Finite-type invariants of Legendrian knots in the 3-space: Maslov index as an order 1 invariant

We consider a contractible closure of the space of Legendrian knots in the standard contact 3-space. We show that in this context the space of finite-type complex-valued invariants of Legendrian knots is isomorphic to that of framed knots in R with an extra order 1 generator (Maslov index) added. The idea to study invariants of generic objects via their degenerations goes back to Poincare. It was very efficiently applied by Vassiliev in his work on knot invariants [10]. It turns out to be very fruitful in numerous other situations. For example, the study of invariants of Legendrian knots from a similar point of view was started by Arnold in [1, 2, 3] where order 1 invariants of plane fronts and immersed curves were extensively investigated. In [8] the complete theory of finite type invariants of plane fronts without dangerous self-tangencies was constructed. The approach was analogous to that used in [6] for the study of invariants of regular plane curves. As in [2, 3], it was based on consideration of only immersed Legendrian curves. The latter causes certain inconvenience since the space of parametrisations of such curves has non-trivial topology. Therefore, to show, for example, that a certain codimension 1 stratum in the space of maps defines a co-cycle, one needs to check that its intersection with infinite order generators of the fundamental group vanishes. This task is sometimes rather tricky (see [3]). On the other hand, introduction of a contractible closure avoids such problems. But this can lead to appearance of some other finite-type invariants corresponding to the strata of the closure and to upgrading some order 0 invariants to positive order. In the present paper we consider the closure of the space of immersed Legendrian 1991 Mathematics Subject Classification: Primary: 32S55. Secondary: 57M25. The paper is in final form and no version of it will be published elsewhere.