On the AJ bracket for tied links

Given a tied link $L$, the invariant $\langle\langle\cdot\rangle\rangle$ generalizes the Kauffman bracket of classical links. However, the analogues of Kauffman states and their relationship to this invariant are not immediately clear. We address this question by defining the Aicardi-Juyumaya states, and show that the contribution of each AJ-state to $\langle\langle\cdot\rangle\rangle$ does not depend on the chosen resolution tree. We also present an algorithm to compute the double bracket of a tied link diagram, and use it to find pairs of examples of (oriented) tied links sharing the same Homflypt polynomial but different tied Jones polynomial.