Unimodality and bimodality of normalized Jones-polynomial defect polynomials

Prove that for every irreducible fraction a>1, the polynomial I_a(q), defined from the normalized Jones polynomial J_a(q) by I_a(q)=(J_a(q)^\vee-J_a(q))/(1-q) when J_a(q) is not palindromic, is unimodal except in the two listed families: polynomials 1+q^n for n\geq2, and polynomials whose coefficient sequence is (1,2,\ldots,k,k-1,k,k-1,k-2,\ldots,2,1) for some k\geq2; in particular, establish that I_a(q) is at most bimodal.

Background

The paper defines I_a(q) as a palindromic polynomial measuring the failure of the normalized Jones polynomial J_a(q) of the rational link associated with an irreducible fraction a>1 to be palindromic. The authors show that I_a(q) has nonnegative coefficients and constant term 1.

Conjecture 4.13 is derived from the preceding conjecture on traces of q-deformed modular-group matrices: the trace representation of I_a(q) transfers the expected unimodality behavior to normalized Jones polynomials of rational links, equivalently to left q-deformed rational numbers. The two exceptional forms correspond to specific matrix families, and the conjecture asserts the weaker general conclusion that I_a(q) is at most bimodal.