Unimodality of the polynomial measuring non-palindromicity of rational-link Jones polynomials

Prove that, for every irreducible fraction α>1 with associated polynomial I_α(q)=(J_α(q)^V−J_α(q))/(1−q) when the normalized Jones polynomial J_α(q) is not palindromic, I_α(q) is unimodal except in the two specified families: 1+q^n for n≥2, and polynomials whose coefficient sequence is (1,2,…,k,k−1,k,k−1,k−2,…,2,1) for some k≥2; in particular, establish that I_α(q) is at most bimodal.

Background

For an irreducible fraction α>1, the paper associates a rational link and its normalized Jones polynomial J_α(q). When J_α(q) is not palindromic, the authors define I_α(q) as the uniquely normalized palindromic polynomial obtained by dividing J_α(q)^V−J_α(q) by 1−q. They prove that I_α(q) has nonnegative coefficients and constant term 1.

Using the trace classification in Remark 4.8, the authors explain that the unimodality conjecture for traces of q-deformed modular-group matrices would imply this conjecture for the polynomials I_α(q). Thus the problem concerns the coefficient-shape behavior of polynomials associated with rational-link Jones polynomials, with two explicitly listed exceptional types.