Constructing doublypointed Heegaard diagrams compatible with (1,1) knots
Abstract
A (1,1) knot K in a 3manifold 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 fourtuple of integers, which they call Schubert's normal form. This article presents an algorithm for constructing a doublypointed Heegaard diagram compatible with K, given a Schubert's normal form for K. The construction, coupled with results of Ozsváth and Szabó, 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 DOI:
 10.48550/arXiv.1110.5675
 arXiv:
 arXiv:1110.5675
 Bibcode:
 2011arXiv1110.5675O
 Keywords:

 Mathematics  Geometric Topology;
 57M25;
 57R58
 EPrint:
 23 pages, 14 figures