Constructing doubly-pointed Heegaard diagrams compatible with (1,1) knots
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 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 e-prints
- Pub Date:
- October 2011
- DOI:
- 10.48550/arXiv.1110.5675
- arXiv:
- arXiv:1110.5675
- Bibcode:
- 2011arXiv1110.5675O
- Keywords:
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- Mathematics - Geometric Topology;
- 57M25;
- 57R58
- E-Print:
- 23 pages, 14 figures