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

Summary:

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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