Complexes of connected graphs

TL;DR

This paper studies the homology of complexes formed by connected graphs, revealing their applications across topology, knot theory, combinatorics, and extending to higher-dimensional analogues in advanced mathematical theories.

Contribution

It describes the homology of the quotient complex of connected graphs and explores its applications and higher-dimensional analogues in various mathematical fields.

Findings

Homology of the quotient complex of connected graphs is characterized.

Applications to braid groups, knot theory, and singularity theory are demonstrated.

Multidimensional analogues are proposed for advanced mathematical contexts.

Abstract

Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory, combinatorics, and singularity theory. The multidimensional analogues of this complex are indicated, which arise naturally in the homotopy theory, higher Chern-Simons theory and complexity theory.