Lissajous-toric knots
Marc Soret, Marina Ville

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 -torus knot in goes times along a vertical circle while this circle rotates times around the vertical axis. In the Lissajous-toric knot , the point goes along a vertical Lissajous curve (parametrized by while this curve rotates times around the vertical axis. Such a knot has a natural braid representation which we investigate here. If , is ribbon; if , is the -th power of a braid which closes in a ribbon knot. We give an upper bound for the -genus of in the spirit of the genus of torus knots; we also give examples of 's which are trivial knots.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory
