Open Collections

Description

I will describe how the knot Floer homology of a knot K can be represented by a decorated collection of immersed curves in the marked torus. The surgery formula for knot Floer homology translates nicely to this setting: the Heegaard Floer homology HF^- of p/q surgery on K is given by the Lagrangian Floer homology of these immersed curves with a line of slope p/q. For a simplified â UV = 0â version of knot Floer homology, the analogous statements follow from earlier work with Rasmussen and Watson by passing through the bordered Floer homology of the knot complement, but a more direct approach allows us to capture the stronger â minusâ invariant by adding decorations to the curves. Often recasting algebraic structures in terms of geometric objects in this way leads to new insights and results; I will mention some applications of this immersed curves framework, including obstructions to cosmetic surgeries.