Abstract:
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.
Citation:
V. G. Bardakov, M. V. Neshchadim, “Knot groups and nilpotent approximability”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 43–51; Proc. Steklov Inst. Math. (Suppl.), 304, suppl. 1 (2019), S23–S30
\Bibitem{BarNes17}
\by V.~G.~Bardakov, M.~V.~Neshchadim
\paper Knot groups and nilpotent approximability
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 4
\pages 43--51
\mathnet{http://mi.mathnet.ru/timm1465}
\crossref{https://doi.org/10.21538/0134-4889-2017-23-4-43-51}
\elib{https://elibrary.ru/item.asp?id=30713958}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2019
\vol 304
\issue , suppl. 1
\pages S23--S30
\crossref{https://doi.org/10.1134/S0081543819020044}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000453521700004}
Linking options:
https://www.mathnet.ru/eng/timm1465
https://www.mathnet.ru/eng/timm/v23/i4/p43
This publication is cited in the following 3 articles:
V. G. Bardakov, O. V. Bryukhanov, M. V. Neshchadim, “On residual nilpotence of group extensions”, Int. J. Algebra Comput., 33:08 (2023), 1617
Valeriy G. Bardakov, Neha Nanda, Mikhail V. Neshchadim, “On the lower central series of some virtual knot groups”, J. Knot Theory Ramifications, 29:09 (2020), 2050065
V. G. Bardakov, Yu. A. Mikhalchishina, M. V. Neshchadim, “Groups of the virtual trefoil and Kishino knots”, J. Knot Theory Ramifications, 27:13, SI (2018), 1842009