Lagrangian Fillings of Legendrian $4$-Plat Knots

TL;DR

This paper characterizes Legendrian 4-plat knots that admit exact orientable Lagrangian fillings and shows that their underlying smooth knot types are positive, contributing to the understanding of their geometric and topological properties.

Contribution

It provides a complete characterization of fillable Legendrian 4-plat knots and establishes positivity of their underlying smooth knot types.

Findings

Legendrian 4-plat knots with fillings are positive knots.

Characterization of fillable Legendrian 4-plat knots.

Underlying smooth knots of fillable Legendrian 4-plats are positive.

Abstract

We characterize which Legendrian $4$-plat knots in the standard contact $3$-space have exact orientable Lagrangian fillings. As a corollary, we show that the underlying smooth knot types of fillable Legendrian $4$-plats are positive.