Alternating Montesinos knots and Conjecture $\mathbb{Z}$

TL;DR

This paper proves that certain classes of knots, specifically alternating Montesinos knots with three tangles and all pretzel knots of the form P(p,q,r), satisfy Conjecture Z, linking knot properties to the Kervaire Conjecture.

Contribution

It establishes that these classes of knots have property Z, providing new evidence and understanding related to the Kervaire Conjecture in knot theory.

Findings

Alternating Montesinos knots with three tangles have property Z.

All pretzel knots P(p,q,r) have property Z.

Supports the conjecture's validity for these knot classes.

Abstract

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that all the pretzel knots of the form $P(p,q,r)$ (not necessarily alternating) have property $\mathbb{Z}$.