Invariants and pictures : low-dimensional topology and combinatorial group theory / Vassily Olegovich Manturov and Denis Fedoseev, Seongjeong Kim, Igor Nikonov.

Format:

Book

Author/Creator:

Manturov, V. O. (Vasiliĭ Olegovich), author.

Fedoseev, Denis, author.

Kim, Seongjeong, author.

Nikonov, Igor (Igor Mikhailovich), author.

Series:

K & E series on knots and everything ; 0219-9769 v. 66.

Series on knots and everything, 0219-9769 ; volume. 66

Language:

English

Subjects (All):

Low-dimensional topology.

Combinatorial group theory.

Invariants.

Physical Description:

xxiv, 357 pages : illustrations ; 24 cm.

Place of Publication:

New Jersey : World Scientific, [2020]

Summary:

"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"-- Provided by publisher.

Contents:

Groups. Small cancellations. Greendlinger theorem

Braid theory

Curves on surfaces. Knots and virtual knots

Two-dimensional knots and links

Parity in knot theories. The parity bracket

Cobordisms

General theory of invariants of dynamical systems

Groups Gk/n and their homomorphisms

Generalisations of the groups Gk/n

Representations of the groups Gk/n

Realisation of spaces with Gk/n action

Word and conjugacy problems in Gk/k+1 groups

The groups Gk/n and invariants of manifolds

The two-dimensional case

The three-dimensional case

Open problems.

Notes:

Includes bibliographical references and index.

ISBN:

9789811220111

9811220115

OCLC:

1157349863