Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory [Hardback]

"This book contains an in-depth overview of the current state of the recently emerged and rapidly growing theory of Gk/n groups, picture-valued invariants, and braids for arbitrary manifolds. Equivalence relations arising in low-dimensional topology and combinatorial group theory inevitably lead to the study of invariants, and good invariants should be strong and apparent. An interesting case of such invariants is picture-valued invariants, whose values are not algebraic objects, but geometrical constructions, like graphs or polyhedra. In 2015, V. O. Manturov defined a two-parametric family of groups Gk/n and formulated the following principle: if dynamical systems describing a motion of n particles possess a nice codimension 1 property governed by exactly k particles then these dynamical systems possess topological invariants valued in Gk/n. The book is devoted to various realisations and generalisations of this principle in the broad sense. The groups Gk/n have many epimorphisms onto free products of cyclic groups; hence, invariants constructed from them are powerful enough and easy to compare. However, this construction does not work when we try to deal with points on a 2-surface, since there may be infinitely many geodesics passing through two points. That leads to the notion of another family of groups - \Gamma_n^k, which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds"--



This book overviews the latest understanding of Gnk groups, picture-valued invariants, and braids for arbitrary manifolds. Part 1 presents basics of knot theory and combinatorial group theory, and Part 2 deals with parity theory and its applications to cobordisms of knots and free knots. Part 3 explains theory of Gnk groups and their relationship to invariants of dynamical systems, while Part 4 looks at manifolds of triangulations and higher dimensional braids. Part 5 presents unsolved problems related to the theories discussed. The book offers a visual approach, with many b&w illustrations and diagrams. Annotation ©2020 Ringgold, Inc., Portland, OR (protoview.com)