On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology

By J. E. Grigsby, S. Wehrli. 2008

49 citations recorded by the source corpus.

Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, \Sigma(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced n-colored Jones polynomial of the mirror of K to the knot Floer homology of (\Sigma(K),\widetilde{K}) (when n odd) and to (S^3, K # K) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.

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

View Paper: https://www.semanticscholar.org/paper/d3b385325fa64a22f59b5eb7fc55fc18c6b81cc7

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Cobordisms of sutured manifolds and the functoriality of link Floer homology (2009, methodology): “bordism. Then there is a well-deﬁned map s: π2(x0,...,xn) → Spin c(W) such that s(Ψ)|Mi,i+1 = s(xi) for every i∈ Zn+1. COBORDISMS OF SUTURED MANIFOLDS 19 Proof. The construction of Grigsby and Wehrli [12], based on the work of Ozsv´ath and Szabo´ [36, Section 8], associates to any Whitney (n+1)-gon ”
A combinatorial spanning tree model for knot Floer homology (2011, background): “homology of Σ(L), the double cover of S3 branched along the mirror of L. Generalizations and applications of this spectral sequence have made for an active area of reseach in recent years; see, e.g., [2, 4, 11, 14, 46, 19]. In forthcoming work, Ozsvath, Szabo´, and the ﬁrst author deﬁne an analogous”
Annular Khovanov homology and knotted Schur–Weyl representations (2015, unknown): “[13], [12]) with Juhász’s sutured version of Heegaard Floer homology [20].”
Khovanov module and the detection of unlinks (2012, background): “A key ingredient of our proof is that the Λ∗H1–module structure on Heegaard Floer homology detects S 1 × S 2 connected summands.”
Categorification of the Jones-Wenzl Projectors (2010, methodology): “It is important to note that our definition disagrees with some previous categorifications based on different axiomatizations of the Jones–Wenzl projectors such as the dimension axiom [17] (for related work see [5] and [11]): Tr.”