| 紀錄類型: |
書目-電子資源
: Monograph/item
|
| 杜威分類號: |
514/.22 |
| 書名/作者: |
Invariants and pictures : low-dimensional topology and combinatorial group theory // Vassily Olegovich Manturov ... [et al.] |
| 其他作者: |
Manturov, V. O. |
| 出版者: |
Hackensack, NJ : : World Scientific,, c2020. |
| 面頁冊數: |
1 online resource (xxiv, 357 p.) : : ill. |
| 標題: |
Low-dimensional topology. |
| 標題: |
Combinatorial group theory. |
| 標題: |
Invariants. |
| ISBN: |
9789811220128 |
| ISBN: |
9811220123 |
| ISBN: |
9789811220135 |
| ISBN: |
9811220131 |
| 書目註: |
Includes bibliographical references and index. |
| 內容註: |
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. |
| 摘要、提要註: |
"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"-- |
| 電子資源: |
https://www.worldscientific.com/worldscibooks/10.1142/11821#t=toc |