Two-fold quasi-alternating links, Khovanov homology and instanton homology

Abstract

We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many computations of framed instanton homology in the presence of a non-trivial real 3-plane bundle. We discuss evidence for a spectral sequence from the twisted Khovanov homology of a link with mod 2 coefficients to the framed instanton homology of the double branched cover. We also discuss the relevant mod 4 gradings. arXiv:1605.05394v1 [math.GT]

Framed instanton homology and concordance

“…This conjecture has only been verified for a small subset of 3-manifolds, due to the difficulty of computing I # . Indeed, framed instanton homology has only been computed for: connected sums of S 1 × S 2 [50], branched double covers of twofold quasi-alternating links [50,51], some Brieskorn spheres including Σ(2, 3, 6k ± 1) [50], a surface times a circle [11], and some Dehn surgeries on instanton L-space knots [4,32]. One of the main goals of our paper is to expand this list.…”

Links all of whose cyclic branched covers are L-spaces

“…For example, L 2,1 is the pretzel knot P (3, −3, −3) which is not even quasi-alternating [Gre10]; see also [Iss18]. We instead show that the links L n,k are two-fold quasi-alternating, a generalization of quasi-alternating links introduced by Scaduto and Stoffregen [SS18]. Two-fold quasi-alternating links are Z/2 reduced Khovanov thin [SS18].…”

“…We instead show that the links L n,k are two-fold quasi-alternating, a generalization of quasi-alternating links introduced by Scaduto and Stoffregen [SS18]. Two-fold quasi-alternating links are Z/2 reduced Khovanov thin [SS18]. Thus Σ 2 (L n,k ) = Σ n (P (3, −3, −2k − 1)) is a Heegaard Floer L-space [OS05] and a framed instanton L-space [Sca15] with Z/2 coefficients.…”

“…In this section we recall the definition of two-fold quasi-alternating (TQA) links, as well as a few key facts which we will use. All of this material can be found in [SS18].…”

Non-left-orderability of cyclic branched covers of pretzel knots $P(3,-3,-2k-1)$

“…, −k)). Because two-fold quasi-alternating links are Khovanov homology thin [8], the double branched covers of them are L-spaces [7]. In this way, they proved that all n-th fold cyclic branched covers Σ n (P (3, −3, −2k − 1)) are L-spaces.…”