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

@article{Ording2011ConstructingDH, title={Constructing doubly-pointed Heegaard diagrams compatible with (1,1) knots}, author={Philip Ording}, journal={arXiv: Geometric Topology}, year={2011} }

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…

## Figures and Tables from this paper

## References

SHOWING 1-10 OF 17 REFERENCES

Parameterizations of 1-Bridge Torus Knots

- Mathematics
- 2001

A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are…

All Strongly-Cyclic Branched Coverings of (1,1)-Knots are Dunwoody Manifolds

- Mathematics
- 2003

It is shown that every strongly-cyclic branched covering of a (1, 1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani,…

Knot Floer Homology of (1, 1)-Knots

- Mathematics
- 2003

We present a combinatorial method for a calculation of the knot Floer homology of (1, l)-knots, and then demonstrate it for nonalternating (1, 1)-knots with 10 crossings and the pretzel knots of type…

A generalized bridge number for links in 3-manifolds

- Mathematics
- 1992

In [Sch] Schubert introduced a new invariant of knots in the 3-sphere, called the bridge number, and showed that, when reduced by 1, it is an additive invariant under the connected sum operation of…

An introduction to Heegaard Floer homology

- 2007

Contents 1. Introduction 1 2. Heegaard decompositions and diagrams 2 3. Morse functions and Heegaard diagrams 7 4. Symmetric products and totally real tori 8 5. Disks in symmetric products 10 6. Spin…

Three-dimensional manifolds and their Heegaard diagrams

- Mathematics
- 1933

One of the outstanding problems in topology today is the classification of n-dimensional manifolds, n >3. Poincare, the founder of modern analysis situs, devoted several papers to it and allied…

Knot polynomials and knot homologies

- Mathematics
- 2005

This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories and give some examples which illustrate…

On train-track splitting sequences

- Mathematics
- 2010

We present a structure theorem for the subsurface projections of train-track splitting sequences. For the proof we introduce induced tracks, efficient position, and wide curves. As a consequence of…

On knot Floer homology of satellite

- knots. PhD thesis, Columbia University,
- 2006