Additional cases of positive twisted torus knots

TL;DR

This paper extends previous work on twisted torus knots by identifying a four-parameter family of positive twisted torus knots and providing new examples with the same surface slope but different representatives.

Contribution

It introduces a new four-parameter family of positive twisted torus knots and finds additional examples with identical surface slopes but distinct representatives.

Findings

Existence of a four-parameter family of positive twisted torus knots.

Presence of twisted torus knots with the same surface slope but different representatives.

Extension of previous results on fibered and primitive/Seifert properties.

Abstract

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean's conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in $F$. In this paper, we extend Doleshal's results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in $F$.