Geodesic
Values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs
Sbornik. Mathematics, Tome 214 (2023) no. 7, pp. 934-951 Cet article a éte moissonné depuis la source Math-Net.Ru

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A weight system is a function on the chord diagrams that satisfies Vassiliev's $4$-term relation. Using the Lie algebra $\mathfrak{sl}_2$ we can construct the simplest nontrivial weight system. The resulting $\mathfrak{sl}_2$ weight system takes values in the space of polynomials of one variable and is completely determined by the Chmutov-Varchenko recurrence relations. Although the definition of the $\mathfrak{sl}_2$ weight system is rather simple, calculations of its values are laborious, and therefore concrete values are only known for a small number of chord diagrams. As concerns the explicit form of values at chord diagrams with complete intersection graphs, Lando stated a conjecture, which initially could only be proved for the coefficients at linear terms of the values of the weight system. We prove this conjecture in full using the Chmutov-Varchenko recurrence relations and the linear operators we introduce for adding a chord to a share, which is the subset of chords of the diagram with endpoints on two selected arcs. Also, relying on the generating function of the values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs, we prove that the quotient space of shares modulo the recurrence relations is isomorphic to the space of polynomials in two variables. Bibliography: 10 titles.
Keywords: chord diagram, $\mathfrak{sl}_2$ weight system, complete graph, share of a chord diagram.
Mots-clés : $4$-term relations 
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P. E. Zakorko. Values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs. Sbornik. Mathematics, Tome 214 (2023) no. 7, pp. 934-951. http://geodesic.mathdoc.fr/item/SM_2023_214_7_a2/

[1] V. A. Vassiliev, “Cohomology of knot spaces”, Theory of singularities and its applications, Adv. Soviet Math., 1, Amer. Math. Soc., Providence, RI, 1990, 23–69 | DOI | MR | Zbl

[2] M. Kontsevich, “Vassiliev's knot invariants”, I. M. Gel'fand seminar, Part 2, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993, 137–150 | DOI | MR | Zbl

[3] D. Bar-Natan, “On the Vassiliev knot invariants”, Topology, 34:2 (1995), 423–472 | DOI | MR | Zbl

[4] S. V. Chmutov and S. K. Lando, “Mutant knots and intersection graphs”, Algebr. Geom. Topol., 7:3 (2007), 1579–1598 | DOI | MR | Zbl

[5] E. Krasilnikov, “An extension of the $\mathfrak{sl}_2$ weight system to graphs with $n\le 8$ vertices”, Arnold Math. J., 7:4 (2021), 609–618 | DOI | MR | Zbl

[6] P. A. Filippova (Zinova), “Values of the $\mathfrak{sl}_2$ weight system on a family of graphs that are not the intersection graphs of chord diagrams”, Sb. Math., 213:2 (2022), 235–267 | DOI | MR | Zbl

[7] S. V. Chmutov and A. N. Varchenko, “Remarks on the Vassiliev knot invariants coming from $\mathfrak{sl}_2$”, Topology, 36:1 (1997), 153–178 | DOI | MR | Zbl

[8] A. Bigeni, “A generalization of the Kreweras triangle through the universal $\mathfrak{sl}_2$ weight system”, J. Combin. Theory Ser. A, 161 (2019), 309–326 | DOI | MR | Zbl

[9] S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev knot invariants, Cambridge Univ. Press, Cambridge, 2012, xvi+504 pp. | DOI | MR | Zbl

[10] P. Flajolet, “Combinatorial aspects of continued fractions”, Discrete Math., 32:2 (1980), 125–161 | DOI | MR | Zbl