Computations of Heegaard-Floer knot homology

By John A. Baldwin, W. D. Gillam. 2006

69 citations recorded by the source corpus.

We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsvath and Sarkar. We review their construction, giving two examples that can be workout out by hand, and we explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, and we formulate a conjecture about the behavior of knot Floer homology under mutation, paying especially close attention to the Kinoshita–Terasaka knot and its Conway mutant. Finally, we discuss a conjecture of Rasmussen on relationship between Khovanov homology and knot Floer homology, and observe that it is consistent with our calculations.

Subfields: Genetics, Geometry and Topology, Neurology

View Paper: https://www.semanticscholar.org/paper/7ba348128001ac9cd12e513a2dd50dc73534cef8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Genetics

1. J. Greene
2. Peter S. Ozsv'ath
3. Zolt'an Szab'o
4. A. Stipsicz
5. Sucharit Sarkar
6. John A. Baldwin

In Geometry and Topology

1. A. Stipsicz
2. John A. Baldwin
3. Peter S. Ozsv'ath
4. Zolt'an Szab'o
5. J. Greene
6. Sucharit Sarkar
7. Ciprian Manolescu
8. Ian Zemke

In Neurology

1. Peter S. Ozsv'ath
2. A. Stipsicz
3. Zolt'an Szab'o
4. John A. Baldwin
5. Sucharit Sarkar
6. J. Greene
7. M. Hedden
8. Ciprian Manolescu

Citations in the corpus

• Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “Of the 42 non-alternating 10-crossing knots, 32 have σ-thin knot Floer homologies, and so are potentially quasi-alternating [2].”
• Grid Homology for Knots and Links (2015, methodology): “Using a direct computer computation, Baldwin and Gillam [5] computed the grid homology of all knots with at most twelve crossings.”
• A combinatorial spanning tree model for knot Floer homology (2011, unknown): “Somewhat surprisingly, however, the computations in [1] support the following conjecture: Conjecture 1.”
• On knot Floer width and Turaev genus (2007, unknown): “G T ] 5 S ep 2”
• AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “y Conway mutation, then ∆K1 = ∆K2. A well-known example of mutant knots, the Conway knot and the Kinoshita-Terasaka knot, have different knot Floer homologies [OS04g]. As can be seen from the table in [BG12], knot Floer homology is generally an effective invariant for distinguishing between two small ”
• Homologically thin, non-quasi-alternating links (2009, methodology): “group ĤFK(K; Z/2Z) was calculated by Baldwin-Gillam [2], and the Khovanov homology groups Kh(K) and Kh ′ (K) by Shumakovitch [30]; their results show that K is homologically thin.”
• Moving basepoints and the induced automorphisms of link Floer homology (2011, background): “By [1], the first spectral sequence collapses at the very next step, ie H .”