Papers › Positive 2-bridge knots and chirally cosmetic surgeries
Positive 2-bridge knots and chirally cosmetic surgeries
Michael Huang, Zelong Li, Rahi Tanaz, Chengyi Zhang
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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 S³ᵣ and S³_(r′) with distinct slopes r and r′ such that S³ᵣ(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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