Computing knot Floer homology in cyclic branched covers

Abstract

We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K S 3 in its m-fold cyclic branched cover † m .K/, and we give computations when m D 2 for over fifty three-bridge knots with up to eleven crossings.

A flexible construction of equivariant Floer homology and applications

“…The involution τ on Σ(K) induces an action on Spin c (Σ(K)) which takes each spin c -structure to its conjugate and fixes only the unique spin structure s 0 [12, p. 1378; 25, Remark 3.4]. Moreover, τ preserves the absolute Alexander grading [25,Proposition 3.4]. Thus, the equivariant cochain complex CF Z/2 ( T α , T β ) splits along Alexander gradings and orbits of spin c -structures.…”

“…Grigsby showed that the absolute Alexander grading of an equivariant lift x ∈ T α ∩ T β of a generator x ∈ T α ∩ T β is the same as the absolute Alexander grading of x [12,Lemma 4.7] (see also [25,Proposition 3.4]). Furthermore, localization isomorphisms for (Σ(K), K) preserve absolute Alexander gradings and spin c structures [14, p. 2144], which in light of the proof of Proposition Theorem 1.14 implies that the isomorphism The theorem is a homology-level reinterpretation of this fact (and induction).…”

“…In fact, the equivariant Floer cohomology is often invariant under even non-equivariant isotopies; see Proposition 3. 25.…”

A spanning tree model for the Heegaard Floer homology of a branched double-cover

“…We point out that Heegaard diagrams presenting Σ(K), and more generally m-fold cyclic branched covers Σ m (K), have appeared in the work of Grigsby [7] and Levine [10]. In their work, the approach is to begin with a doubly or multiply pointed Heegaard diagram presenting K ⊂ S 3 , form the appropriate cyclic branched cover of the Heegaard surface, and thereby obtain a Heegaard diagram presenting the preimage of the linkK ⊂ Σ m (K).…”

A Combinatorial Description of the U2=0 Version of Heegaard Floer Homology

“…The Heegaard diagram considered in this paper is obtained from a triple branched cover of a link, represented by a grid diagram. As such, our constructions here were partly motivated by work of Levine [5], who studied branched covers of grid diagrams in a slightly different context. The construction of the Heegaard diagrams equips them with some extra structures, most notably with a projection to a grid diagram.…”