Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial

TL;DR

This paper provides a geometric proof that links the Alexander polynomial's leading coefficient to the uniqueness of minimal genus Seifert surfaces, extending the result to homogeneous links and exploring broader implications.

Contribution

It generalizes Juhasz's result from alternating knots to homogeneous links, connecting Alexander polynomial coefficients with Seifert surface properties.

Findings

Unique minimal genus Seifert surface for certain homogeneous links

Extension of results from alternating to homogeneous links

Implications for the topology of alternating links

Abstract

We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompressible' and `alternating' with `homogeneous'. We also examine the implications of our proof for alternating links in general.