Link–knot defect proportion comparison conjecture

For a prime alternating knot or link $K$, let $\gamma(K)$ denote its crosscap number and let $\Gamma(K)$ denote its unoriented genus. At crossing number $c$, compare the proportions of prime alternating links and knots satisfying $\gamma>\Gamma$.

Defect proportion comparison conjecture. For all even $c\geq 4$ and all odd $c\geq 11$, the proportion of $c$-crossing prime alternating links $L$ with $\gamma(L)>\Gamma(L)$ is larger than the proportion of $c$-crossing prime alternating knots $K$ with $\gamma(K)>\Gamma(K)$.

The conjecture is motivated by the heuristic that links have more two-sided state surfaces, together with an observed parity-dependent emergence in the data. Its status is open.