Geodesic
Values of the $\mathfrak{sl}_2$ weight system on a family of graphs that are not the intersection graphs of chord diagrams
Sbornik. Mathematics, Tome 213 (2022) no. 2, pp. 235-267 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Chmutov-Lando theorem claims that the value of a weight system (a function on the chord diagrams that satisfies the four-term Vassiliev relations) corresponding to the Lie algebra $\mathfrak{sl}_2$ depends only on the intersection graph of the chord diagram. We compute the values of the $\mathfrak{sl}_2$ weight system at the graphs in several infinite series, which are the joins of a graph with a small number of vertices and a discrete graph. In particular, we calculate these values for a series in which the initial graph is the cycle on five vertices; the graphs in this series, apart from the initial one, are not intersection graphs. We also derive a formula for the generating functions of the projections of graphs equal to the joins of an arbitrary graph and a discrete graph to the subspace of primitive elements of the Hopf algebra of graphs. Using the formula thus obtained, we calculate the values of the $\mathfrak{sl}_2$ weight system at projections of the graphs of the indicated form onto the subspace of primitive elements. Our calculations confirm Lando's conjecture concerning the values of the $\mathfrak{sl}_2$ weight system at projections onto the subspace of primitives. Bibliography: 17 titles.
Keywords: chord diagram, $\mathfrak{sl}_2$ weight system, intersection graph, join of graphs, Hopf algebra. 
@article{SM_2022_213_2_a4,
     author = {P. A. Filippova},
     title = {Values of the $\mathfrak{sl}_2$ weight system on a~family of graphs that are not the intersection graphs of chord diagrams},
     journal = {Sbornik. Mathematics},
     pages = {235--267},
     year = {2022},
     volume = {213},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2022_213_2_a4/}
}
TY  - JOUR
AU  - P. A. Filippova
TI  - Values of the $\mathfrak{sl}_2$ weight system on a family of graphs that are not the intersection graphs of chord diagrams
JO  - Sbornik. Mathematics
PY  - 2022
SP  - 235
EP  - 267
VL  - 213
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_2022_213_2_a4/
LA  - en
ID  - SM_2022_213_2_a4
ER  - 
%0 Journal Article
%A P. A. Filippova
%T Values of the $\mathfrak{sl}_2$ weight system on a family of graphs that are not the intersection graphs of chord diagrams
%J Sbornik. Mathematics
%D 2022
%P 235-267
%V 213
%N 2
%U http://geodesic.mathdoc.fr/item/SM_2022_213_2_a4/
%G en
%F SM_2022_213_2_a4
P. A. Filippova. Values of the $\mathfrak{sl}_2$ weight system on a family of graphs that are not the intersection graphs of chord diagrams. Sbornik. Mathematics, Tome 213 (2022) no. 2, pp. 235-267. http://geodesic.mathdoc.fr/item/SM_2022_213_2_a4/

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

[2] 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

[3] A. Bouchet, “Circle graph obstructions”, J. Combin. Theory Ser. B, 60:1 (1994), 107–144 | DOI | MR | Zbl

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

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

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

[7] P. A. Filippova, “Values of the $\mathfrak{sl}_2$ weight system on complete bipartite graphs”, Funct. Anal. Appl., 54:3 (2020), 208–223 | DOI | DOI | MR | Zbl

[8] S. A. Joni, G.-C. Rota, “Coalgebras and bialgebras in combinatorics”, Stud. Appl. Math., 61:2 (1979), 93–139 | DOI | MR | Zbl

[9] 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

[10] E. Kulakova, S. Lando, T. Mukhutdinova, G. Rybnikov, “On a weight system conjecturally related to $\mathfrak{sl}_2$”, European J. Combin., 41 (2014), 266–277 | DOI | MR | Zbl

[11] S. K. Lando, “On a Hopf algebra in graph theory”, J. Combin. Theory Ser. B, 80:1 (2000), 104–121 | DOI | MR | Zbl

[12] S. K. Lando, “On primitive elements in the bialgebra of chord diagrams”, Topics in singularity theory, v. 1, Amer. Math. Soc. Transl. Ser. 2, 180, Adv. Math. Sci., 34, Amer. Math. Soc., Providence, RI, 1997, 167–174 | DOI | MR | Zbl

[13] S. K. Lando, A. K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia Math. Sci., 141, Low-Dimensional Topology, II, Springer-Verlag, Berlin, 2004, xvi+455 pp. | DOI | MR | Zbl

[14] S. Lando, V. Zhukov, “Delta-matroids and Vassiliev invariants”, Mosc. Math. J., 17:4 (2017), 741–755 | DOI | MR | Zbl

[15] J. W. Milnor, J. C. Moore, “On the structure of Hopf algebras”, Ann. of Math. (2), 81:2 (1965), 211–264 | DOI | MR | Zbl

[16] W. R. Schmitt, “Incidence Hopf algebras”, J. Pure Appl. Algebra, 96:3 (1994), 299–330 | DOI | MR | Zbl

[17] V. 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