Uniqueness of reduced alternating rational 3-tangles

TL;DR

This paper introduces a Kauffman bracket-based invariant for rational n-tangles, proving its effectiveness for classifying rational 2-tangles and reduced alternating rational 3-tangles, with a conjecture extending this to all rational 3-tangles.

Contribution

It presents a new invariant derived from the Kauffman bracket that classifies certain rational tangles, advancing understanding in tangle theory.

Findings

Invariant classifies rational 2-tangles.

Invariant classifies reduced alternating rational 3-tangles.

Conjecture: invariant classifies all rational 3-tangles.

Abstract

In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.