Lagrangian Subspaces, Delta-Matroids, and Four-Term Relations

V. I. Zhukov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 2, pp. 15-24.

Résumé

Finite-order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams (embedded graphs with a single vertex) satisfying the four-term relations. Weight systems have graph analogues, the so-called 4-invariants of graphs, i.e., functions on graphs that satisfy the four-term relations for graphs. Each 4-invariant determines a weight system. The notion of a weight system is naturally generalized to the case of embedded graphs with an arbitrary number of vertices. Such embedded graphs correspond to links; to each component of a link there corresponds a vertex of an embedded graph. Recently, two approaches have been suggested to extend the notion of 4-invariants of graphs to the case of combinatorial structures corresponding to embedded graphs with an arbitrary number of vertices. The first approach is due to V. Kleptsyn and E. Smirnov, who considered functions on Lagrangian subspaces in a $2n$-dimensional space over $F_2$ endowed with a standard symplectic form and introduced four-term relations for them. The second approach, due to V. Zhukov and S. Lando, gives four-term relations for functions on binary delta-matroids.

Détail

Keywords: Vassiliev invariants, weight system, chord diagrams, symplectic spaces, Lagrangian subspaces, binary delta-matroids, Hopf algebra, embedded graphs.
Mots-clés : 4-invariants

Bibliographie

[1] R. F. Booth, A. V. Borovik, I. M. Gelfand, D. A. Stone, “Lagrangian matroids and cohomology”, Ann. Comb., 4:2 (2000), 171–182 | DOI | MR | Zbl

[2] A. Bouchet, “Greedy algorithm and symmetric matroids”, Math. Programm., 38:2 (1987), 147–159 | DOI | MR | Zbl

[3] A. Bouchet, “Representability of $\Delta$-matroids”, Combinatorics (Eger, 1987), Colloq. Math. Soc. János Bolyai, 52, 1988, 167–182 | MR | Zbl

[4] A. Bouchet, “Maps and delta-matroids”, Discrete Math., 78:1–2 (1989), 59–71 | DOI | MR | Zbl

[5] S. Chmutov, “Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial”, J. Combin. Theory Ser. B, 99:3 (2009), 617–638 | DOI | MR | Zbl

[6] C. Chun, I. Moffatt, S. D. Noble, R. Rueckriemen, Matroids, delta-matroids and embedded graphs, arXiv: 1403.0920

[7] V. Kleptsyn, E. Smirnov, “Ribbon graphs and bialgebra of Lagrangian subspaces”, J. Knot Theory Ramifications, 25:12 (2016), 1642006 | DOI | MR | Zbl

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

[9] S. K. Lando, “$J$-invarianty ornamentov i osnaschennye khordovye diagrammy”, Funkts. analiz i ego pril., 40:1 (2006), 1–13 | DOI | MR | Zbl

[10] S. Lando, V. Zhukov, Delta-matroids and Vassiliev invariants, arXiv: 1602.00027 | MR

[11] S. Lando, A. Zvonkin, Graphs on surfaces and their applications, Springer-Verlag, Berlin, 2004 | MR | Zbl

[12] G. Malic, An action of the Coxeter group $BC_n$ on maps on surfaces, Lagrangian matroids and their representations, arXiv: 1507.01957

[13] I. Moffatt, E. Mphako-Banda, Handle slides for delta-matroids, arXiv: 1510.07224 | MR

[14] 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 | MR