On constraints for knots to admit chirally cosmetic surgeries and their calculations

Abstract

We discuss various constraints for knots in $S^3$ to admit chirally cosmetic surgeries, derived from invariants of 3-manifolds, such as, the quantum $SO(3)$-invariant, the rank of the Heegaard Floer homology, and finite type invariants. We apply them to show that a large portion (roughly 75$\%$) of knots which are neither amphicheiral nor $(2,p)$-torus knots with less than or equal to 10 crossings admits no chirally cosmetic surgeries.

21 pages, 4 figures; Added Remark 1.13 on hyperbolic geometry computations by M. Kegel showing the non-existence of chirally cosmetic surgeries up to 10 crossings