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

Area of Science:

• Cognitive Neuroscience
• Theoretical Mathematics
• Computational Neuroscience

Background:

• The long-standing goal of unifying diverse knowledge domains has persisted across history.
• Cognitive neuroscience reveals distinct, distributed brain activation patterns for human symbolic and cultural activities.
• An algebraic neural representation could offer a unifying framework for interdisciplinary knowledge dialogue.

Purpose of the Study:

• To investigate associative memory models as a potential unifying algebraic neural representation.
• To identify neural representations for abstract, algebraic, and topological mathematical structures.
• To develop a novel matrix representation for knots inspired by associative memory principles.

Main Methods:

• Review of existing representations for graphs and finite groups within associative memory frameworks.
• Introduction of a novel tensor product-based matrix representation for knot structures.
• Analysis of the relationship between the proposed knot representation, Gauss codes, and Seifert circles.

Main Results:

• A novel 'associative matrix' representation for knots is developed using tensor products of crossing states.
• This representation is shown to be closely related to the knot's Gauss code.
• The adjacency matrix of the Gauss diagram is identified as a construct that enhances knot classification.

Conclusions:

• Associative memory models offer a promising foundation for an algebraic neural representation unifying knowledge.
• The novel matrix representation provides a new method for encoding and analyzing knot topology.
• This research opens avenues for integrating mathematical structures within a cognitive neuroscience framework.