Knot Floer homology, genus bounds, and mutation

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2003

77 citations recorded by the source corpus.

Abstract In an earlier paper, we introduced a collection of graded Abelian groups HFK (Y,K) associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the Kinoshita–Terasaka knots and their “Conway mutants”. These results show that HFK contains more information than the Alexander polynomial and the signature of these knots; and they also illustrate the fact that HFK detects mutation. We also calculate HFK for certain pretzel knots, and knots with small crossing number ( n ⩽9). Our calculations give obstructions to certain Seifert fibered surgeries on the knots considered here.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fefe1fa8bc4507f5effdcb01763869ce189d0096

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Citations in the corpus

On knot Floer homology and cabling (2004, background): “AMS Classification 57M27; 57R58 Keywords Knots, Floer homology, cable, satellite, Heegaard diagrams”
Knot Floer homology and the four-ball genus (2003, methodology): “We give also calculations for some ten-crossing knots in Section 4. The calculations rest on the combinatorial techniques developed in [19] and extended in [ 20 ].”
Holomorphic disks and knot invariants (2002, background): “ In fact, it is easy to see that they do – as graded Abelian groups, HFK\ distinguishes the right-handed from the left-handed trefoil. More interestingly, building on the material in [23], we show in [24] that HFK\ distinguishes all non-trivial pretzel knots of the form P(2a+1,2b+1,2c+1)from the unk”
Twisting quasi-alternating links (2007, background): “In addition, thick Khovanov homology or knot Floer homology precludes some of these links from being quasi-alternating (see [ 17 , 7, 21]).”
The slice-ribbon conjecture for 3-stranded pretzel knots (2007, result): “n Donaldson’s diagonalization theorem. It is not hard, however, to come up with a purely Heegaard Floer theoretic proof of their result: it follows from work of C. Livingston [13] and Ozsv´ath-Szab´o [21] that the concordance invariant τ(cf. [17]) for three stranded pretzel knots P(p,q,r) with p,q,r”
Heegaard Floer homology and alternating knots. (2002, methodology): “More calculations of knot homology groups are given [ 29 ].”
The Thurston norm, fibered manifolds and twisted Alexander polynomials (2005, methodology): “knot is two. The computation of the genus for all 11–crossing knots was done by Jacob Rasmussen, using a computer assisted computation of the Oszv´ath–Szab´o knot Floer homology (cf. also [OS04a] and [OS04b]). We ﬁrst consider the Conway knot K = 11401 whose diagram is given in Figure 3. This knot h”
Computations of Heegaard-Floer knot homology (2006, background): “n be distinguished by their Seifert genuses, however - KT2,1 has genus 2, whereas C2,1 has genus 3 (c.f. [4], Theorem 1.2). The homology in extremal Alexander gradings was computed for these knots in [8], Theorem 1.1. There are a surprisingly large number of knots with arc-index at most 12. Non-alte”
Knot polynomials and knot homologies (2005, methodology): “hese are (1,1) knots [8]. Of the remainder, skein theory can be used to show that 10140,10143, 10144, 10146–10149,10151,10155, 10158, and 10163– 10166 are δ-thin with δ = σ/2. Finally, the methods of [23] can be used to show that 10141, 10156,10157,10159, and 10160 are δ-thin with δ = σ/2, and to co”
Knot Floer Homology of (1, 1)-Knots (2003, methodology): “That of prime knots up to nine crossings are treated in [26].”
The decategorification of sutured Floer homology (2009, methodology): “It is an offshoot of the Heegaard Floer homology of Ozsváth and Szabó [26], and contains knot Floer homology [28, 33] as a special case.”