Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

j, k, l, m,

      (4) for distinct i, j, k, l, m.

      Just like we formulated the

principle, here we could formulate the principle in whole generality, but we restrict ourselves with several examples.

      It turns out that groups have nice presentation coming from the Ptolemy relation and the cluster algebras. The Ptolemy relation

      says that the product of diagonals of an inscribed quadrilateral equals the sum of products of its opposite faces, see Fig. 0.8.

      Fig. 0.8The Ptolemy relation

      We can use it when considering triangulations of a given surface: when performing a flip, we replace one diagonal (x) with the other diagonal (y) by using this relation. It is known that if we consider all five triangulations of the pentagon and perform five flips all the way around, we return to the initial triangulation with the same label, see Fig. 0.6.

      This well known fact gives rise to presentations of .

      Thus, by analysing the groups , we can get

      (1)Invariants of braids on 2-surfaces valued in polytopes;

      (2)Invariants of knots;

      (3)Relations to groups

;

      (4)Braids on

³.

      Going slightly beyond, we can investigate braids in

³ and the configuration space of polytopes.

      We will not say much about the groups for k > 4. The main idea is:

      (1)Generators (codimension 1) correspond to simplicial (k − 2)-polytopes with k vertices;

      (2)The most interesting relations (codimension 2) correspond to (k − 2)-polytopes with k + 1 vertices.

      It would be extremely interesting to establish the connection between

with the Manin–Schechtmann “higher braid groups” [Manin and Schechtmann, 1990], where the authors study the fundamental group of complements to some configurations of complex hyperplanes.

      It is also worth mentioning, that the relations in the group

resemble the relations in Kirillov–Fomin algebras, see [Fomin and Kirillov, 1999]. For that reason it seems interesting to study the interconnections between those objects.

      Finally, our invariants may not be just group-valued: some variations of

admit simplicial group structures, which is studied now in a joint work with S. Kim, F. Li and J. Wu.

      Note that the present book is very much open-ended. On one hand, the invariants of manifolds are calculated in some explicit cases. On the other hand, one can vary “the

-principle” and “-principle” together with the groups itself and try to find invariants of manifolds depending on complex structures, spin-structures, and other structures by looking at some “good codimension 1 properties” and creating interesting configuration spaces.

      The present book has the following structure. In Part 1 we review basic notions of knot theory and combinatorial group theory: groups and their presentations, van Kampen diagrams, braid theory, knot theories and the theory of 2-dimensional knots. Part 2 is devoted to the parity theory and its applications to cobordisms of knots and free knots. In Part 3 we present the theory of

groups and their relations to invariants of dynamical systems. Part 4 deals with the notion of manifold of triangulations, higher dimensional braids, and investigates the groups . In the final Part 5 we present a list of unsolved problems in the theories discussed in the present book.

       Vassily Olegovich Manturov 2019

      _____________________________

      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

and Kirillov–Fomin algebras.

      The first named author would like to express his special thanks to (11 Sep. 1952–4 Sep. 2019):

      “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).