๐Ÿ”— Gauss Linking Number โ€” Two Curves in 3D

Lk(ฮณโ‚,ฮณโ‚‚) = (1/4ฯ€) โˆฎโˆฎ (แน™โ‚ร—แน™โ‚‚)ยท(rโ‚โˆ’rโ‚‚)/|rโ‚โˆ’rโ‚‚|ยณ dsโ‚ dsโ‚‚ โ€” topological invariant of two closed curves
Linking Number Lk
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Gauss integral
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Crossings (signed)
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Gauss (1833): Lk is an integer, topological invariant โ€” unchanged by smooth deformations.

Computed as double line integral. Also equals half the sum of signed crossings in a projection.

Applications: DNA supercoiling, electromagnetic inductance (Neumann formula), knot theory foundation.