Geodesic
Parity, free knots, groups, and invariants of finite type
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 72 (2011) no. 2, pp. 207-222.

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In this paper, on the basis of the notion of parity introduced recently by the author, for each positive integer $m$ we construct invariants of long virtual knots with values in some simply defined group $\mathcal G_m$; conjugacy classes of this group play a role as invariants of compact virtual knots. By construction, each of the invariants is unaltered by the move of virtualization. Factorization of the group algebra of the corresponding group leads to invariants of finite order of (long) virtual knots that do not change under virtualization. The central notion used in the construction of the invariants is parity: the crossings of diagrams of free knots is endowed with an additional structure — each crossing is declared to be even or odd, where even crossings behave regularly under Reidemeister moves. 
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V. O. Manturov. Parity, free knots, groups, and invariants of finite type. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 72 (2011) no. 2, pp. 207-222. http://geodesic.mathdoc.fr/item/MMO_2011_72_2_a1/

[Af] Afanasev D. M., “Ob usilenii invariantov virtualnykh uzlov s pomoschyu chetnosti”, Matem. sb., 201:6 (2010), 3–18 | MR

[AM] Andersen J. E., Mattes J., Configuration space integrals and universal Vassiliev invariants over closed surfaces, arXiv: q-alg/9704019

[BL] Birman J., Lin X-S., “Knot Polynomials and Vassiliev's Invariants”, Inventiones Mathematicae, 111 (1993), 225–270 | DOI | MR | Zbl

[Chr] Chrisman M., “Twist Lattices and the Jones–Kauffman Polynomial for Long Virtual Knots”, J. Knot Theory Ramif., 19:5 (2010), 644–675 | MR

[ChM] Chrisman M., Manturov V. O., Combinatorial Formulae for Finite-Type Invariants via Parities, 2010, arXiv: 1002.0539 [math.GT]

[FKM] Fenn R. A., Kauffman L. H., Manturov V. O., “Virtual Knots: Unsolved Problems”, Fund. Math., 2005, no. 188, 293–323 | DOI | MR | Zbl

[Gib] Gibson A., Homotopy invariants of Gauss words, arXiv: 0902.0062 [math.GT]

[Gib2] Gibson A., Finite type invariants of nanowords and nanofrases, arXiv: 1007.1693 [math.GT]

[GPV] Goussarov M., Polyak M., Viro O., “Finite type invariants of classical and virtual knots”, Topology, 39 (2000), 1045–1068 | DOI | MR | Zbl

[Гус] Gusarov M., “Novaya forma polinoma Dzhonsa–Konveya dlya orientirovannykh zatseplenii”, Geometriya i topologiya, v. 1, Zap. nauchn. sem. LOMI, 193, 1991, 4–9 | MR | Zbl

[Ka] Kauffman L. H., “Virtual Knot Theory”, Eur. J. Combinatorics, 20:7 (1999), 662–690 | DOI | MR

[Kon] Kontsevich M., “Vassiliev's Knot Invariants”, Adv. Sov. Math., 16:2 (1993), 137–150 | MR | Zbl

[KrM] Kronheimer P., Mrowka T., Khovanov Homology is an Unknot Detector, arXiv: 1005.4346 [math.GT]

[Ma] Manturov V. O., “Chetnost v teorii uzlov”, Mat. sb., 201:5 (2010), 65–110 | MR | Zbl

[Ma2] Manturov V. O., “O dlinnykh virtualnykh uzlakh”, DAN, 401:5 (2005), 595–598 | MR

[Ma3] Manturov V. O., Teoriya uzlov, RKhD, M.; Izhevsk, 2005, 512 pp.

[Ma4] Manturov V. O., “Gomologii Khovanova dlya virtualnykh uzlov s proizvolnymi koeffitsientami”, Izvestiya RAN. Ser. matem., 71:5 (2007), 111–148 | MR | Zbl

[Ma5] Manturov V. O., “Chetnost i kobordizmy svobodnykh uzlov”, Matem. sb., 203:2 (2012), 45–76

[MM] Manturov V. O., Manturov O. V., “Svobodnye uzly i gruppy”, DAN, 434:1 (2010), 1–4

[MM2] Manturov O. V., Manturov V. O., “Free Knots and Groups”, J. Knot Theory Ramif., 19:2 (2010), 181–186 ; arXiv: 0912.2694 [math.GT] | DOI | MR | Zbl

[Tu] Turaev V. G., “Topology of words”, Proc. Lond. Math. Soc. (3), 95:2 (2007), 360–412 | DOI | MR | Zbl

[Vas] Vassiliev V. A., “Cohomology of knot spaces”, Theory of singularities and its applications, Adv. Sov. Math., 1, 1990, 23–70 | MR