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arXiv · 2401.11073

On an invariant for colored classical and singular links

Audrey Baumheckel·Carmen Caprau·Conor Righetti
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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.

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Audrey Baumheckel, Carmen Caprau, Conor Righetti. 2024-01-20. On an invariant for colored classical and singular links. https://arxiv.org/abs/2401.11073

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