Median and mode conjecture for crosscap number and unoriented genus of prime alternating knots

Let cc be the crossing number of a prime alternating knot. For each cc, consider the distributions of its crosscap number 4γ4\gamma and unoriented genus Γ\Gamma among all cc-crossing prime alternating knots.

Median and mode conjecture. For any c12c\geq 12, the medians and modes for γ\gamma and Γ\Gamma among all cc-crossing prime alternating knots equal

c21.\left\lfloor\frac{c}{2}\right\rfloor-1.

This conjecture extrapolates a pattern observed in the tabulated data; its status is open, and no proof or disproof is supplied.

Sources & referencesView supporting material

Primary source

Isaias Bahena, Thomas Kindred and Jason Parsley, “Crosscap numbers of alternating links via state codes”, arXiv:2512.09887 (2025).

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