### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 032, 29 pages      math.GT/0605164      http://dx.doi.org/10.3842/SIGMA.2010.032

### A Euclidean Geometric Invariant of Framed (Un)Knots in Manifolds

Jérôme Dubois a, Igor G. Korepanov b and Evgeniy V. Martyushev b
a) Institut de Mathématiques de Jussieu, Université Paris Diderot-Paris 7, UFR de Mathématiques, Case 7012, Bâtiment Chevaleret, 2, place Jussieu, 75205 Paris Cedex 13, France
b) South Ural State University, 76 Lenin Avenue, Chelyabinsk 454080, Russia

Received October 09, 2009, in final form April 07, 2010; Published online April 15, 2010

Abstract
We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory.

Key words: Pachner moves; Reidemeister torsion; framed knots; differential relations in Euclidean geometry; topological quantum field theory.

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