Mathematical Modeling of the Knot Theory of Mind: A Topological Framework for Psychological Dynamics and Abusive Behavior

This technical report presents a formal mathematical model for the Knot Theory of Mind (Petric, 2022), integrating classical knot theory, braid groups, and topological data analysis with contemporary psychological dynamics. The psychological state space Ψ is defined as a smooth manifold equipped with an introspective involution map ι: Ψ → Ψ, and model recurrent cognitive-emotional patterns as embeddings of S¹ into a reconstructed three-dimensional state space. Mind knots are formalized as ambient isotopy classes of such embeddings, with topological invariants (crossing number, unknotting number, linking number, Jones polynomial, and braid entropy) serving as quantitative descriptors of psychological complexity, entrenchment, and interdependence. Abuse operators are introduced—Ĝ (gaslighting), Ď (defamation), B̂ (bullying), and M̂ (mobbing)—that act on psychological trajectories to increase topological complexity, break introspective symmetry, and create multi-component links. The Broken-Mirror Syndrome is modeled as a symmetry-breaking bifurcation of the introspective involution. Therapeutic interventions are formalized as sequences of Reidemeister moves and crossing changes that reduce unknotting number. Synthetic state-space trajectories illustrate the transition from healthy unknots to trauma-induced trefoil and figure-eight knots. The cyclist syndrome is developed as a chain complex of N linked components and provide formal proofs in the Appendix. This framework unifies clinical Knot Theory of Mind with the dynamical topological psychology of Diamantis & Okamoto (2024), offering a structural language for trauma, recovery, and psychotherapeutic change.