Geometric Emergence of Z3 Colour Structure from Hopf Preimages on S3
Year: 2026
Type: article
Abstract: We report a numerical study of preimage links arising from the composition of odd equivariant lifted maps F:S3→S3 with the Hopf bration πH:S3→S2. For the cubic lift F3(z1,z2)=(z3 1,z3 2)/∥(z3 1,z3 2)∥, the preimage of a regular target valuec is a stable three component link. We show that the three components are separated by exactly 2π/3=120◦ in the invariant relative Hopf phase ψ=ξ1−ξ2, and that this separation is stable across four levels of grid refinement (48000 to 384000 sample points).This provides a continuous geometric origin for the discrete Z3 colour structure of the Topological InversionModel [1]. The three components are pairwise unlinked but carry nontrivial self-framing diagnostics under natural normal fields. The bridge from continuous Z3 phase to integer framing vectors may be provided by the discrete braid transfer rule [2,3]; this connection is conjectured but not yet derived.
Author: Janse van Rensburg, Kobie
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7140042586
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