SL(2,C) Floer Cohomology for Surgeries on Some Knots

Abstract

We establish a relationship between HP(Y), the Abouzaid–Manolescu sheaf-theoretic SL(2,C) Floer cohomology for a surgery Y on a small knot in S3 and Curtis’ SL(2,C) Casson invariant. We use this to compute HP for most surgeries on two-bridge knots. We also compute HP for surgeries on two nonsmall knots, the granny and square knots. We provide a partial calculation of the framed sheaf-theoretic SL(2,C) Floer cohomology HP#(Y) for surgeries on two-bridge knots and apply the data we obtain to show the nonexistence of a surgery exact triangle for HP#.