Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Aicardi-Juyumaya bracket for tied links

O'Bryan Cárdenas-Andaur

TL;DR

The paper introduces the Aicardi-Juyumaya bracket $\langle\langle\cdot\rangle\rangle$ for tied links, addressing the lack of a Kauffman-state framework in this setting by defining AJ-states. It proves that the AJ-state contribution to $\langle\langle D\rangle\rangle$ is independent of resolution trees and provides a termination-guaranteed resolution framework via a complexity measure. An explicit algorithm to compute $\langle\langle D\rangle\rangle$ is developed, and the authors demonstrate the approach with examples that separate tied links beyond what Homflypt or the classical Kauffman bracket detect, including refinements to the tied Jones polynomial. The results open avenues for potential categorifications of tied-link invariants and strengthen the toolbox for distinguishing tied links, with computational checks that both validate and challenge prior computations.

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.

On the Aicardi-Juyumaya bracket for tied links

TL;DR

The paper introduces the Aicardi-Juyumaya bracket for tied links, addressing the lack of a Kauffman-state framework in this setting by defining AJ-states. It proves that the AJ-state contribution to is independent of resolution trees and provides a termination-guaranteed resolution framework via a complexity measure. An explicit algorithm to compute is developed, and the authors demonstrate the approach with examples that separate tied links beyond what Homflypt or the classical Kauffman bracket detect, including refinements to the tied Jones polynomial. The results open avenues for potential categorifications of tied-link invariants and strengthen the toolbox for distinguishing tied links, with computational checks that both validate and challenge prior computations.

Abstract

Given a tied link , the invariant 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 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.

Paper Structure

This paper contains 10 sections, 3 theorems, 11 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tied links and Aicardi-Juyumaya bracket
  3. Tied links
  4. Aicardi-Juyumaya bracket
  5. AJ-states and computation of $\langle\langle\cdot\rangle\rangle$
  6. Complexity of tied link diagrams
  7. Contribution of an $AJ$-state to $\langle\langle D\rangle\rangle$
  8. A regular isotopy invariant
  9. Algorithm to compute $\langle\langle\cdot\rangle\rangle$
  10. Some key examples and explicit calculation of $\langle\langle\cdot\rangle\rangle$

Key Result

Theorem 2.3

There exists a unique function defined by the following axioms: Here, $\delta=A+A^{-1}$ and, $D_+, D_-, D_0, D_{\infty}$ and $D_{+,\sim}$ are diagrams differing in the neighborhood of a given crossing, as indicated in Figure multidiagram, where we distinguish whether the two arcs' components belong to the same block (same colors), or to different blocks (differe

Figures (7)

  • Figure 1: Examples of tied link diagrams.
  • Figure 2: Two interpretations of the same tied link diagram.
  • Figure 3: Local diagrams illustrating axioms of Theorem \ref{['invariante']}.
  • Figure 4: Types of smoothings, asumming color black and red represent colors $i$ and $j$, with $i<j$.
  • Figure 5: Diagrams $D_5$ to $D_8$ are AJ-states but $D_1$ to $D_4$ are not.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Definition 3.5
  • ...and 2 more