problems in the theories discussed in the present book.
Vassily Olegovich Manturov 2019
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1After I constructed such invariant, I learnt that free knots were invented by Turaev five years before that and thought to be trivial [Turaev, 2007], hence, I disproved Turaev’s conjecture without knowing that.
2I.M. Nikonov coauthored my first published paper about
Acknowledgments
The authors would like to express their heartfelt gratitude to L. A Bokut’, H. Boden, J. S. Carter, A. T. Fomenko, S. G. Gukov, Y. Han, D. P. Ilyutko, A. B. Karpov, R. M. Kashaev, L. H. Kauffman, M. G. Khovanov, A. A. Klyachko, P. S. Kolesnikov, I. G. Korepanov, S. V. Matveev, A. Yu. Olshanskii, W. Rushworth, G. I. Sharygin, V. A. Vassiliev, Zheyan Wan, Jun Wang, J. Wu and Zerui Zhang for their interest and various useful discussions on the present work. We are grateful to Efim I. Zelmanov for pointing out the resemblance between the groups
The first named author would like to express his special thanks to
“I learnt a lot about word problems and conjugacy problems in the braid group from him. Meeting him many times during the last twenty years increased my knowledge in braid group theory. It is a great loss for the mathematical community that he passed away in 2019.”
— Vassily Olegovich Manturov
The first named author was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (grant No. 14.Y26.31.0025 of the government of the Russian Federation). The second named author was supported by the program “Leading Scientific Schools” (grant no. NSh-6399.2018.1, Agreement No. 075-02-2018-867) and by the Russian Foundation for Basic Research (grant No. 19-01-00775-a). The fourth named author was supported by the program “Leading Scientific Schools” (grant no. NSh-6399.2018.1, Agreement No. 075-02-2018-867) and by the Russian Foundation for Basic Research (grant No. 18-01-00398-a).
Contents
Introduction
1.Groups. Small Cancellations. Greendlinger Theorem
1.1.2The notion of a diagram of a group
1.2.1Small cancellation conditions
1.3Algorithmic problems and the Dehn algorithm
2.1Definitions of the braid group
2.2The stable braid group and the pure braid group
2.3The curve algorithm for braids recognition
2.3.1Construction of the invariant
2.3.2Algebraic description of the invariant
2.4.1Definitions of virtual braids
2.4.2Invariants of virtual braids
3.Curves on Surfaces. Knots and Virtual Knots
3.1Basic notions of knot theory
3.2Curve reduction on surfaces
3.2.2Minimal curves in an annulus
3.2.3Proof of Theorems 3.3 and 3.4
3.2.4Operations on curves on a surface
3.3.1Classical case
3.3.3An analogue of Markov’s theorem in the virtual case
4.Two-dimensional Knots and Links
