A Complete Computable Knot Invariant
Year: 2025
Type: article
Abstract: We introduce the LDI-Chirality invariant and prove it is a complete and computable knotinvariant, distinguishing all non-degenerate knots in O(N ) time while correctly identifying chi-rality. The invariant Λ(K) = (LDIκ , LDI|τ | , χ) is derived from the curvature and torsion of aknot via the Frenet-Serret apparatus, combining Local Divergence Intensity (LDI) with a chiralindex given by mean torsion.This invariant characterizes geometric knots, distinct from topological knot-types. Knot-types—equivalence classes of knots under ambient isotopy—intentionally discard the geometricstructure (curvature, torsion, handedness) essential to physical and scientific applications. Thepresent work addresses knots as geometric objects.We provide a rigorous proof of completeness for generic knots and validate it empirically on8,372 knots across seven families, performing 35 million pairwise comparisons with zero collisions.The proof rests on three pillars: (1) for smooth periodic functions, identical LDI values in thefine-discretization limit imply identical normalized shapes (Shape Determination Lemma); (2)the closure constraint for space curves is rigid under affine transformations of curvature (ClosureRigidity Proposition); (3) the chiral index χ is anti-invariant under reflection, resolving mirrorambiguity.
Author: Taylor, John
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7111018537
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