Constructing doubly-pointed Heegaard diagrams compatible with (1,1) knots

TL;DR

This paper introduces an algorithm to construct doubly-pointed Heegaard diagrams for (1,1) knots using Schubert's normal form, enabling practical computation of knot Floer homology groups.

Contribution

It provides a novel algorithm for constructing compatible Heegaard diagrams for (1,1) knots based on Schubert's normal form, facilitating knot Floer homology calculations.

Findings

Algorithm successfully constructs Heegaard diagrams for (1,1) knots.

Enables computation of knot Floer homology groups.

Uses train tracks inspired by previous work of Goda, Matsuda, and Morifuji.

Abstract

A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parameterization of this family of knots by a four-tuple of integers, which they call Schubert's normal form. This article presents an algorithm for constructing a doubly-pointed Heegaard diagram compatible with K, given a Schubert's normal form for K. The construction, coupled with results of Ozsv\'ath and Szab\'o, provides a practical way to compute knot Floer homology groups for (1,1) knots. The construction uses train tracks, and its method is inspired by the work of Goda, Matsuda and Morifuji.