Almost alternating diagrams and fibered links in S^3

TL;DR

This paper extends the characterization of fibered links via Seifert surfaces from alternating diagrams to almost alternating diagrams, showing the equivalence with Hopf plumbing, but finds this does not hold for 2-almost alternating diagrams.

Contribution

It generalizes the known criterion for fibered links from alternating to almost alternating diagrams and provides counterexamples for 2-almost alternating diagrams.

Findings

Seifert surfaces from almost alternating diagrams are fiber if and only if they are Hopf plumbing.

The characterization does not extend to 2-almost alternating diagrams.

Counterexamples are constructed using Melvin-Morton's criterion.

Abstract

Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ is obtained by successively plumbing finite number of Hopf bands to a disk. In this paper, we discuss its extension so that we show the following theorem. Let $R$ be a Seifert surface obtained by applying Seifert's algorithm to an almost alternating diagrams. Then $R$ is a fiber surface if and only if $R$ is a Hopf plumbing. We also show that the above theorem can not be extended to 2-almost alternating diagrams, that is, we give examples of 2-almost alternating diagrams for knots whose Seifert surface obtained by Seifert's algorithm are fiber surfaces that are not Hopf plumbing. This is shown by using a criterion of Melvin-Morton.