On detection probabilities of link invariants

Abstract

We prove that, for many standard link invariants, both the proportion of distinct invariant values and the detection probability among prime alternating links with at most n crossings decay exponentially in n, with an explicit universal rate. In fact, almost every such link belongs to an invariant fiber whose size is itself exponential in n. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. The companion website gives a much more detailed view of the data, including complete distributions of fiber sizes, separate alternating and non-alternating data, and topological data analysis.