On knot Floer homology and cabling

By M. Hedden. 2004

149 citations recorded by the source corpus.

This paper is devoted to the study of the knot Floer homology groups \ HFK(S 3 ,K2,n), where K2,n denotes the (2,n) cable of an arbitrary knot, K. It is shown that for sufficiently large|n|, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of \(K). A precise formula for this relationship is presented. In fact, the homology groups in the top 2 filtration dimensions for the cabled knot are isomorphic to the original knot's Floer homology group in the top filtration dimension. The results are extended to (p,pn ±1) cables. As an example we compute \ HFK((T2,2m+1)2,2n+1) for all sufficiently large|n|, where T2,2m+1 denotes

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

View Paper: https://www.semanticscholar.org/paper/c3945c9c4f851c711e69a23e0f490c99260d4509

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT (2005, background): “3 but does not admit a lens space surgery by the examination of its knot Floer homology groups found in [6].”
The decategorification of sutured Floer homology (2009, background): “lows that τ(S3(R)) = ∆ K(t)· tn −1 t−1 . Note that when n= 0, the torsion vanishes, regardless of what K is. The group HFK\(K(n)) has been studied by Eftekhary [Ef05] (in the case n= 0) and by Hedden [He05], who gives a complete calculation in terms of the groups HFK−(K). In particular, HFK\(K(0)) i”
Bordered Heegaard Floer homology and the tau‐invariant of cable knots (2012, background): “The first such result appears in [3], and is strengthened in [5], where Hedden proves the following inequality for τ of the (p, pn + 1)-cable of a knot K:”
Knot polynomials and knot homologies (2005, methodology): “of thumb, it tends to be eﬀective at computing HFK\(K,i) when i is close to g(K), but rather less so when i is close to 0. Other special Heegaard splittings have been used by Eftekhary [4] and Hedden [18] to make some computations for Whitehead doubles and cabled knots. We close this section by ment”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “Hedden [Hed05, Hed07, Hed09] and Eftekhary [Eft05] studied the knot Floer homology of cables and Whitehead doubles.”
HEEGAARD FLOER HOMOLOGY AND RATIONAL CUSPIDAL CURVES (2013, background): “If K is a knot admitting an L–space surgery, in particular an algebraic knot (see [8]), then it has Alexander polynomial of the form ∆K(t) = ∑2m i=0(−1)t .”
A note on cabling and $L$–space surgeries (2009, background): “Hedden [1] and Van Cott [11] have studied the behavior of under cabling, giving bounds and, in special cases, formulas for .”
Knot Floer homology of Whitehead doubles (2006, unknown): “See Figure 1 foran illustration.”
Link Floer homology and the Thurston norm (2006, background): “We are particularly indebted to Jake for his many valuable suggestions following a thorough reading of an early version of this paper.”
On Knot Floer Homology and Cabling: 2 (2008, methodology): “2 of [8]) Let K ⊂ S3 be a knot of genus g.”
Link Floer homology detects the Thurston norm (2006, methodology): “We use the “cabling trick” from Ozsváth and Szabó [14], as well as the techniques from Hedden [4], to reduce the general case of our main theorem to the case proved in Section 3.”
On the geography and botany of knot Floer homology (2014, background): “Of relevance here is the fact that there are known pairs of knots which admit L-space surgeries and share the same Floer homology, the simplest being the (2, 3)-cable of the trefoil and the (3, 4)-torus knot [27].”
The reduced knot Floer complex (2013, methodology): “By combining work of Hedden and Hom [10, 14], the (p, q)-cable of a knot K is an L-space knot if and only if K is an L-space knot and q p ≥ 2g(K)− 1, where g is the Seifert genus.”