# Traceless Character Varieties, the Link Surgeries Spectral Sequence, and Khovanov Homology

## TL;DR

This paper develops the structural properties of symplectic instanton homology, including functoriality under cobordisms and a spectral sequence linking Khovanov homology to instanton homology of branched covers.

## Contribution

It proves functoriality of symplectic instanton homology under 4D cobordisms and establishes a spectral sequence connecting Khovanov homology with instanton homology of branched double covers.

## Key findings

- Functoriality of symplectic instanton homology under cobordisms.
- Spectral sequence from Khovanov homology to instanton homology.
- Generalization of surgery exact triangle to a spectral sequence.

## Abstract

In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly satisfy the axioms of a topological quantum field theory (TQFT), so that in particular they should be functorial with respect to cobordisms. Following a strategy used by Ozsv\'ath and Szab\'o in the context of Heegaard Floer homology, we prove that our theory is functorial with respect to connected $4$-dimensional cobordisms, so that cobordisms induce homomorphisms between symplectic instanton homologies. We also generalize the surgery exact triangle by proving that Dehn surgeries on a link $L$ in a $3$-manifold $Y$ induce a spectral sequence of symplectic instanton homologies -- the $E^2$-page is isomorphic to a direct sum of symplectic instanton homologies of all possible combinations of $0$- and $1$-surgeries on the components of $L$, and the spectral sequence converges to $\mathrm{SI}(Y)$. For the branched double cover $\Sigma(L)$ of a link $L \subset S^3$, we show there is a link surgery spectral sequence whose $E^2$-page is isomorphic to the reduced Khovanov homology of $L$ and which converges to the symplectic instanton homology of $\Sigma(L)$.