Classical graph theory describes networks combinatorially but lacks the tools to capture global topological congurations such as knots and spatial graphs. This paper completes the foundational computational platform of dynamical knot theory within the Tensor Model of Discrete Dynamics (TMDD) by introducing a rigorous variational apparatus for lifting graphs to knots and spatial graphs. Unlike heuristic geometric approaches, lifting is formulated as a minimization problem for the action functional S⃗λ in the conguration space of spectral parameters Mcons ⊂ R6 under the Morse stability condition µ = 0. The fourth coordinate h4d emerges from the spectral height of θ-cells, and crossing data Σ = σk are derived from temporal activation asymmetries or phase shifts, with active crossings determined by the structural risk index K (t) ≥ Kthr ≈ 0. 301. The algorithm comprises seven steps: trajectory discretization, crossing extraction, introduction of h4d, variational optimization with L-BFGS-B under barrier constraints, computation of topological invariants (Lk, W r, Jones polynomial VL (q) ), and dynamic segmentation of the phase space. Key theoretical results include: (i) a rigorous correspondence between Reidemeister moves (RI, RII, RIII) and threshold-driven recongurations in Mcons, with RIII occurring entirely within the µ = 0 basin via cooperative stabilization; (ii) invariance of Lk, Wr, and VL (q) under variational isotopy; and (iii) introduction of the cross-domain phase plane (K, ln Me/Mbase) as a universalnormalization tool. This work, together with its predecessors on TMDD ontology and the PRCS phenomenological pipeline, establishes a unied framework that transforms discrete observational data into a topologically protected knot conguration, bridging combinatorial graph theory, low-dimensional topology, and complex systems diagnostics.
Sergey Aleksandrovich Mazein (Sun,) studied this question.