A prismatic classifying space

TL;DR

This paper introduces a homology theory and classifying space for qualgebras, algebraic structures related to knots, which generalize existing classifying spaces and provide invariants for knotted graphs and foams.

Contribution

It develops a new homology theory for qualgebras and constructs a classifying space that unifies and extends simplicial and cubical classifying spaces.

Findings

Defines a homology theory for qualgebras.

Constructs a prismatic classifying space from $G$-colored prisms.

Provides invariants for knotted graphs and foams via homotopy classes.

Abstract

A qualgebra $G$ is a set having two binary operations that satisfy compatibility conditions which are modeled upon a group under conjugation and multiplication. We develop a homology theory for qualgebras and describe a classifying space for it. This space is constructed from $G$-colored prisms (products of simplices) and simultaneously generalizes (and includes) simplicial classifying spaces for groups and cubical classifying spaces for quandles. Degenerate cells of several types are added to the regular prismatic cells; by duality, these correspond to "non-rigid" Reidemeister moves and their higher dimensional analogues. Coupled with $G$-coloring techniques, our homology theory yields invariants of knotted trivalent graphs in $\mathbb{R}^3$ and knotted foams in $\mathbb{R}^4$. We re-interpret these invariants as homotopy classes of maps from $S^2$ or $S^3$ to the classifying space of $G$.