Ribbonlength of twisted torus knots

TL;DR

This paper establishes upper bounds for the ribbonlength of twisted torus knots, relating it to their parameters, which advances understanding of their geometric properties.

Contribution

It provides new upper bounds for the ribbonlength of twisted torus knots based on their defining parameters.

Findings

Ribbonlength of $T_{p,q;r,s}$ is at most $2( ext{max}igracevert p,q,r igracevert +|s|r)$.

If $r \

The bounds depend on the parameters $p, q, r, s$, with a tighter bound when $r \

Abstract

The ribbonlength Rib$(K)$ of a knot $K$ is the infimum of the ratio of the length of any flat knotted ribbon with core $K$ to its width. A twisted torus knot $T_{p,q;r,s}$ is obtained from the torus knot $T_{p,q}$ by twisting $r$ adjacent strands $s$ full twists. In this paper, we show that the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(\max \{ p, q, r \} +|s|r)$ where $p$ and $q$ are positive. Furthermore, if $r \leq p-q$, then the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(p+(|s|-1)r)$.