Several spectral sequences are known to exist relating the Khovanov homology of a link to Floer-theoretic invariants such as Heegaard Floer, monopole Floer or instanton homology. The present paper constructs another such spectral sequence, proving several related results and calculations.
For a link \(L \subset S^3\), Ozsváth-Szabó found a spectral sequence with \(E^2\) page \(\overline{\mathrm{Kh}}(L; \mathbb{F}_2)\), the reduced Khovanov homology of \(L\) with coefficients in the field with 2 elements \(\mathbb{F}_2\), converging to \(\widehat{HF}( \overline{\Sigma(L)} ; \mathbb{F}_2 )\), the “hat” Heegaard Floer homology with \(\mathbb{F}_2\) coefficients of the double branched cover \(\Sigma(L)\) over \(L\), with reversed orientation [P. Ozsváth and Z. Szabó, Adv. Math. 194, No. 1, 1–33 (2005; Zbl 1076.57013)]. J. M. Bloom found a spectral sequence from \(\overline{\mathrm{Kh}}(L; \mathbb{F}_2)\), to the monopole Floer homology \(\widetilde{HM}(\overline{\Sigma(L)};\mathbb{F}_2)\) of the same manifold [Adv. Math. 226, No. 4, 3216–3281 (2011; Zbl 1228.57016)]. And P. B. Kronheimer and T. S. Mrowka found a spectral sequence from \(\mathrm{Kh}(L)\), the (unreduced) Khovanov homology of \(L\), to the reduced singular instanton homology \(I^\# (L)\) [Publ. Math., Inst. Hautes Étud. Sci. 113, 97–208 (2011; Zbl 1241.57017)].
The main theorem of the present paper constructs a new spectral sequence, from the reduced odd Khovanov homology \(\overline{\mathrm{Kh}'}(L)\) of an oriented link \(L \subset S^3\), to the framed instanton homology \(I^\# (\overline{\Sigma(L)})\) of the double branched cover with reversed orientation. Both groups have \(\mathbb{Z}\) coefficients, and each page of the spectral sequence has a \(\mathbb{Z}/4\) grading, which interpolates with the usual grading on Khovanov homology.
While instanton homology \(I(Y)\) of an admissible \(\mathrm{SO}(3)\)-bundle \(\mathbb{Y}\) over a 3-manifold \(Y\) is defined by counting suitable instantons over \(Y \times \mathbb{R}\), framed instanton homology \(I^\# (Y)\) is defined by counting suitable \(\mathrm{SO}(3)\)-instantons over \(\mathbb{R} \times (Y \# T^3)\) in an appropriate bundle which is nontrivial over \(T^3\). There is also a twisted version \(I^\# (Y; \lambda)\).
The appearance of the odd version of Khovanov homology is perhaps not too surprising: this variant was in fact introduced by P. S. Ozsváth et al. in order to extend their spectral sequence from \(\mathbb{F}_2\) to \(\mathbb{Z}\) coefficients [Algebr. Geom. Topol. 13, No. 3, 1465–1488 (2013; Zbl 1297.57032)].
In certain cases the spectral sequence of the present paper collapses. This occurs for instance when \(L\) is a quasi-alternating link, and in this case the author shows \(I^\# (\Sigma(L))\) is a free abelian group, with the rank in each grading given explicitly.
The author also gives a new proof of Floer’s surgery exact triangle for instanton homology, adapting a proof of Kronheimer and Mrowka in singular instanton knot homology. This leads to a link surgery spectral sequence, similar to one found by Ozsváth and Szabó in the context of Heegaard Floer homology.
Several results are given relating framed instanton homology \(I^\# (Y)\) to other instanton invariants, making various computations along the way. When \(Y\) is a homology 3-sphere, the author relates \(I^\# (Y)\) and \(I^\# (Y; \lambda)\) to \(I(Y)\), by using the endomorphism \(u\) of the reduced groups \(\widehat{I}(Y)\) defined by K. A. Frøyshov [Topology 41, No. 3, 525–552 (2002; Zbl 0999.57032)]. These results are used to calculate the Euler characteristic \(\chi(I^\# (Y; \lambda)) = | H_1 (Y; \mathbb{Z}) |\). This description simplifies when an admissible bundle \(\mathbb{Y}\) restricts nontrivially to a surface of genus \(\leq 2\), which occurs for instance when \(Y\) is the result of \(\pm 1\) surgery on a knot \(K \subset S^3\). In this way the author is able to give explicit descriptions of the framed instanton homology of the branched double covers of the \((3, 6k \pm 1)\) torus knots; these are also realised as \(\pm 1\) surgeries on twist knots.