Reconnectads

TL;DR

This paper introduces reconnectads, a new operad-like structure based on graphs, and explores their theoretical properties and applications to toric varieties and graph associahedra.

Contribution

It develops the theory of reconnectads and applies it to the study of complex toric varieties associated with graph associahedra.

Findings

Defined the structure of reconnectads and their compositions.

Connected reconnectads to toric varieties of graph associahedra.

Analyzed the properties of the wonderful reconnectad from homology groups.

Abstract

We introduce a new operad-like structure that we call a reconnectad; the ``input'' of an element of a reconnectad is a finite simple graph, rather than a finite set, and ``compositions'' of elements are performed according to the notion of the reconnected complement of a subgraph. The prototypical example of a reconnectad is given by the collection of toric varieties of graph associahedra of Carr and Devadoss, with the structure operations given by inclusions of orbits closures. We develop the general theory of reconnectads, and use it to study the ``wonderful reconnectad'' assembled from homology groups of complex toric varieties of graph associahedra.