The Kakimizu complex of a connected sum of links

TL;DR

This paper establishes a multiplicative relation for the Kakimizu complex of a connected sum of links, extending understanding of Seifert surfaces in knot theory.

Contribution

It proves that the Kakimizu complex of a connected sum of non-split, non-fibred links is the product of the individual complexes times the real line, including for incompressible surfaces.

Findings

Kakimizu complex of connected sum equals product of individual complexes times ℝ

Result applies to taut and incompressible Seifert surfaces

Advances understanding of Seifert surfaces in link theory

Abstract

We show that $|MS(L_1 # L_2)|=|MS(L_1)|\times|MS(L_2)|\times\mathbb{R}$ when $L_1$ and $L_2$ are any non-split and non-fibred links. Here $MS(L)$ denotes the Kakimizu complex of a link $L$, which records the taut Seifert surfaces for $L$. We also show that the analogous result holds if we study incompressible Seifert surfaces instead of taut ones.