On Fox's trapezoidal conjecture

TL;DR

This paper investigates Fox's trapezoidal conjecture for alternating links, proving it for certain classes of links and exploring related inequalities and extensions involving the Alexander polynomial and link signatures.

Contribution

It extends the validity of Fox's trapezoidal conjecture to diagrammatic Murasugi sums of special alternating links and twist-concentrated links, and analyzes related inequalities and conjectures.

Findings

Fox's conjecture holds for diagrammatic Murasugi sums with sums less than three

The conjecture is valid for links with large twist regions

The paper establishes bounds on the Alexander polynomial's stable length using signature

Abstract

We investigate Fox's trapezoidal conjecture for alternating links. We show that it holds for diagrammatic Murasugi sums of special alternating links, where all sums involved have length less than three (which includes diagrammatic plumbing). It also holds for links containing a large twist region, which we call twist-concentrated. Furthermore, we show some weaker inequalities between consecutive coefficients of the Alexander polynomial of an alternating 3-braid closure, and extend this to arbitrary alternating links. We then study an extension of the trapezoidal conjecture due to Hirasawa and Murasugi, which states that the stable length of the Alexander polynomial of an alternating link can be bounded from above using the signature. We estabilish this and determine when equality holds for diagrammatic Murasugi sums of special alternating knots where each sum has length less than three, and also for twist-concentrated 3-braids. Finally, we study the behavior of the Hirasawa-Murasugi inequality under concordance.