Positivity of coefficients in h(y) for positive 3-braids (Conjecture RH)

Prove that, for any positive word w ∈ Br_3 whose closure is a knot and with length #w=2m, the coefficients b_k of the polynomial h(y)=Re(i^{−m} f(i y)) are all strictly positive for 0≤k≤m/2, where f(x) is the real monic degree-m polynomial defined by t^{−m} det(B_t(w)−I)=f(t+1/t).

Background

To analyze roots in the right half-plane, the authors recast det(B_t(w)−I) into a symmetric polynomial f(t+1/t) and then define h(y) via a rotation to imaginary arguments. The positivity of h’s coefficients would imply strong root-location constraints, notably that all roots with Re(t)>0 lie on the unit circle for such knots.

This conjecture is motivated by extensive computational evidence and is used as an assumption in propositions deriving root-free regions and unit-circle localization under positive braid conditions.