Positive 2-bridge knots and chirally cosmetic surgeries

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 $S_r^3$ and $S_{r^{\prime }}^3$ with distinct slopes $r$ and $r^{\prime }$ such that $S_r^3(K)\cong -S_{r^{\prime }}^3(K)$, 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.

25 pages, 22 figures, code developed can be found at https://github.com/zl830/chiral_cosmetic_surgery_for_pos_2_bridge_knots