The generalized cosmetic crossing conjecture for knots

Let KS3K\subset S^3 be a knot, and let cc be a crossing of KK. A crossing is nugatory if its crossing circle bounds a disk in the complement of KK; 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 cc yields a knot isotopic to KK, then cc 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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