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Operator invariants of tangles, and R-matrices. (English. Russian original) Zbl 0707.57003
Math. USSR, Izv. 35, No. 2, 411-444 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1073-1107 (1989).
Let R be an R-matrix, that is a matrix which satisfies the (quantum) Yang-Baxter equation. Then there is a natural homomorphism from a braid group \(B_ n\) into \(Aut(V^{\otimes n})\) induced by R. The goal of the paper is to generalize the above construction to tangles and to construct operator invariants of tangles (morphisms from the category of tangles into a category of modules); compare also D. Yetter [Contemp. Math. 78, 705-730 (1988; Zbl 0665.57004)] and P. Freyd and D. Yetter [Adv. Math. 77, 156-182 (1989; Zbl 0679.57003)]. In particular the author constructs in that way the Jones-Conway (called also skein or Homfly) polynomial, the Kauffman polynomial and finds skein modules of a tangle [compare H. Morton and P. Traczyk, “Knots, skeins and algebras”, preprint (1987)]. The author extends his approach to colored tangles and further to colored thickened graphs (i.e. a graph is considered together with a surface which can be contracted onto the graph). The method of the paper has been used in the author’s sequel paper [Zap. Nauchn. Semin. LOMI 167, 79-89 (1988; Zbl 0673.57004)] to find the Conway and Kauffman skein modules of a solid torus [see also J. Hoste and M. Kidwell, Trans. Am. Math. Soc. 321, No.1, 197-229 (1990; Zbl 0702.57002)]. The last result has been generalized by the reviewer to any surface cross interval [“Skein module of links in a handlebody”, Proc. OSU Research Seminar (to appear)]. On the other hand the refinement of the method used in the reviewed paper (involving Hopf algebras) lead to the paper by the author and N. Reshetikhin [Commun. Math. Phys. 127, No.1, 1-26 (1990)] and finally to new invariants of 3-manifolds (conjectured by E. Witten) in the paper by the author and N. Reshetikhin: “Invariants of 3-manifolds via link polynomials and quantum groups” [Invent. Math. (to appear)].
Reviewer: J.H.Przytycki

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
57N10 Topology of general \(3\)-manifolds (MSC2010)
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