Positive oriented Thompson links

TL;DR

This paper shows that positivity for elements of the oriented Thompson group $\\vec{F}$ is compatible with positive link diagrams: for any $g \in \\vec{F}_+$ the oriented link $\\vec{\\mathcal{L}}(g)$ admits a positive diagram. The authors exploit the isomorphism $\\vec{F} \cong F_3$ via the map $\\alpha$, converting ternary trees to binary trees and performing local Reidemeister-type moves to remove negative crossings while preserving upper-tangle positivity; the crossing count is bounded by the number of right leaves of the transformed top tree. They establish an explicit unknotting-number bound for $\\vec{\\mathcal{L}}(g)$ by a seven-move reduction on 4-valent tree diagrams and show optimality through explicit examples, including chain-link families arising from $\\prod_{i}\\varphi^{7i}(g)$. Together these results clarify how positive elements in $\\vec{F}$ yield positive oriented Thompsons links and provide constructive controls on diagram complexity and unknotting.

Abstract

We prove that the links associated with positive elements of the oriented subgroup of the Thompson group are positive.

Figures (16)

• Figure 1: The $4$-regular rooted tree $T$ and its binary transform $\alpha(T)$.
• Figure 2: Positive oriented Thompson link associated with $T$, and its knotted component $5_2$.
• Figure 3: The generators of $F$.
• Figure 4: A pair of opposing carets and a pair of trees equivalent to $x_0$.
• Figure 5: A positive and a negative crossing.

Theorems & Definitions (22)

• Theorem 1
• Remark 1
• Remark 2
• Lemma 1
• proof
• Proposition 1
• Definition 1
• Example 1
• Example 2
• Example 3