A classification of rational 3-tangles

TL;DR

This work extends Conway's rational $2$-tangle classification to rational $3$-tangles by developing an arc-system framework on the fixed six-punctured sphere $\\Sigma_{0,6}$, using Dehn's parametrization to encode bridge arcs. It introduces normal forms and normal jump moves, builds the bridge-arc complex and its normal subcomplex $\\mathcal{N}(T)$, and proves $\\mathcal{N}(T)$ is contractible, enabling a canonical representative via minimal normal coordinates. Classification of rational $3$-tangles up to isotopy is achieved by their Dehn parameters $(p_1,q_1,p_2,q_2,p_3,q_3)$ after normalization, providing a constructive geometric criterion in place of a simple invariant for $n\ge3$. The approach links arc-system parametrization with a deformation-theoretic view of the normal forms and yields an effective method for isotopy classification within this broader tangle family.

Abstract

In this paper, we define the \textit{normal form} and \textit{normal coordinate} of a rational 3-tangle $T$ with respect to $\partial E_1$, where $E_1$ is the fixed two punctured disk in $Σ_{0,6}$. Among all normal coordinates of $T$ with respect to $\partial E_1$, we investigate the collection of \textit{minimal} normal coordinates of $T$. We show that the simplicial complex constructed with normal forms of the rational 3-tangle is contractible. As an effectiveness of the contractibility of the simplicial complex by normal forms of $T$, we would choose a minimal normal coordinate of $T$ with a certain rule for the representative for the rational $3$-tangle $T$. This classifies rational $3$-tangles up to isotopy.

Figures (17)

• Figure 1: Two punctured disks $E_1, E_2$ and $E_3$
• Figure 2: Barycentric disk $E_1'$
• Figure 3: A jump move
• Figure 4: A diagram with $k_i^2$
• Figure 5: Outlets having $k_i^2$

Theorems & Definitions (35)

• Theorem 1: Special case II of Dehn's theorem
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof