{"product_id":"invariants-pictures-9789811220111-new","title":"Invariants and Pictures","description":"\u003cp\u003eThis book contains an in-depth overview of the current state of the recently emerged and rapidly growing theory of \u003cem\u003eGnk\u003c\/em\u003e 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.\u003c\/p\u003e\u003cp\u003eIn 2015, V O Manturov defined a two-parametric family of groups \u003cem\u003eGnk\u003c\/em\u003e 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 \u003cem\u003eGnk\u003c\/em\u003e.\u003c\/p\u003e\u003cp\u003eThe book is devoted to various realisations and generalisations of this principle in the broad sense. The groups \u003cem\u003eGnk\u003c\/em\u003e 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 -- \u003cem\u003eΓnk\u003c\/em\u003e, which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds.\u003c\/p\u003e","brand":"WORLD SCIENTIFIC PUB CO INC","offers":[{"title":"New","offer_id":51544447254818,"sku":"9789811220111-new","price":123.85,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0893\/4755\/5618\/files\/9789811220111.jpg?v=1776404048","url":"https:\/\/www.albakireads.com\/products\/invariants-pictures-9789811220111-new","provider":"AlbakiReads","version":"1.0","type":"link"}