On Rational Knots and Links in the Solid Torus

TL;DR

This paper introduces rational links in the solid torus, characterizes them via rational tangles and continued fractions, and develops new invariants within the Kauffman bracket skein module for colored diagrams.

Contribution

It defines rational links in the solid torus, establishes their characterization through rational tangles and continued fractions, and introduces a family of ambient isotopy invariants.

Findings

Rational links in the solid torus are fully characterized by rational tangles.

A new family of ambient isotopy invariants is constructed within the Kauffman bracket skein module.

The invariants apply to colored diagrams in an oriented surface.

Abstract

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.