  In this paper, we present new obstructions to the existence of Lagrangian
cobordisms in $\mathbb{R}^4$ that depend only on the enriched knot diagrams of
the boundary knots or links, using holomorphic curve techniques. We define
enriched knot diagrams for generic smooth links. The existence of Lagrangian
cobordisms gives a well-defined transitive relation on equivalence classes of
enriched knot diagrams that is a strict partial order when restricted to exact
enriched knot diagrams To establish obstructions we study $1$-dimensional
moduli spaces of holomorphic disks with corners that have boundary on
Lagrangian tangles - an appropriate immersed Lagrangian closely related to
embedded Lagrangian cobordisms. We adapt existing techniques to prove
compactness and transversality, and compute dimensions of these moduli spaces.
We produce obstructions as a consequence of characterizing all boundary points
of such moduli spaces. We use these obstructions to recover and extend results
about ``growing" and ``shrinking" Lagrangian slices. We hope that this
investigation will open up new directions in studying Lagrangian surfaces in
$\mathbb{R}^4$.
