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Quantum invariants of templates. (English) Zbl 1046.57012

A template is a \(2\)-dimensional complex in a \(3\)-manifold, used in studying knotting and linking of periodic orbits of flows. Each template contains some knots and links, and there is a set of template moves (switch move and splitting move) that does not change the set of knots and links contained in templates. Thus the following question arises: when are two given templates equivalent under template moves?
The paper under review studies invariants of templates in the \(3\)-sphere using some ideas from quantum invariants. It is shown that each bialgebra (over a field \(k\)) yields an invariant of templates if it is equipped with an antiautomorphism satisfying a certain set of identities. (The relation between templates and bialgebras has also been discovered by D. Hillman [Combinatorial spacetimes, Ph.D. Dissertation, University of Pittsburgh (1995)] independently.) In particular, a Hopf algebra such that the square of the antipode is the identity is such. It is shown that a special case of this invariant is related to the B. Parry and D. Sullivan invariant [Topology 14, 297–299 (1975; Zbl 0314.54045)]. Also, the paper gives a method to associate to each equivalence class of templates a Kirby-move equivalence class of framed links, and thus a homeomorphism class of closed oriented \(3\)-manifolds. Hence any closed oriented \(3\)-manifold produces a template invariant.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
37C27 Periodic orbits of vector fields and flows

Citations:

Zbl 0314.54045
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References:

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