Tangle Equations, the Jones conjecture, slopes of surfaces in tangle complements, and q-deformed rationals

TL;DR

This paper explores the uniqueness of solutions to tangle equations, introduces a new ratio invariant related to the Jones polynomial, and connects these concepts to major conjectures in topology and knot theory.

Contribution

It proves the uniqueness of rational solutions to framed tangle equations, introduces the Kauffman bracket ratio as a topological invariant, and relates tangle systems to significant conjectures like the Jones Unknot conjecture.

Findings

Every framed tangle equation system has at most one rational solution.

The Kauffman bracket ratio is conjectured to represent slopes of certain surfaces in tangle complements.

For algebraic and rational tangles, the bracket ratio aligns with known q-rationals.

Abstract

We study systems of $2$-tangle equations which play an important role in the analysis of enzyme actions on DNA strands. We show that every system of framed tangle equations has at most one framed rational solution. Furthermore, we show that the Jones Unknot conjecture implies that if a system of tangle equations has a rational solution then that solution is unique among all $2$-tangles. This result potentially opens a door to a purely topological disproof of the Jones Unknot conjecture. We introduce the notion of the Kauffman bracket ratio $\{T\}_q\in \mathbb Q(q)$ of any $2$-tangle $T$ and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic $T$. We also prove that for rational $T$, the brackets $\{T\}_q$ coincide with the $q$-rationals of Morier-Genoud-Ovsienko. Additionally, we relate systems of tangle equations to the Cosmetic Surgery Conjecture and the Nugatory Crossing Conjecture.