Shadow links and face models of statistical mechanics
type: Item
instance of: scholarly article
title: Shadow links and face models of statistical mechanics
zbMATH Open document ID: 0773.57012
author: Vladimir Turaev
published in: Journal of Differential Geometry
publication date: 1993-04-01T00:00:00Z
review text: The author develops geometric techniques which reduce the study of isotopy of links in the total space \(N\) of an oriented circle fibration over a closed oriented surface \(F\) to the study of certain equivalence classes of (purely 2-dimensional) objects on \(F\), which he calls shlinks (shadow links). More precisely, for each abelian group \(A\supset\mathbb{Z}\), the author defines a shadow on \(F\) over \(A\) to be a finite family of immersed closed curves on \(F\) with only double transversal crossings, together with a labelling of the components of the complement by elements of \(A\) (``gleams''). A shlink is an equivalence class of shadows modulo suitably defined ``Reidemeister moves''. (It should be noted that crossings are treated as if labelled by \((-2)\in\mathbb{Z}\subset A\).) The sum over all labels of a shadow is called the total gleam, and it is an invariant of the shlink. The author shows how to associate to each (isotopy class of) a link \(K\) in \(N\) a shlink \(S(K)\) on \(F\). Then he proves that it is possible to almost reconstruct \(K\) from \(S(K)\). More precisely, the map \(K\mapsto S(K)\) establishes a bijective correspondence between the set of isotopy classes of links in \(N\), modulo a natural action of \(H_ 1(F)\), and the set of integral shlinks on \(F\) with total gleam equal to the negative of the Euler number of the bundle \(N\to F\). An interesting example is the Hopf fibration \(S^ 3\to S^ 2\), where in fact the transition from links to shlinks is faithful. The author goes on to discuss framed and colored shlinks. Then he sets up an IRF model for colored complex shlinks, which enables to define invariants of shlinks from the representation theory of \(U_ q(sl_ 2(\mathbb{C}))\). For the Hopf fibration example, existence of the Jones polynomial and the Jones type invariants of links in \(\mathbb{R}^ 3\) colored by irreducible finite dimensional representations of \(U_ q(sl_ 2)\) [\textit{N. Yu. Reshetikhin} and the author, Commun. M…
reviewed by: Uwe Kaiser
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 81T99
zbMATH DE Number: 147456
zbMATH Keywords: quantum group
zbMATH Keywords: isotopy of links in the total space of an oriented circle fibration over a closed oriented surface
zbMATH Keywords: shadow links
zbMATH Keywords: Reidemeister moves
zbMATH Keywords: total gleam
zbMATH Keywords: shlink
zbMATH Keywords: Hopf fibration
zbMATH Keywords: IRF model
zbMATH Keywords: Jones polynomial
zbMATH Keywords: integer shlinks
zbMATH Keywords: complex shlinks
Wikidata QID: Q115180712
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214448442
OpenAlex ID: W1573696277
DOI: 10.4310/JDG/1214448442
description: scientific article; zbMATH DE number 147456