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arXiv · 2510.02075

Explicit formulae for the Aicardi-Juyumaya bracket of tied links

O'Bryan Cárdenas-Andaur·Juan González-Meneses·Marithania Silvero
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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.

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O'Bryan Cárdenas-Andaur, Juan González-Meneses, Marithania Silvero. 2025-10-02. Explicit formulae for the Aicardi-Juyumaya bracket of tied links. https://arxiv.org/abs/2510.02075

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