Symplectic instanton knot homology

Abstract

There have been a number of constructions of Lagrangian Floer homology invariants for $3$-manifolds defined in terms of symplectic character varieties arising from Heegaard splittings. With the aim of establishing an Atiyah-Floer counterpart of Kronheimer and Mrowka's singular instanton homology, we generalize one of these, due to H. Horton, to produce a Lagrangian Floer invariant of a knot or link $K \subset Y$ in a closed, oriented $3$-manifold, which we call symplectic instanton knot homology ($\mathrm{SIK}$). We use a multi-pointed Heegaard diagram to parametrize the gluing together of a pair of handlebodies with properly embedded, trivial arcs to form $(Y, K)$. This specifies a pair of Lagrangian embeddings in the traceless $\mathrm{SU}(2)$-character variety of a multiply punctured Heegaard surface, and we show that this has a well-defined Lagrangian Floer homology. Portions of the proof of its invariance are special cases of Wehrheim and Woodward's results on the quilted Floer homology associated to compositions of so-called elementary tangles, while others generalize their work to certain non-elementary tangles.