On Conway mutation and link homology

Abstract

We give a new, elementary proof that Khovanov homology with Z/2Z-coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ-graded knot Floer homology is mutation-invariant. Using the Clifford module structure on HFK induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let L ′ be a link obtained from L by mutating the tangle T . Suppose some rational closure of T corresponding to the mutation is the unlink on any number of components. Then L and L ′ have isomorphic δ-graded HFK groups over Z/2Z as well as isomorphic Khovanov homology over Q. We apply these results to establish mutation-invariance for the infinite families of Kinoshita-Terasaka and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be mutation-invariant.2010 Mathematics Subject Classification. 57M27; 57R58.The complex (CKh(D), d Kh ) possess two gradings, the quantum and homological grading, and the differential preserves the quantum grading and increases the homological grading by one. Thus the homology Kh(D) splits into the direct sum of bigraded modules Kh i,j (D) where i denotes the homological grading and j the quantum grading.Let p be a fixed basepoint in the plane contained in the diagram D. The basepoint p determines a chain map X p : CKh(D) → CKh(D) that squares to 0. The kernel of X p is a subcomplex and reduced Khovanov homology is its homology, with a shift in the quantum grading:It is an invariant of L up to isotopies supported away from p. While over Z the reduced homology depends on the component containing the basepoint, this is not true over Z/2Z.for every link L and any basepoint p. In particular, Kh(L) is well-defined independent of p.Consequently, when discussing reduced Khovanov homology over Z/2Z we will suppress any mention of the basepoint p.Khovanov homology satisfies Kunneth-type formulas for disjoint unions and connected sums. The following properties are well-known and the proofs are easy deductions from the definition of Khovanov homology and Proposition 2.1.Lemma 2.2. Let L 1 , L 2 be arbitrary links. Over Z/2Z there are isomorphismsfor any choice of connected sum.Khovanov homology satisfies an unoriented skein exact triangle. Fix a crossing of D and let D 0 , D 1 denote the 0-and 1-resolutions of D at this crossing. Since Khovanov homology is computed from a cube of resolutions, the complex CKh(D) is, up to a grading shift, the mapping cone of a chain map

Skein relations for tangle Floer homology

“…One idea involves computing the maps in the skein relation combinatorially: The second author [38] gives a version of the skein sequence for grid homology, generalizing the results on quasi-alternating links [24,25] to Z-coefficients and giving a spectral sequence from a cube-of-resolutions complex with no diagonal maps. Lambert-Cole [21] exploits the computability in [38] to show that δ-graded knot Floer homology is invariant under Conway mutation by a large class of tangles.…”

On symmetries of peculiar modules; or, $δ$-graded link Floer homology is mutation invariant

“…However, based on computations from [BG12], Baldwin and Levine conjectured that HFL(L) is mutation invariant if the bigrading is collapsed to a single grading known as the δ-grading [BL12,Conjecture 1.5]. This conjecture was confirmed for a number of families of mutant links by Lambert-Cole [LC18,LC19] and the author [Zib20]. In this paper, we prove the conjecture in general: Theorem 0.1.…”

Peculiar modules for 4‐ended tangles

“…It would be interesting to see how these two approaches can be merged. Finally, I also want to mention some impressive work of Lambert‐Cole , where he confirms Conjecture for various families of mutant pairs (different from the one in Theorem ), using entirely different techniques.…”

“…Definition 2. 16. Given a 4-ended tangle T with n closed components in a Z-homology 3-ball M with spherical boundary and an (admissible) peculiar Heegaard diagram for T , let us define a generalised peculiar module CFT − (T ) := CFT − (T, M ) in gpqMod PT ,n whose underlying relatively bigraded right I ∂ -module agrees with CFT(T ).…”