Classical graph theory describes a neural network as a set of vertices and edges but does not distinguish topologically inequivalent embeddings of the same graph in space. Two isomorphic graphs may be projections of topologically distinct knots. In this paper, we apply the variational method of lifting graphs to R4 with verification of the Morse index µ = 0 to analyze the structural connectomes of two model organisms: the nematode Caenorhabditis elegans (277 neurons) and the medulla of Drosophila melanogaster (1781 neurons).Two qualitatively distinct topological regimes of neural organization are identified: C. elegans — compact, achiral entanglement (Lk ~ 102, W r = 0) with a distributed core and deep stability; Drosophila medulla — extreme, chiral hyper-entanglement (|Lk| ~106, Wr ~103) with a single super-hub and precritical dynamics. Both systems satisfy the topological protection condition µ = 0. A superquadratic scaling law |Lk| ∝ Nα with α ≈ 4.3 is established empirically, significantly exceeding linear and quadratic growth.The results indicate that the evolution of nervous systems involves transitions between qualitatively distinct topological regimes: from a minimally sufficient connectivity to an excessive chiral entanglement. The method opens the possibility of topological diagnostics of neural networks, including the prediction of epileptic seizures from EEG/MEG time series.Keywords: spatial graphs, knot theory, connectome, C. elegans, Drosophila medulla, lifting to R4, Morse index, linking number, topological stability.
Sergey Aleksandrovich Mazein (Mon,) studied this question.