On the Aicardi-Juyumaya bracket for tied links

TL;DR

This paper introduces the Aicardi-Juyumaya states for tied links, providing a new way to compute the tied link invariant and demonstrating that different tied links can share the same Homflypt polynomial but differ in their tied Jones polynomial.

Contribution

It defines Aicardi-Juyumaya states, clarifies their role in the tied link invariant, and presents an algorithm for computing the double bracket, revealing new distinctions among tied links.

Findings

The contribution of each AJ-state is resolution-tree independent.

An algorithm for computing the tied link double bracket is provided.

Examples of tied links with identical Homflypt but different tied Jones polynomials are found.

Abstract

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.