Exponential growth of comparisons-to-equality for quantum invariants on random knot pairs

Determine, for each quantum invariant Q in {A2, Alexander, B1 (the sl2 symmetric-color-2 invariant), Jones, KhovanovT1}, whether there exists a constant γ(Q) > 1 such that the expected number of random pair selections of distinct prime knots with n crossings needed to obtain equal Q-values grows exponentially like γ(Q)^n up to a multiplicative constant. Additionally, assess whether the ordering γ(B1) > γ(A2) > γ(Jones or KhovanovT1) > γ(Alexander) holds.

Background

Section 13F introduces Q(n)% ,% as the expected number of random selections of two distinct prime knots with n crossings required to find a pair whose images under Q coincide. Empirical plots suggest this quantity grows roughly exponentially and that the B1 invariant requires more comparisons, indicating larger growth rate than the other invariants considered.

The invariants include A2, the Alexander polynomial, B1 (color-2 symmetric sl2 invariant), the Jones polynomial, and Khovanov’s t=1 specialization; the authors conjecture explicit exponential growth and a hierarchy among the bases γ(Q).