Topological Neural Coding: The associative memory representation of graphs, groups and knots

The idea of having a representation capable of unifying all branches of knowledge has been envisioned throughout different eras. Cognitive neuroscience has shown that human symbolic and cultural activities consistently evoke distinct activation patterns distributed across multiple brain regions. An algebraic neural representation shared by all cognitive domains could provide a unifying framework and enable dialogue between the structures underlying diverse fields of knowledge. Associative memory models-characterized by vector representations of neural states, matrix operators for memory retrieval, and tensor compositions of inputs to neuronal assemblies-exhibit promising features as candidates for such a unifying algebraic representation. In this work, we present a research program aimed at identifying neural representations of abstract, algebraic, and topological structures in mathematics, within the framework of context-dependent associative memories. After reviewing prior representations for graphs and finite groups, we introduce a novel matrix representation of knots inspired by associative memory models. In this representation, each "position" in the knot (i.e., over or under at a given crossing) is encoded by the tensor product of vectors representing crossings and their respective states (over or under). The resulting "associative matrix" of the knot is closely related to its Gauss code and, in the case of alternating knot diagrams, it finds the Seifert circles. Finally, we demonstrate that a related construct-the adjacency matrix of the Gauss diagram-enhances the classification capabilities of this representation.