A Seifert algorithm for integral homology spheres

TL;DR

This paper extends Seifert's algorithm from classical knot theory in $S^3$ to integral homology spheres, providing a method to construct Seifert surfaces using handle decompositions and planar diagrams.

Contribution

It introduces a natural generalization of Seifert's algorithm applicable to any closed integral homology 3-sphere, utilizing handle structures and planar diagrams.

Findings

Provides a constructive method for Seifert surfaces in homology spheres.

Uses handle decompositions to represent the handle structure as a planar diagram.

Generalizes classical Seifert algorithm beyond $S^3$.

Abstract

From classical knot theory we know that every knot in $S^3$ is the boundary of an oriented, embedded surface. A standard demonstration of this fact achieved by elementary technique comes from taking a regular projection of any knot and employing Seifert's constructive algorithm. In this note we give a natural generalization of Seifert's algorithm to any closed integral homology 3-sphere. The starting point of our algorithm is presenting the handle structure of a Heegaard splitting of a given integral homology sphere as a planar diagram on the boundary of a $3$-ball. (For a well known example of such a planar presentation, see the Poincar\'e homology sphere planar presentation in {\em Knots and Links} by D. Rolfsen \cite{Rolfsen}.) An oriented link can then be represented by the regular projection of an oriented $k$-strand tangle. From there we give a natural way to find a ``Seifert circle" and associated half-twisted bands.