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Moscow Mathematical Journal

Volume 24, Issue 1, January–March 2024  pp. 125–140.

On Values of $\mathfrak{sl}_3$ Weight System on Chord Diagrams whose Intersection Graph is Complete Bipartite

Authors:  Zhuoke Yang (1)
Author institution:(1) Faculty of Mathematics, National Research University Higher School of Economics, Usacheva str., 6, 119048 Moscow, Russian Federation


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 the so-called $4$-term relations. Given a Lie algebra $\mathfrak{g}$ equipped with a nondegenerate 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 $K_{2,n}$.

2020 Math. Subj. Class. 57K16, 05C10.

Keywords: Weight system, chord diagram, Jacobi diagram, complete bipartite graph.

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