V.G. Bardakov, M.V. Neshchadim. Knot groups and nilpotent approximability ... P. 43-51

We study groups of classical links, welded links, and virtual links. For classical braids, it is proved that a braid and its automorphic image are weakly equivalent. This implies the affirmative answer to the question of the coincidence of the groups constructed from a braid and from its automorphic image. We also study the problem of approximability of groups of virtual knots by nilpotent groups. It is known that in a classical knot group the commutator subgroup coincides with the third term of the lower central series, and hence the factorization by the terms of the lower central series yields nothing. We prove that the situation is different for virtual knots. A nontrivial homomorphism of the virtual trefoil group to a nilpotent group of class 4 is constructed. We use the Magnus representation of a free group by power series to construct a homomorphism of the virtual trefoil group to a finite-dimensional algebra. This produces the nontrivial linear representation of the virtual trefoil group by unitriangular matrices of order 8.

Keywords: virtual knots, links, groups.

The paper was received by the Editorial Office on June 21, 2017.

Valeriy Georgievich Bardakov, Dr. Phys.-Math. Sci., docent,
Sobolev Institute of Mathematics,  Novosibirsk, 630090 Russian;
Novosibirsk State University, Novosibirsk, 630090 Russian;
Novosibirsk State Agrarian University, Novosibirsk, 630039 Russian,
email: bardakov@math.nsc.ru

Mikhail Vladimirovich Neshchadim, Dr. Phys.-Math. Sci., docent,
Sobolev Institute of Mathematics,  Novosibirsk, 630090 Russian;
Novosibirsk State University, Novosibirsk, 630090 Russian,
email:neshch@math.nsc.ru

REFERENCES

1.   Dyer J.,  Grossman E. The automorphism groups of the braid groups. Amer. J. Math., 1981, vol. 103, no. 6, pp. 1151-1169. doi: 10.2307/2374228.

2.   Kauffman L.H. Virtual knot theory. Eur. J. Comb., 1999, vol. 20, no. 7, pp. 663-690. doi: 10.1006/eujc.1999.0314.

3.   Bardakov V.G. Virtual and welded links and their invariants. Sib. Elektron. Mat. Izv., 2005, vol. 2, pp. 196-199 (electronic).

4.   Manturov V.O. On the recognition of virtual braids.J. Math. Sci. (N.Y.), 2005, vol. 131, no. 1,  pp. 5409-5419. doi: 10.1007/s10958-005-0415-5.

5.   Carter J.S.,  Silver D.,  Williams S. Invariants of links in thickened surfaces. Algebr. Geom. Topol., 2014, vol. 14, no. 3, pp. 1377-1394. doi: 10.2140/agt.2014.14.1377.

6.   Bardakov V.G.,   Mikhalchishina Yu.A.,  Neshchadim M.V. Representations of virtual braids by automorphisms and virtual knot groups. J. Knot Theory Ramifications, 2017, vol. 26, no. 1,  1750003, 17p. doi: 10.1142/S0218216517500031.

7.   Bardakov V.G.,  Bellingeri P. Groups of virtual and welded links. J. Knot Theory Ramifications, 2014, vol. 23, no. 3, 1450014, 23 p. doi: 10.1142/S021821651450014X.

8.   Bardakov V.G., Mikhailov R.V. On the residual properties of link groups. Siberian Math. J., 2007, vol. 48, no. 3, pp. 387-394. doi: 10.1007/s11202-007-0042-0.

9.   Lyndon R.C., Schupp P.E. Combinatorial group theory. Berlin, Springer-Verlag,  1977, 339 p. ISBN: 3-540-07642-5.

10.   Fenn R.,  Rimanyi R.,  Rourke C. The braid-permutation group.Topology, 1997, vol. 36, no. 1, pp. 123-135. doi: 10.1016/0040-9383(95)00072-0.

11.   McCool J. On basis-conjugating automorphisms of free groups. Can. J. Math., 1986, vol. 38, no. 6, pp. 1525-1529. doi: 10.4153/CJM-1986-073-3.

12.   Birman J.S. Braids, links, and mapping class groups. Princeton,  Princeton University Press, 1974, Ser. Annals of Math. Studies, vol. 82. 228 p. ISBN: 0691081492.

13.   Markov A.A. Fundamentals of algebraic theory of braid groups. Trudy Mat. Inst. Akad. Nauk SSSR, 1945, vol. 16, pp. 1-54 (in Russian).

14.   The Kourovka notebook. Unsolved problems in group theory. Ed. by V. D. Mazurov and E. I. Khukhro. 14th augm. ed. Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 1999, 134 p.