Exact Lagrangian caps of Legendrian knots

TL;DR

This paper proves that after enough stabilizations, any Legendrian knot in the standard 3-sphere bounds an exact Lagrangian surface in four-dimensional space, using combinatorial moves on knot projections.

Contribution

It introduces a new combinatorial framework of moves on knot projections that correspond to Lagrangian cobordisms, enabling the construction of exact Lagrangian caps.

Findings

Any Legendrian knot can be capped with an exact Lagrangian surface after stabilizations.

Develops a set of combinatorial moves representing Lagrangian cobordisms.

Provides a method to construct Lagrangian caps for Legendrian knots.

Abstract

We prove that any Legendrian knot in $(S^3,\xi_{std})$ bounds an exact Lagrangian surface in $\mathbb{R}^4\setminus B^4$ after a sufficient number of stabilizations. In order to show this, we construct a family combinatorial moves on knot projections with some additional data that correspond to Lagrangian cobordisms between knots.