Ja n 20 07 EUCLIDEAN GEOMETRIC INVARIANTS OF LINKS IN 3 - SPHERE

We present a new link invariant which depends on a representation of the link group in SO(3). The computer calculations indicate that an abelian version of this invariant is expressed in terms of the Alexander polynomial of the link. On the other hand, if we use non abelian representation, we get the squared non abelian Reidemeister torsion (at least for some torus knots). Introduction In this paper we consider a new link invariant. Its construction is naturally divided into three main parts. First, on a given representation of the link group we define a covering of 3-sphere branched along the link. Then, we map the covering space into 3-dimensional Euclidean space according to the representation. In the last, algebraic part, we build an acyclic complex; the torsion of this complex is the main ingredient of our invariant. Such an invariant was first constructed by I.G. Korepanov for 3-manifolds in the paper [3]. There the simplest version of the invariant was considered corresponding to the trivial covering of a manifold. Calculations showed that this version raised to the (−1/6)th power equals the order of the torsion subgroup of the first homology group. However, when we use the universal cover, which corresponds to the unity of the fundamental group, we get more interesting version of the invariant associated with the (abelian) Reidemeister torsion, see [6, 8]. We have written a few computer programs which allow to calculate the invariant for a given manifold M (or link L) and for a given representation ρ of the fundamental group (or the link group) in the group of orientation preserving motions of R . With the help of these programs we made two conjectures. The first one states a connection of the abelian version of our invariant with the Alexander polynomial of link. The second conjecture states that for a torus knot and non abelian representation of its group, our invariant is the squared non abelian Reidemeister torsion investigated and calculated in [1]. The paper is organized as follows. In section 1 we define the invariant for a given link and representation of its group. In section 2 we propose an example of calculation of abelian and non abelian versions of the invariant for the trefoil knot. In the last section we suggest our conjectures. Acknowledgements. I am glad to thank I.G. Korepanov for proposing me the problem and numerous helpful discussions and remarks. The work is partially supported by Russian Foundation for Basic Research, Grant no. 04-0196010. 1 2 EVGENIY V. MARTYUSHEV