2025/10/02 by Cárdenas-Andaur, O'Bryan, González-Meneses, Juan, Silvero, Marithania
#57K10 #57K12 #57K14 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2510.02075
The double bracket ⟨ ⟨ ⋅ ⟩ ⟩ (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 ⟨ ⟨ ⋅ ⟩ ⟩. 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.