P, except perhaps at a collection of added crosscaps and/or handles. Furthermore, S must be connected if L is non-splittable. 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.