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A Computable Knot Invariant

Year: 2025

Type: article

Abstract: We introduce a computable invariant for non-degenerate knots based on Local DivergenceIntensity (LDI), a measure derived from the topological theory of distinguishability. Theinvariant is constructed by encoding a knot’s Frenet-Serret curvature and torsion functionsas symbolic sequences, then computing the LDI signature—a 4-dimensional vector capturingcoordination structure at multiple scales. We prove that distinct non-degenerate knots havedistinct LDI signatures, establishing the invariant’s injectivity. The construction is fullyalgorithmic: given a smooth knot representative, the signature can be computed to arbitraryprecision in polynomial time. We demonstrate the invariant on classical examples includingthe trefoil and figure-eight knots, and discuss the open question of completeness

Source: Zenodo (CERN European Organization for Nuclear Research)

DOI: 10.5281/zenodo.17764167

Authors: Taylor [email protected], John, Taylor [email protected], John

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7107943779

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