On Coxeter mapping classes and fibered alternating links

TL;DR

This paper explores the relationship between Coxeter transformations and fibered alternating links, providing new insights into their Alexander polynomial zeros, group properties, and monodromy dilatations, with implications for several conjectures.

Contribution

It establishes a connection between Coxeter transformations and fibered alternating links, leading to new results on Alexander polynomial zeros, group bi-orderability, and monodromy dilatation bounds.

Findings

Supports a strong case of Hoste's conjecture.

Confirms the trapezoidal conjecture for these links.

Provides a sharp lower bound for homological dilatation.

Abstract

Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a strong case of Hoste's conjecture, the trapezoidal conjecture, bi-orderability of the link group, and a sharp lower bound for the homological dilatation of the monodromy of the fibration. The results extend to more general hyperbolic fibered 3-manifolds associated to alternating-sign Coxeter graphs.