Geodesic
Integrable 3D statistical models on six-valent graphs
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 214-233.

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The paper is devoted to the study of a special statistical model on graphs with vertices of degrees $6$ and $1$. We show that this model is invariant with respect to certain Roseman moves if one regards the graph as the singular point set of the diagram of a $2$-knot. Our approach is based on the properties of the tetrahedron cohomology complex. 
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     author = {I. G. Korepanov and D. V. Talalaev and G. I. Sharygin},
     title = {Integrable {3D} statistical models on six-valent graphs},
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I. G. Korepanov; D. V. Talalaev; G. I. Sharygin. Integrable 3D statistical models on six-valent graphs. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 214-233. http://geodesic.mathdoc.fr/item/TM_2018_302_a9/

[1] Aschieri P., Cantini L., Jurčo B., “Nonabelian bundle gerbes, their differential geometry and gauge theory”, Commun. Math. Phys., 254:2 (2005), 367–400, arXiv: hep-th/0312154 | DOI | MR

[2] Baez J., Schreiber U., Higher gauge theory: 2-connections on 2-bundles, 2004, arXiv: hep-th/0412325

[3] Carter J. S., Elhamdadi M., Saito M., “Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles”, Fundam. math., 184 (2004), 31–54, arXiv: math/0206255 [math.GT] | DOI | MR

[4] Carter J. S., Jelsovsky D., Kamada S., Langford L., Saito M., “Quandle cohomology and state-sum invariants of knotted curves and surfaces”, Trans. Amer. Math. Soc., 355:10 (2003), 3947–3989 | DOI | MR

[5] Cattaneo A. S., Cotta-Ramusino P., Fröhlich J., Martellini M., “Topological $BF$ theories in 3 and 4 dimensions”, J. Math. Phys., 36:11 (1995), 6137–6160, arXiv: hep-th/9505027 | DOI | MR

[6] Hopfield J. J., “Neural networks and physical systems with emergent collective computational abilities”, Proc. Natl. Acad. Sci. USA, 79:8 (1982), 2554–2558 | DOI | MR

[7] I. G. Korepanov, “Geometric torsions and an Atiyah-style topological field theory”, Theor. Math. Phys., 158:3 (2009), 344–354 | DOI | DOI | MR | MR

[8] Korepanov I. G., Sharygin G. I., Talalaev D. V., “Cohomologies of $n$-simplex relations”, Math. Proc. Cambridge Philos. Soc., 161:2 (2016), 203–222 | DOI | MR

[9] Little W. A., “The existence of persistent states in the brain”, Math. Biosci., 19 (1974), 101–120 | DOI

[10] S. V. Matveev, “Distributive groupoids in knot theory”, Math. USSR, Sb., 119:1 (1984), 73–83 | DOI | MR

[11] Mnëv P., “Notes on simplicial BF theory”, Moscow Math. J., 9:2 (2009), 371–410 | MR

[12] Roseman D., “Reidemeister-type moves for surfaces in four-dimensional space”, Knot theory, Banach Cent. Publ., 42, Pol. Acad. Sci., Inst. Math., Warszawa, 1998, 347–380 | DOI | MR

[13] Talalaev D. V., “Zamolodchikov tetrahedral equation and higher Hamiltonians of $2d$ quantum integrable systems”, SIGMA, 13 (2017), 031 | MR

[14] Witten E., “Quantum field theory and the Jones polynomial”, Commun. Math. Phys., 121:3 (1989), 351–399 | DOI | MR

[15] A. B. Zamolodchikov, “Tetrahedra equations and integrable systems in three-dimensional space”, Sov. Phys. JETP, 52:2 (1980), 325–336 | MR

[16] Zeeman E. C., “Twisting spun knots”, Trans. Amer. Math. Soc., 115 (1965), 471–495 | DOI | MR