Exponential decay of unique-values rate for quantum invariants on prime knots

Establish the existence of constants δ(Q) in (0,1) for each of the quantum invariants Q in {A2, Alexander, B1 (the sl2 symmetric-color-2 invariant), Khovanov} such that the percentage Q(n)% of distinct values attained by Q on the set of prime knots with n crossings satisfies an inequality of the form Q(n)% ≤ C·(δ(Q))^n for some constant C independent of n. Additionally, determine whether the ordering δ(B1) > δ(A2) > δ(Khovanov) > δ(Alexander) holds, and quantify the rates.

Background

Section 13 compares how different quantum invariants distinguish prime knots as crossing number n grows. The authors define Q(n)% as the percentage of distinct values obtained by an invariant Q on all prime knots with n crossings and present empirical evidence that Q(n)% decays rapidly for several invariants. They formulate an explicit conjecture asserting exponential decay with invariant-specific rates and an ordering among these rates.

The invariants considered include A2 (type A2), the Alexander polynomial, B1 (the sl2 symmetric-color-2 invariant), and Khovanov homology (with Jones polynomial and Khovanov’s t=1 specialization noted to behave similarly).