Research ArticlesNo Access

Groups of virtual and welded links

Valeriy G. Bardakov

Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk 630090, Russia

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and

Paolo Bellingeri

Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, Université de Caen BP 5186, F-14032 Caen, France

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https://doi.org/10.1142/S021821651450014XCited by:16 (Source: Crossref)

Abstract

We define a new notion of group of virtual and welded links. We use two approaches for defining it: via representations of (generalized) braids by automorphisms of free groups and via Wirtinger-like labeling on virtual and welded diagrams. In the case of virtual links our invariant is stronger than the notion of fundamental group of a virtual link introduced by Kauffman. In the case of welded links our invariant coincides with Kauffman invariant, but using a generalization of Wada representations we provide new families of invariants of welded links.

Keywords:

AMSC: 57M25, 57M27

References

J. C. Baez, Lett. Math. Phys. 26, 43 (1992), DOI: 10.1007/BF00420517. Crossref, Web of Science, Google Scholar
V. G. Bardakov, Fund. Math. 184, 1 (2004), DOI: 10.4064/fm184-0-1. Crossref, Web of Science, Google Scholar
V. G. Bardakov, Sib. Elektron. Mat. Izv. 2, 196 (2005). Google Scholar
V. G. Bardakov and P. Bellingeri, Topol. Appl. 156, 1071 (2009), DOI: 10.1016/j.topol.2008.10.002. Crossref, Web of Science, Google Scholar
V. G. Bardakov, R. Mikhailov, V. Vershinin and J. Wu, On the pure virtual braid group PV3, preprint (2009) , arXiv:0906.1743 . Google Scholar
D. Bar Natan and Z. Dancso, Finite type invariants of W-knotted objects: from Alexander to Kashiwara and Vergne, preprint, http://www.math.toronto.edu/ drorbn/papers/WKO/ . Google Scholar
B. Berceanu and S. Papadima, J. Knot Theory Ramifications 18, 999 (2009), DOI: 10.1142/S0218216509007257. Link, Web of Science, Google Scholar
T. Brendle and A. Hatcher, Comment. Math. Helv. 88, 131 (2013), DOI: 10.4171/CMH/280. Crossref, Web of Science, Google Scholar
J. Crisp and L. Paris, Pacific J. Math. 221, 1 (2005), DOI: 10.2140/pjm.2005.221.1. Crossref, Web of Science, Google Scholar
M. Elhamdadiet al., J. Knot Theory Ramifications 18, 957 (2009). Web of Science, Google Scholar
R. Fenn, R. Rimányi and C. Rourke, Topology 36, 123 (1997), DOI: 10.1016/0040-9383(95)00072-0. Crossref, Web of Science, Google Scholar
B. Gemein, J. Knot Theory Ramifications 6, 441 (1997), DOI: 10.1142/S0218216597000297. Link, Web of Science, Google Scholar
E. Godelle and L. Paris, Math. Z. 272, 1339 (2012), DOI: 10.1007/s00209-012-0989-9. Crossref, Web of Science, Google Scholar
D. Goldsmith, Michigan Math. J. 28, 3 (1981). Web of Science, Google Scholar
M. Goussarov, M. Polyak and O. Viro, Topology 39, 1045 (2000), DOI: 10.1016/S0040-9383(99)00054-3. Crossref, Web of Science, Google Scholar
K. W. Gruenberg, Proc. London Math. Soc. (3) 7, 29 (1957). Google Scholar
J. Hempel, Combinatorial Group Theory and Topology, Annals of Mathematics Studies 111 (Princeton University Press, Princeton, 1987) pp. 379–396. Crossref, Google Scholar
S. Kamada, Kobe J. Math. 21, 33 (2004). Web of Science, Google Scholar
S. Kamada, Osaka J. Math. 44, 441 (2007). Web of Science, Google Scholar
L. H. Kauffman, European J. Combin. 20, 663 (1999), DOI: 10.1006/eujc.1999.0314. Crossref, Web of Science, Google Scholar
V. Lundsgaard Hansen , Braids and Coverings , London Mathematical Society Student Texts 18 ( Cambridge University Press , 1989 ) . Crossref, Google Scholar
V. O. Manturov, On the recognition of virtual braids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 299 (Geom. i Topol. 8) (2003) 267–286, 331–332 (in Russian); translation in J. Math. Sci. (N. Y.) 131 (2005) 5409–5419 . Google Scholar
A. G. Savushkina, On a group of conjugating automorphisms of a free group, Mat. Zametki 60 (1996) 92–108, 159 (in Russian); translation in Math. Notes 60 (1996) 68–80 . Google Scholar
D. Silver and S. Williams, Virtual knot groups, Knots in Hellas '98 (World Scientific, 2000) pp. 440–451. Google Scholar
V. V. Vershinin, J. Knot Theory Ramifications 10, 795 (2001), DOI: 10.1142/S0218216501001165. Link, Web of Science, Google Scholar
M. Wada, Topology 31, 399 (1992), DOI: 10.1016/0040-9383(92)90029-H. Crossref, Web of Science, Google Scholar
B. Winter, J. Knot Theory Ramifications 18, 1287 (2009), DOI: 10.1142/S0218216509007476. Link, Web of Science, Google Scholar
www.katlas.math.toronto.edu/drorbn/AcademicPensieve/Projects/WKO/nb/The_ Kishino_Braid.pdf . Google Scholar