On the classification of rational tangles

TL;DR

This paper presents two new combinatorial topological proofs for classifying rational tangles, introducing methods based on canonical forms and integral coloring, and proves related coloring conjectures.

Contribution

It provides novel proofs of rational tangle classification using canonical forms and coloring techniques, advancing understanding of their topological invariants.

Findings

Rational tangles are isotopic to canonical alternating forms.

The tangle fraction can be defined via canonical form and coloring.

The coloring method proves the Kauffman-Harary conjecture for rational knots and links.

Abstract

This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rational tangles). This paper forms the basis for a sequel on the classification of rational knots and links.