Explicit formulae for the Aicardi-Juyumaya bracket of tied links
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arXiv
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| Format: | Preprint |
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2025
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| author | Cárdenas-Andaur, O'Bryan González-Meneses, Juan Silvero, Marithania |
| author_facet | Cárdenas-Andaur, O'Bryan González-Meneses, Juan Silvero, Marithania |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02075 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit formulae for the Aicardi-Juyumaya bracket of tied links Cárdenas-Andaur, O'Bryan González-Meneses, Juan Silvero, Marithania Geometric Topology 57K10, 57K12, 57K14 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. |
| title | Explicit formulae for the Aicardi-Juyumaya bracket of tied links |
| topic | Geometric Topology 57K10, 57K12, 57K14 |
| url | https://arxiv.org/abs/2510.02075 |