Classical homological invariants are not determined by knot Floer homology and Khovanov homology

2016 ◽  
Vol 25 (07) ◽  
pp. 1650039
Author(s):  
Jae Choon Cha ◽  
Toshifumi Tanaka
Keyword(s):  
Floer Homology ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Homological Invariants

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

scholarly journals Khovanov homology and knot Floer homology for pointed links

2017 ◽  
Vol 26 (02) ◽  
pp. 1740004 ◽  
Author(s):  
John A. Baldwin ◽  
Adam Simon Levine ◽  
Sucharit Sarkar
Keyword(s):  
Spectral Sequence ◽  
Floer Homology ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Text Page ◽  
Text Component

A well-known conjecture states that for any [Formula: see text]-component link [Formula: see text] in [Formula: see text], the rank of the knot Floer homology of [Formula: see text] (over any field) is less than or equal to [Formula: see text] times the rank of the reduced Khovanov homology of [Formula: see text]. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose [Formula: see text] page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field [Formula: see text].

scholarly journals QUASI-ALTERNATING MONTESINOS LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1459-1469 ◽  
Author(s):  
TAMARA WIDMER
Keyword(s):  
Open Problem ◽  
Floer Homology ◽  
Natural Generalization ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Interesting Open Problem ◽  
Alternating Links

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.

scholarly journals COMPUTATIONS OF HEEGAARD-FLOER KNOT HOMOLOGY

2012 ◽  
Vol 21 (08) ◽  
pp. 1250075 ◽  
Author(s):  
JOHN A. BALDWIN ◽  
WILLIAM D. GILLAM
Keyword(s):  
Floer Homology ◽  
Khovanov Homology ◽  
Combinatorial Approach ◽  
Close Attention ◽  
Knot Floer Homology ◽  
Knot Homology ◽  
Heegaard Floer

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, Ozsváth 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.

scholarly journals A note on quasi-alternating Montesinos links

2015 ◽  
Vol 24 (09) ◽  
pp. 1550048 ◽  
Author(s):  
Abhijit Champanerkar ◽  
Philip Ording
Keyword(s):  
Recent Work ◽  
Floer Homology ◽  
Khovanov Homology ◽  
Joint Work ◽  
Knot Floer Homology ◽  
Rational Tangles ◽  
Heegaard Floer ◽  
Alternating Links

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results.

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

scholarly journals An introduction to knot Floer homology and curved bordered algebras

2020 ◽  
Vol 80 (2) ◽  
pp. 211-236
Author(s):  
Antonio Alfieri ◽  
Jackson Van Dyke
Keyword(s):  
Floer Homology ◽  
Knot Floer Homology

scholarly journals Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

2008 ◽  
Vol 2008 ◽  
Author(s):  
Kenneth L. Baker ◽  
J. Elisenda Grigsby ◽  
Matthew Hedden
Keyword(s):  
Floer Homology ◽  
Lens Spaces ◽  
Knot Floer Homology ◽  
Grid Diagrams

scholarly journals A combinatorial spanning tree model for knot Floer homology

2012 ◽  
Vol 231 (3-4) ◽  
pp. 1886-1939 ◽  
Author(s):  
John A. Baldwin ◽  
Adam Simon Levine
Keyword(s):  
Spanning Tree ◽  
Floer Homology ◽  
Tree Model ◽  
Knot Floer Homology
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