Quantifying the hierarchical topological decay of fluid knots.

An entanglement index based on adapted Jones polynomials accurately detects the decreasing topological complexity in superfluid vortex tangle decay across prime knots, links, and composites up to 8 crossings.

• Why it matters: Quantifying how knot and link topologies cascade during superfluid vortex tangles is challenging, and existing complexity metrics can be less reliable for tracking that hierarchical decay.
• What they did: The authors reformulate knot polynomial invariants algebraically, model evolutionary paths as geodesic flows in a knot-polynomial space, and compute an entanglement index as a probability density over topological states.
• The result: In simulations and laboratory experiments, the index matches the observed topological simplification and, compared with a complexity degree measure, is more robust and reliable for quantifying the cascade.

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