Constructing doubly-pointed Heegaard diagrams compatible with (1,1) knots
Geometric Topology
2011-10-27 v1
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.
Keywords
Cite
@article{arxiv.1110.5675,
title = {Constructing doubly-pointed Heegaard diagrams compatible with (1,1) knots},
author = {Philip Ording},
journal= {arXiv preprint arXiv:1110.5675},
year = {2011}
}
Comments
23 pages, 14 figures