AN L²-ALEXANDER INVARIANT FOR KNOTS

Abstract

In this paper, in an attempt to extend an earlier work of Lück, we construct a knot invariant with parameter in C* by using the fundamental L 2 -representation of the fundamental group of the knot complement, which may be thought of as an L 2 -analogue of the usual Alexander polynomial of the knot in S 3 . When restricted to U(1) parameters, we interpret this invariant in terms of the U(1) twisted L 2 -Reidemeister torsion. We also show that this L 2 -invariant depends only on the norm |t| for t ∈ C*. In particular, this implies an unexpected rigidity property of the U(1) twisted L 2 -torsion on a knot complement. A possible relationship with the volume conjecture is discussed.