Non-integral boundary slopes of alternating knots

TL;DR

This paper demonstrates that for every positive integer, there exists an alternating knot with a boundary slope having that denominator, expanding understanding of boundary slopes in knot theory.

Contribution

It introduces a method to construct alternating knots with boundary slopes of any given denominator using Kabaya's approach and layered solid torus techniques.

Findings

For each positive integer n, an alternating knot with boundary slope denominator n exists.

Utilizes Kabaya's method combined with layered solid torus construction.

Provides a systematic way to realize all possible boundary slope denominators.

Abstract

We show, for every positive integer $n$, there is an alternating knot having a boundary slope with denominator $n$. We make use of Kabaya's method for boundary slopes and the layered solid torus construction introduced by Jaco and Rubinstein and further developed by Howie et al.