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Virtual tangles and fiber functors

    https://doi.org/10.1142/S0218216519500445Cited by:1 (Source: Crossref)

    We define a category v𝒯 of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to v𝒯 which induces an equivalence of categories. On the other hand, we show that v𝒯 is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

    AMSC: 17B37, 57M27

    References

    • 1. D. Bar-Natan, Non-associative tangles, in Geometric Topology (Athens, GA, 1993), AMS/IP Studies in Advanced Mathematics, Vol. 2 (American Mathematical Society, Providence, RI, 1997), pp. 139–183. Google Scholar
    • 2. D. Bar-Natan, On associators and the Grothendieck-Teichmuller group. I, Selecta Math. (N.S.) 4(2) (1998) 183–212. Google Scholar
    • 3. D. Bar-Natan, Facts and dreams about v-knots and Etingof-Kazhdan (2011), https://www.math.toronto.edu/drorbn/Talks/SwissKnots-1105/index.html. Google Scholar
    • 4. D. Bar-Natan and Z. Dancso, Homomorphic expansions for knotted trivalent graphs, J. Knot Theory Ramifications 22(1) (2013) 1250137. Link, Web of ScienceGoogle Scholar
    • 5. D. Bar-Natan and Z. Dancso, Finite type invariants of w-knotted objects II: Tangles, Foams and the Kashiwara-Vergne problem, preprint (2014), arXiv:1405.1955. Google Scholar
    • 6. P. Cartier, Construction combinatoire des invariants de Vassiliev-Kontsevich des nœuds, Prépublicationde l’ Institut Recherche Mathematique Avancée, in R.C.P. 25, Vol. 45 (French) (Strasbourg, 1992–1993), Vol. 1993/42 (University Louis Pasteur, Strasbourg, 1993), pp. 1–10. Google Scholar
    • 7. B. A. Cisneros de la Cruz, Virtual braids from a topological viewpoint, J. Knot Theory Ramifications 24(6) (2015) 1550033. Link, Web of ScienceGoogle Scholar
    • 8. V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q¯/Q), Leningrad Math. J. 2(4) (1990) 829–860. Google Scholar
    • 9. P. Etingof and D. Kazhdan, Quantization of Lie bialgebras, I. Selecta Math. (N.S.) 2(1) (1996) 1–41. Google Scholar
    • 10. L. H. Kauffman, Virtual knot theory, European J. Combin. 20(7) (1999) 663–691. Web of ScienceGoogle Scholar
    • 11. C. Kassel and V. Turaev, Chord diagram invariants of tangles and graphs, Duke Math. J. 92(3) (1998) 497–552. Web of ScienceGoogle Scholar
    • 12. M. Kontsevich, Vassiliev’s knot invariants, in I. M. Gelfand Seminar, Advances in Soviet Mathematics, Vol. 16 (American Mathematical Society, Providence, RI, 1993), pp. 137–150. Google Scholar
    • 13. G. Kuperberg, What is a virtual link?, Algebraic Geom. Topol. 3 (2003) 587–591. Web of ScienceGoogle Scholar
    • 14. M. B. McCurdy, Graphical methods for Tannaka duality of weak bialgebras and weak Hopf algebras, Theory Appl. Categ. 26(9) (2012) 233–280. Google Scholar
    • 15. T. Pantev, B. Toën, M. Vaquie and G. Vezzosi, Shifted symplectic structures, preprint (2011), arXiv:1111.3209. Google Scholar
    • 16. N. Y. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127(1) (1990) 1–26. Web of ScienceGoogle Scholar
    • 17. J. Scott Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11(3) (2002) 311–322. Link, Web of ScienceGoogle Scholar
    • 18. M. C. Shum, Tortile tensor categories, J. Pure Appl. Algebra 93(1) (1994) 57–110. Web of ScienceGoogle Scholar
    • 19. D. P. Thurston, The algebra of knotted trivalent graphs and Turaev’s shadow world, Geom. Topol. Monogr. 4 (2002) 337–362. Google Scholar