Relative Contact Homology - Higher-dimensional Legendrian Submanifolds

Higher-dimensional Legendrian Submanifolds

In the work of Lenhard Ng, relative SFT is used to obtain invariants of smooth knots: a knot or link inside a topological three-manifold gives rise to a Legendrian torus inside a contact five-manifold, consisisting of the unit conormal bundle to the knot inside the unit cotangent bundle of the ambient three-manifold. The relative SFT of this pair is a differential graded algebra; Ng derives a powerful knot invariant from a combinatorial version of the zero-th degree part of the homology. It has the form of a finitely presented tensor algebra over a certain ring of multivariable Laurent polynomials with integer coefficients. This invariant assigns distinct invariants to (at least) knots of at most ten crossings, and dominates the Alexander polynomial and the A-polynomial (and thus distinguishes the unknot).