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Invariants and pictures = low-dimensional topology and combinatorial group theory /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Invariants and pictures/ Vassily Olegovich Manturov ... [et al.]
Reminder of title:	low-dimensional topology and combinatorial group theory /
other author:	Manturov, V. O.
Published:	Hackensack, NJ :World Scientific, : c2020.,
Description:	1 online resource (xxiv, 357 p.) :ill. :
Subject:	Low-dimensional topology. -
Online resource:	https://www.worldscientific.com/worldscibooks/10.1142/11821#t=toc
ISBN:	9789811220128

Invariants and pictures = low-dimensional topology and combinatorial group theory /
Invariants and pictureslow-dimensional topology and combinatorial group theory /[electronic resource] :Vassily Olegovich Manturov ... [et al.] - 1st ed. - Hackensack, NJ :World Scientific,c2020. - 1 online resource (xxiv, 357 p.) :ill. - Series on knots and everything,vol. 660219-9769 ;. - Series on knots and everything,vol. 66..

Includes bibliographical references and index.

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.

"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"--

ISBN: 9789811220128

LCCN: 2020012113Subjects--Topical Terms:

1099032
Low-dimensional topology.

LC Class. No.: QA612.14 / .M36 2020

Dewey Class. No.: 514/.22

Invariants and pictures = low-dimensional topology and combinatorial group theory /
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490 1 $a Series on knots and everything, $x 0219-9769 ; $v vol. 66
504 $a Includes bibliographical references and index.
505 0 $a 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.
520 $a "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"-- $c Provided by publisher.
588 $a Description based on print version record.
650 0 $a Low-dimensional topology. $3 1099032
650 0 $a Combinatorial group theory. $3 672565
650 0 $a Invariants. $3 527918
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830 0 $a Series on knots and everything, $x 0219-9769 ; $v vol. 66. $3 1406804
856 4 0 $u https://www.worldscientific.com/worldscibooks/10.1142/11821#t=toc