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Topological Quantum Field Theory and Strong Shift Equivalence

Published online by Cambridge University Press:  20 November 2018

Patrick M. Gilmer
Patrick M. Gilmer*
Affiliation:
Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 USA, email: gilmer@math.lsu.edu
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Abstract

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Given a $\text{TQFT}$ in dimension $d\,+\,1$, and an infinite cyclic covering of a closed ($d\,+\,1$)-dimensional manifold $M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a $\text{TQFT}$ and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an ${{S}^{1}}$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a $\text{TQFT}$ associated to a finite group $G$ which has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the $\text{TQFT}$ associated to $G$ in its unmodified form.

Keywords

57R9957M9954H20knotlinkTQFTsymbolic dynamicsshift equivalence
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 2 , 01 June 1999 , pp. 190 - 197
Copyright
Copyright © Canadian Mathematical Society 1999

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