The generalized cosmetic crossing conjecture for knots
The generalized cosmetic crossing conjecture for knots
Let be a knot, and let be a crossing of . A crossing is nugatory if its crossing circle bounds a disk in the complement of ; a generalized crossing change is a Dehn surgery operation along the crossing circle that changes the crossing by an arbitrary nonzero order. Generalized cosmetic crossing conjecture. If a generalized crossing change on yields a knot isotopic to , then is nugatory. This is a generalized form of the cosmetic crossing conjecture, which asks whether a non-nugatory crossing change can preserve the knot type. The source states that the corresponding cosmetic crossing question is open, and gives no resolution of this generalized formulation.
Sources & referencesView supporting material
Primary source
Marion Campisi, Brandy Doleshal and Eric Staron, “The n-adjacency graph for knots”, arXiv:2603.08597 (2026).
Additional references
7 papers in this index state this conjecture (2015–2026). The statement above is taken from the most recent of them; the others are arXiv:2410.15593, arXiv:2103.15277, arXiv:2102.09116, arXiv:1908.05701, arXiv:1603.09039, arXiv:1507.07996.
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