An algorithm for Seifert surfaces in 3-manifolds via surgery presentations

TL;DR

This paper extends the classical Seifert algorithm to construct Seifert surfaces for null-homologous links in any 3-manifold using surgery presentations, broadening its applicability beyond $S^3$ and integral homology spheres.

Contribution

It introduces a new algorithm for Seifert surfaces in arbitrary 3-manifolds via surgery, generalizing previous methods for $S^3$ and homology spheres.

Findings

Provides an explicit construction method for Seifert surfaces in general 3-manifolds.

Extends classical algorithms to a broader class of 3-manifolds.

Facilitates new topological analyses of links in complex 3-manifolds.

Abstract

The classical Seifert algorithm provides an explicit construction of a Seifert surface for any link in $S^3$. Alegria and Menasco extended this construction to integral homology $3$-spheres using Heegaard splittings. In this paper, we extend the Seifert algorithm to null-homologous links in arbitrary $3$-manifolds via surgery on framed links in $S^3$.