Symplectic Instanton Homology of Knots and Links in 3-manifolds

Powerful homology invariants of knots in 3-manifolds have emerged from both the gauge-theoretic and the symplectic kinds of Floer theory: on the gauge-theoretic side is the instanton knot homology of Kronheimer-Mrowka, and on the symplectic the (Heegaard) knot Floer homology developed independently by Ozsváth-Szabó and by Rasmussen. These theories are conjecturally equivalent, but a precise connection between the gauge-theoretic and symplectic sides here remains to be understood. We describe a construction designed to translate singular instanton knot homology more directly into the symplectic domain, a so-called symplectic instanton knot homology: We define a Lagrangian Floer homology invariant of knots and links which extends a 3-manifold invariant developed by H. Horton. The construction proceeds by using specialized Heegaard diagrams to parametrize an intersection of traceless SU(2) character varieties. The latter is in fact an intersection of Lagrangians in a symplectic manifold, giving rise to a Lagrangian Floer homology. We discuss its relation to singular instanton knot homology, as well as the formal properties which this suggests and methods to prove these properties.