Tangle contact homology

Abstract

Knot contact homology is an ambient isotopy invariant of knots and links in R 3 which is defined as the Chekanov-Eliashberg dg-algebra of the unit conormal Legendrian in the unit cotangent bundle of R 3 with homology coefficients. The purpose of this paper is to extend this definition to an ambient isotopy invariant of tangles of any dimension and prove that gluing of tangles gives a gluing formula for knot contact homology. The gluing formula follows from a gluing formula for the Chekanov-Eliashberg dg-algebra with loop space coefficients. As a consequence of the gluing formula we obtain that the tangle contact homology detects the 1-dimensional untangle up to ambient isotopy.