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On an invariant for colored classical and singular links

Audrey Baumheckel, Carmen Caprau, Conor Righetti

TL;DR

The paper addresses invariants for colored links and their colored singular counterparts by formulating a graphical, skein-based framework. It extends the Aicardi-Juyumaya invariant $F$ to colored singular links via a new invariant $[\cdot]$ and establishes a robust set of skein relations, including moves $R4$ and $R5$, ensuring invariance. A central contribution is a state-sum model built from oriented colored 4-valent planar graphs that recovers $F$ on colored links and computes colored singular links through a graphical calculus. This work unifies knot and singular-link invariants under a graphical, color-aware formalism and provides practical computational tools while linking to the HOMFLY-PT polynomial in special cases.

Abstract

A colored link, as defined by Francesca Aicardi, is an oriented classical link together with a coloration, which is a function defined on the set of link components and whose image is a finite set of colors. An oriented classical link can be regarded as a colored link with its components colored with a sole color. Aicardi constructed an invariant $F(L)$ of colored links $L$ defined via skein relations. When the components of a colored link are colored with the same color or when the colored link is a knot, $F(L)$ is a specialization of the HOMFLY-PT polynomial. Aicardi also showed that $F(L)$ is a stronger invariant than the HOMFLY-PT polynomial when evaluated on colored links whose components have different colors. In this paper, we provide a state-sum model for the invariant $F(L)$ of colored links using a graphical calculus for oriented, colored, 4-valent planar graphs. We also extend $F(L)$ to an invariant of oriented colored singular links.

On an invariant for colored classical and singular links

TL;DR

The paper addresses invariants for colored links and their colored singular counterparts by formulating a graphical, skein-based framework. It extends the Aicardi-Juyumaya invariant to colored singular links via a new invariant and establishes a robust set of skein relations, including moves and , ensuring invariance. A central contribution is a state-sum model built from oriented colored 4-valent planar graphs that recovers on colored links and computes colored singular links through a graphical calculus. This work unifies knot and singular-link invariants under a graphical, color-aware formalism and provides practical computational tools while linking to the HOMFLY-PT polynomial in special cases.

Abstract

A colored link, as defined by Francesca Aicardi, is an oriented classical link together with a coloration, which is a function defined on the set of link components and whose image is a finite set of colors. An oriented classical link can be regarded as a colored link with its components colored with a sole color. Aicardi constructed an invariant of colored links defined via skein relations. When the components of a colored link are colored with the same color or when the colored link is a knot, is a specialization of the HOMFLY-PT polynomial. Aicardi also showed that is a stronger invariant than the HOMFLY-PT polynomial when evaluated on colored links whose components have different colors. In this paper, we provide a state-sum model for the invariant of colored links using a graphical calculus for oriented, colored, 4-valent planar graphs. We also extend to an invariant of oriented colored singular links.
Paper Structure (3 sections, 6 theorems, 40 equations, 2 figures)

This paper contains 3 sections, 6 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An extension of the Aicardi-Juyumaya invariant to colored singular links
  3. A state-sum model for the Aicardi-Juyumaya invariant for colored links

Key Result

Theorem 1

If $\tilde{D}_1$ and $\tilde{D}_2$ are diagrams representing ambient isotopic colored singular links, then $[\tilde{D}_1] = [\tilde{D}_2]$. That is, $[ \, \cdot \,]$ is an invariant for colored singular links.

Figures (2)

  • Figure 1: Extended Reidemeister moves
  • Figure 2: Examples of colored classical and singular links

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • ...and 4 more