Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory
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A01=Denis Fedoseev
A01=Igor Nikonov
A01=Seongjeong Kim
A01=Vassily Olegovich Manturov
Author_Denis Fedoseev
Author_Igor Nikonov
Author_Seongjeong Kim
Author_Vassily Olegovich Manturov
AZA"nk Group
Braid
Category=PBPH
Coxeter Groups
Diagram
Diamond Lemma
Dynamical System
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Gale Diagram
Gnk Group
Group
Invariant
Kirillov-Fomin Algebra
Knot
Manifold
Manifold of Triangulations
Pachner Move
Planarity
Regular Triangulation
Regular Triangulations
Small Cancellation
Γnk Group
Product details
- ISBN 9789811220111
- Publication Date: 11 May 2020
- Publisher: World Scientific Publishing Co Pte Ltd
- Publication City/Country: SG
- Product Form: Hardback
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This book contains an in-depth overview of the current state of the recently emerged and rapidly growing theory of Gnk 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 Gnk 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 Gnk.The book is devoted to various realisations and generalisations of this principle in the broad sense. The groups Gnk 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 — Γnk, which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds.
