# Smoothly non-isotopic Lagrangian disk fillings of Legendrian knots

## TL;DR

This paper constructs new examples of Lagrangian ribbon disks in the symplectic 4-ball with identical Legendrian knot boundaries that are not smoothly isotopic, revealing novel distinctions in Lagrangian fillings.

## Contribution

It introduces the first families of non-isotopic Lagrangian ribbon disks with the same boundary Legendrian knots and non-homeomorphic exteriors.

## Key findings

- Existence of multiple non-isotopic Lagrangian ribbon disks with identical boundary knots
- Disks have non-homeomorphic exteriors, indicating topological diversity
- First explicit constructions of such Lagrangian fillings

## Abstract

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.