Lissajous-toric knots

TL;DR

This paper introduces Lissajous-toric knots, explores their braid representations, and investigates their properties such as ribbon status, genus bounds, and triviality, expanding understanding of their topological characteristics.

Contribution

It defines a new class of knots called Lissajous-toric knots, analyzes their braid structures, and provides results on their genus bounds and conditions for triviality.

Findings

Lissajous-toric knots are ribbon when gcd(q,p)=1.

The braid representation $B_{N,q,p}$ is a power of a simpler braid when gcd(q,p)>1.

Upper bounds for the 4-genus of these knots are established.

Abstract

A point in the $(N,q)$-torus knot in $\mathbb{R}^3$ goes $q$ times along a vertical circle while this circle rotates $N$ times around the vertical axis. In the Lissajous-toric knot $K(N,q,p)$, the point goes along a vertical Lissajous curve (parametrized by $t\mapsto(\sin(qt+\phi),\cos(pt+\psi)))$ while this curve rotates $N$ times around the vertical axis. Such a knot has a natural braid representation $B_{N,q,p}$ which we investigate here. If $gcd(q,p)=1$, $K(N,q,p)$ is ribbon; if $gcd(q,p)=d>1$, $B_{N,q,p}$ is the $d$-th power of a braid which closes in a ribbon knot. We give an upper bound for the $4$-genus of $K(N,q,p)$ in the spirit of the genus of torus knots; we also give examples of $K(N,q,p)$'s which are trivial knots.