Parametric satellites and connected-sums in the space of Legendrian embeddings

TL;DR

This paper develops parametric satellite and connected-sum constructions in the space of Legendrian embeddings, creating new invariants and infinite families of Legendrian loops with non-trivial monodromy.

Contribution

It introduces parametric Legendrian satellite and connected-sum constructions, extending the toolkit for studying Legendrian embedding spaces at higher homotopy levels.

Findings

New invariants at higher homotopy levels of Legendrian embeddings

Construction of infinite families of Legendrian loops with non-trivial invariants

Extension of satellite operation to parametric families

Abstract

This article introduces two new constructions at the higher homotopy level in the space of Legendrian embeddings in $(\mathbb{R}^3, \xi_{\operatorname{std}})$. We first introduce the parametric Legendrian satellite construction, showing that the satellite operation works for parametric families of Legendrian embeddings. This yields new invariants at the higher-order homotopy level. We then introduce the parametric connected-sum construction. This operation takes as inputs two $n$-spheres based at Legendrian embeddings $K_1$ and $K_2$, respectively, and produces a new $n$-sphere based at $K_1\# K_2$. As a main application we construct new infinite families of loops of Legendrian embeddings with non-trivial LCH monodromy invariant.