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arXiv · 2102.00888

On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite

Zhuoke Yang
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Abstract

Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord diagram encodes the order of double points along a singular knot. A Vassiliev invariant of order $n$ gives rise to a function on chord diagrams with $n$ chords. Such a function should satisfy some conditions in order to come from a Vassiliev invariant. A weight system is a function on chord diagrams that satisfies so-called 4-term relations. Given a Lie algebra $\mathfrak{g}$ equipped with a non-degenerate invariant bilinear form, one can construct a weight system with values in the center of the universal enveloping algebra $U(\mathfrak{g})$. In this paper, we calculate $\mathfrak{sl}_3$ weight system for chord diagram whose intersection graph is complete bipartite graph $K_{2,n}$.

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Zhuoke Yang. 2021-02-01. On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite. https://doi.org/10.17323/1609-4514-2024-24-1-125-140

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