Chirally cosmetic surgery conjecture for knots

Let KK be a knot in S3S^3. Two surgeries on KK with distinct slopes r,rQ{}r,r'\in\mathbb{Q}\cup\{\infty\} are chirally cosmetic surgeries when Sr3(K)Sr3(K)S^3_r(K)\cong -S^3_{r'}(K), where the minus sign denotes orientation reversal. Chirally cosmetic surgery conjecture. A knot KK does not admit chirally cosmetic surgeries if it is not the unknot, a (2,2n+1)(2,2n+1) torus knot, or an amphichiral knot. The conjecture identifies the unknot, the specified torus knots, and amphichiral knots as the possible exceptions to the absence of chirally cosmetic surgeries; the source does not establish a general resolution.

Sources & referencesView supporting material

Primary source

Michael Huang, Zelong Li, Rahi Tanaz and Chengyi Zhang, “Positive 2-bridge knots and chirally cosmetic surgeries”, arXiv:2308.10126 (2023).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.