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

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

2011-10-25

arxiv.org/abs/1110.5675v1

Mathematics

Geometric Topology

23 pages, 14 figures

Scientific paper

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.

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