An algorithm to classify rational 3-tangles

TL;DR

This paper presents an algorithm for determining whether two rational 3-tangles are isotopic, utilizing a modified Dehn's method to classify simple closed curves on surfaces, advancing the classification techniques in knot theory.

Contribution

It introduces a novel algorithm that effectively checks isotopy of rational 3-tangles using a modified Dehn's approach, enhancing classification methods.

Findings

Algorithm successfully distinguishes non-isotopic rational 3-tangles.

Provides a practical method for classifying rational 3-tangles.

Advances the computational tools in knot and tangle theory.

Abstract

A $3$-$tangle$ $T$ is the disjoint union of $3$ properly embedded arcs in the unit 3-ball; it is called rational if there is a homeomorphism of pairs from $(B^3,T)$ to $(D^2\times I,\{x_1,x_2,x_3\}\times I)$. Two rational 3-tangles $T$ and $T'$ are isotopic if there is an orientation-preserving self-homeomorphism $h: (B^3, T)\rightarrow (B^3,T')$ that is the identity map on the boundary. In this paper, we give an algorithm to check whether or not two rational 3-tangles are isotopic by using a modified version of Dehn's method for classifying simple closed curves on surfaces.