We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route, we provide a new proof of Floer's surgery exact triangle for instanton homology using metric stretching maps, and generalize the exact triangle to a link surgeries spectral sequence. Finally, we relate framed instanton homology to Floer's instanton homology for admissible bundles. (primary), 57R57 (secondary).
INSTANTONS AND ODD KHOVANOV HOMOLOGY
745Corollary 1.2. If L is a quasi-alternating link, then I # (Σ(L)) is free abelian of rank det(L) and is supported in even gradings. The rank in grading j ∈ {0, 2} ⊂ Z/4 is given bywhere #L is the number of components of L.If rational coefficients are assumed, of the 250 prime knots that have at most 10 crossings, only 7 of them have potentially non-trivial differentials after the E 2 -page of Theorem 1.1. This follows from the computations of odd Khovanov homology in [33].To put Theorem 1.1 into context, we relate framed instanton homology to previously studied instanton invariants. To start, framed instanton homology is a special case of Floer's instanton homology for admissible bundles.This latter condition guarantees that Y does not support any reducible flat connections.AThe non-trivial admissibility condition for Y amounts to the existence of an oriented surface Σ ⊂ Y that intersects ω in an odd number of points, or, equivalently, to the conditions that [ω] is non-zero and lifts to a non-torsion class in H 1 (Y ; Z).For admissible Y, Floer defined in [13, 14] a relatively Z/8-graded abelian group I(Y). When Y is a homology 3-sphere and Y is trivial, we write I(Y ) for this group, and it comes equipped with an absolute Z/8-grading. The isomorphism class of I(Y) depends only on the oriented homeomorphism type of Y and w 2 (Y). Now let Y be any SO(3)-bundle over a closed, connected, oriented 3-manifold Y geometrically represented by λ. Making some inessential choices, we can construct a bundle Y # over Y #T 3 by gluing together Y and a non-trivial bundle over T 3 . The bundle Y # is always admissible, and the group I(Y # ) is always 4-periodic. The framed instanton homology twisted by λ, written as I # (Y ; λ), is relatively Z/4-graded and isomorphic to four consecutive gradings of I(Y # ). When λ is mod 2 null-homologous, we recover the framed instanton homology I # (Y ).When Y is a homology 3-sphere, we relate I # (Y ) to Floer's Z/8-graded I(Y ). It is convenient to employ Frøyshov's reduced groups I(Y ) from [15], which are obtained from I(Y ) by considering interactions with the trivial connection. They come equipped with an absolute Z/8-grading and a degree 4 endomorphism u. Frøyshov's Theorem 10 says (u 2 − 64) n = 0 for some n > 0, when the coefficient ring used contains an inverse for 2.If Y is non-trivial and admissible, then the situation i...
