Positive 2-bridge knots and chirally cosmetic surgeries

2023/08/19 by Huang, Michael, Li, Zelong, Tanaz, Rahi +1
#57K10 (primary) 57K18 #57K31 (secondary) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2308.10126

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 S³_r and S³_r' with distinct slopes r and r' such that S³_r(K) ≅ -S³_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 a₂, a₄, v₃, det, and g of a positive 2-bridge knot.

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