On the Lagrangian cobordism relation on Legendrian links

Lagrangian cobordism induces a preorder on the set of Legendrian links in any contact 3-manifold. We show that any finite collection of null-homologous Legendrian links in a tight contact 3-manifold with a common rotation number has an upper bound with respect to the pre-order. While a similar result due to Lazarev in higher dimensions requires the use of an h-principle, we are able to work "by hand" in dimension 3. In particular, we give concrete constructions of an exact Lagrangian cobordism from each element of the collection to a common Legendrian link. This construction allows us to define a notion of minimal Lagrangian genus between any two null-homologous Legendrian links with common rotation number, a notion that I will explore further towards the end of the talk. Most of this material is joint work with Shea Vela-Vick and C.-M. Michael Wong.