The representativity of pretzel knots

TL;DR

This paper characterizes when certain pretzel knots have a specific property called representativity 3, and shows that large algebraic knots have representativity at most 3, contributing to knot theory classification.

Contribution

It provides a complete characterization of pretzel knots with representativity 3 and bounds the representativity of large algebraic knots, advancing understanding in knot invariants.

Findings

Pretzel knots with parameters ±(-2,3,3) or ±(-2,3,5) have representativity 3.

Large algebraic knots have representativity ≤ 3.

The results classify knots based on their representativity values.

Abstract

In the present paper, we will show that a $(p,q,r)$-pretzel knot has the representativity 3 if and only if $(p,q,r)$ is either $\pm(-2,3,3)$ or $\pm(-2,3,5)$. We also show that a large algebraic knot has the representativity less than or equal to 3.