In-Person and Online Talk

A basic question for a knot invariant asks which knots the invariant detects (if any). For example, it is famously open whether the Jones polynomial detects the unknot. I'll focus in this talk on the detection question for knot invariants coming from Floer theory and the Khovanov--Rozansky link homology theories. I'll survey the progress made on this question over the past twenty years, and will gesture at some of the topological ideas that go into my recent work with Sivek on this question. I'll end with applications of these knot detection results to problems in Dehn surgery, explaining in particular how we use them to extend some of Gabai's celebrated work from 1987.