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Shadow links and face models of statistical mechanics. (English) Zbl 0773.57012

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 \(U_ q(sl_ 2)\) [N. Yu. Reshetikhin and the author, Commun. Math. Phys. 127, 1-26 (1990; Zbl 0768.57003)] is proved. The important philosophical point of the work is that shlinks over \(A=\mathbb{Q},\mathbb{R},\mathbb{C},\dots\) should be treated as interesting geometric objects on its own. So, in the Hopf fibration example, the transition from integer shlinks to complex shlinks, can be understood as a “completion” of the set of isotopy classes of links in \(S^ 3\), which needs to be explored in the future.
Reviewer: U.Kaiser (Siegen)

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81T99 Quantum field theory; related classical field theories

Citations:

Zbl 0768.57003
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