Salem-Boyd sequences and Hopf plumbing

TL;DR

This paper introduces a new construction called iterated plumbing for fibered links, analyzing the asymptotic behavior of roots of their characteristic polynomials, which relate to Salem and Boyd's work on special algebraic numbers.

Contribution

It defines a novel method to generate sequences of fibered links and explores their polynomial roots, linking topological link invariants with Salem and P-V number distributions.

Findings

Sequence of characteristic polynomials resembles Salem and Boyd's distributions

Asymptotic behavior of large roots is characterized

A new poset structure for Salem fibered links is proposed

Abstract

Given a fibered link, consider the characteristic polynomial of the monodromy restricted to first homology. This generalizes the notion of the Alexander polynomial of a knot. We define a construction, called iterated plumbing, to create a sequence of fibered links from a given one. The resulting sequence of characteristic polynomials has the same form as those arising in work of Salem and Boyd in their study of distributions of Salem and P-V numbers. From this we deduce information about the asymptotic behavior of the large roots of the generalized Alexander polynomials, and define a new poset structure for Salem fibered links.