Hoste’s conjecture for generalized Fibonacci polynomials

Abstract

One very long-standing theme in the theory of knots and links in S³ is the description of Alexander polynomials of alternating links. Hoste, based on computer verification, made the following conjecture: If $z\in \mathbb{C}$ is a root of the Alexander polynomial of an alternating knot, then $\mathfrak{R}e\hspace{0.17em}z>-1$. For 15 years Hoste’s problem has remained open for any moderately noteworthy subclass of alternating knots. We prove that for every sequence of polynomials $P_0=0,P_1=1$, $P_i(t)=a_{i}t{P}_{i-1}(t)+P_{i-2}(t)$ for integers a_i, the complex roots of $P_i(z^{1/2}-z^{-1/2})$ satisfy $\mathfrak{R}e\hspace{0.17em}z\hspace{0.17em}>-1$. This confirms the conjecture of Hoste for 2-bridge knots. The resolution even for 2-bridge knots requires us to bring up a substantial new technical framework. Our approach will be to work numerically and fundamentally rely on the capacity of MATHEMATICA. It draws on methods and motivation from several fields (it proves numerically an algebraic statement with knot theory background.)

Keywords:

• Fibonacci polynomial
• polynomial root
• alternating knot
• Alexander polynomial
• 2-bridge link

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F20
• 65D20
• 57M25
• 12D10
• 30A10 (secondary)