Real-part conjecture for roots of alternating-knot Q-polynomials

Prove that, for every alternating knot K, every root of its Q-polynomial Q_K(x) has real part less than 1.

Background

The paper reports that numerical computations for alternating prime knots with at most 13 crossings found all roots of Q_K(x) to have real part less than 1, whereas roots with real part greater than 1 occur for some non-alternating prime knots in the same crossing range.

This computational observation motivates the stated conjecture, which proposes a universal real-part bound for the roots of Q-polynomials of alternating knots. The conjecture is presented as analogous to Hoste’s conjecture for Alexander polynomials, although that analogous conjecture was subsequently disproved.