Cosmetic surgery and the $SL(2,\mathbb{C})$ Casson invariant for two-bridge knots

TL;DR

This paper investigates the cosmetic surgery problem for two-bridge knots, demonstrating that most such knots do not admit purely cosmetic surgeries, especially when considering the $SL(2,bC)$ Casson invariant as a key tool.

Contribution

It establishes new results on the non-existence of cosmetic surgeries for specific classes of two-bridge knots using the $SL(2,bC)$ Casson invariant.

Findings

Most two-bridge knots with up to 9 crossings admit no purely cosmetic surgeries.

Certain two-bridge knots with specific Conway forms also admit no cosmetic surgeries yielding homology 3-spheres.

The $SL(2,bC)$ Casson invariant is effective in detecting the absence of cosmetic surgeries.

Abstract

We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. It is seen that all the two-bridge knots at most 9 crossings other than $9_{27} = S(49,19)=C[2,2,-2,2,2,-2]$ admits no purely cosmetic surgery pairs. Then we show that any two-bridge knot of the Conway form $[2x,2,-2x,2x,2,-2x]$ with $x \ge 1$ admits no cosmetic surgery pairs yielding homology 3-spheres, where $9_{27}$ appears for $x=1$. Our advantage to prove this is using the $SL(2,\mathbb{C})$ Casson invariant.