Explicit formulae for the Aicardi-Juyumaya bracket of tied links

TL;DR

This work delivers the first closed-form state-sum expressions for the Aicardi-Juyumaya bracket on tied links by a careful, case-by-case analysis of AJ-states in 2- and 3-tied diagrams. The authors deploy resolution trees and a meticulous accounting of passable leaf types to derive explicit formulas that express $\\langle\\langle D \\\rangle\\rangle$ in terms of classical Kauffman brackets of sublinks and the functions $H_k$. The results provide concrete combinatorial tools toward categorifying the tied Jones polynomial and illuminate the structure of AJ-states, with potential extensions to tied Khovanov-type constructions. Overall, the paper advances the understanding of how partitions of components interact with the AJ-bracket and enables practical computations for low-tied cases.

Abstract

The double bracket $\langle \langle \cdot \rangle \rangle$ (also known as the AJ-bracket) is an invariant of framed tied links that extends the Kauffman bracket of classical links. Unlike the classical setting, little is known about the structure of AJ-states (analogous to classical Kauffman states) of a given tied link diagram, and no general state-sum formula for the AJ-bracket is currently available. In this paper we analyze the AJ-states of $2$- and $3$-tied link diagrams, and provide a complete description of their associated resolution trees leading to a computation of $\langle \langle \cdot \rangle \rangle$. As a result, we derive explicit state-sum formulas for the AJ-bracket. These are the first closed-form expressions of this kind, and they constitute a concrete step toward a combinatorial categorification of the tied Jones polynomial.

Figures (5)

• Figure 1: Local diagrams with colors $i$ (black) and $j$ (red) such that $i<j$.
• Figure 2: Resolution trees encoding axioms $(5)$ (left) and $(6)$ (right), where colors $i$ (black) and $j$ (red) satisfy $i<j$.
• Figure 3: The first steps in the construction of a resolution tree of $D$. Diagrams $D_5$ to $D_8$ are AJ-states, whereas $D_1$ to $D_4$ are not.
• Figure 4: A sketch of the resolution tree of $D$ used in the proof of \ref{['teobicolor']}.
• Figure 5:

Theorems & Definitions (22)

• Definition 1
• Theorem 2.1
• Remark 1
• Definition 2
• Remark 2
• Definition 3
• Remark 3
• Theorem 3.1
• proof
• Definition 4