Positive 2-bridge knots and chirally cosmetic surgeries

TL;DR

This paper investigates chirally cosmetic surgeries in positive 2-bridge knots, confirming their rarity among knots up to 31 crossings except for specific torus knots, using computational and theoretical tools.

Contribution

It verifies the non-existence of chirally cosmetic surgeries in positive 2-bridge knots up to 31 crossings, except for certain torus knots, by developing a computational approach based on an obstruction formula.

Findings

Most positive 2-bridge knots up to 31 crossings do not admit chirally cosmetic surgeries.

The exception is the family of (2, 2n+1) torus knots.

A Python program was developed to compute relevant knot invariants for this verification.

Abstract

In this paper we verify that with the exception of the $(2, 2n+1)$ torus knots, positive 2-bridge knots up to 31 crossings do not admit chirally cosmetic surgeries. A knot $K$ admits chirally cosmetic surgeries if there exist surgeries $S^3_r$ and $S^3_{r'}$ with distinct slopes $r$ and $r'$ such that $S^3_r(K) \cong -S^3_{r'}(K)$, where the negative represents an orientation reversal. To verify this, we use the obstruction formula from arXiv:2112.03144 which relates classical knot invariants to the existence of chirally cosmetic surgeries. To check the formula, we develop a Python program that computes the classical knot invariants $a_2$, $a_4$, $v_3$, $\det$, and $g$ of a positive 2-bridge knot.