Georgia Institute of Technology College of Sciences

A basic question for any knot invariant asks which knots the invariant detects. 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 over the past twenty years, and will describe some of the topological ideas that go into my recent work with Sivek on these questions. Time permitting, I'll end with applications of these knot detection results to problems in Dehn surgery, explaining in particular how we use them to dramatically extend some of Gabai's celebrated results from the 80's.