Exponential growth of maximal coefficient size for quantum invariants across prime knots

Prove that for each quantum invariant Q in {A2, Alexander, B1 (the sl2 symmetric-color-2 invariant), Khovanov}—and hence also for Jones or KhovanovT1—the maximum coefficient magnitude coeff_n of Q over all prime knots with n crossings grows exponentially, i.e., coeff_n ≥ c·y^n for some y > 1 and constant c. In particular, verify that the same exponential growth applies to the maximum absolute sum of coefficients ev_n, and compare growth rates across invariants (with B1 expected to have the largest base among A2, Alexander, Jones, and KhovanovT1).

Background

Section 13G studies how the sizes of coefficients of knot polynomials grow with crossing number. The quantities coeff_n (maximum absolute coefficient) and ev_n (maximum absolute sum of coefficients) are examined empirically for several invariants and appear to grow roughly exponentially.

Based on these observations, the authors conjecture exponential growth uniformly across invariants and suggest a comparative hierarchy of the growth rates, with B1 exhibiting the fastest growth.