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}
}
  • Philip Ording
  • Published 25 October 2011
  • Mathematics
  • arXiv: Geometric Topology
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… Expand

References

SHOWING 1-10 OF 17 REFERENCES
Parameterizations of 1-Bridge Torus Knots
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 areExpand
All Strongly-Cyclic Branched Coverings of (1,1)-Knots are Dunwoody Manifolds
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,Expand
Knot Floer Homology of (1, 1)-Knots
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 typeExpand
A generalized bridge number for links in 3-manifolds
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 ofExpand
An introduction to Heegaard Floer homology
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. SpinExpand
Three-dimensional manifolds and their Heegaard diagrams
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 alliedExpand
Knot polynomials and knot homologies
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 illustrateExpand
On train-track splitting sequences
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 ofExpand
On knot Floer homology of satellite
  • knots. PhD thesis, Columbia University,
  • 2006
...
1
2
...