Unimodality of the normalized Jones-derived polynomial I_a(q)

Prove that for every irreducible fraction a>1, the polynomial I_a(q), defined from the normalized Jones polynomial J_a(q) of the associated rational link by I_a(q)=(J_a(q)^\vee-J_a(q))/(1-q) when J_a(q) is not palindromic, is unimodal except for the two listed types: 1+q^n for n≥2, and coefficient sequences of the form (1,2,...,k,k−1,k,k−1,k−2,...,2,1) for k≥2; in particular, establish that I_a(q) is at most bimodal.

Background

For an irreducible fraction a>1, the paper associates a rational link and its normalized Jones polynomial J_a(q). When J_a(q) is not palindromic, the authors define I_a(q) as the normalized difference between its reciprocal polynomial and itself. They prove that I_a(q) has nonnegative coefficients and constant term 1.

The conjecture is derived from the circular-fence trace conjecture: the authors explain that Conjecture 4.9 would imply the asserted behavior of I_a(q). The proposed conclusion allows two explicit exceptional families and states the weaker general consequence that I_a(q) is at most bimodal.