Lagrangian Cobordisms between Enriched Knot Diagrams

TL;DR

This paper introduces new obstructions to Lagrangian cobordisms between knots in four-dimensional space, based on enriched knot diagrams and holomorphic curve techniques, advancing understanding of Lagrangian surfaces.

Contribution

It develops novel obstructions using holomorphic disks and enriched diagrams, establishing a partial order on equivalence classes of knots via Lagrangian cobordisms.

Findings

Obstructions derived from moduli space boundary analysis

Extension of results on Lagrangian slices

New partial order on enriched knot diagrams

Abstract

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 $\mathbb{R}^4$.