Knot Theory — Invariants, Crossings & Reidemeister Moves

🪢 Knot Theory

A mathematical knot is a closed loop in 3D space that cannot be untangled to a circle without cutting. Knots are classified by invariants that don't change under continuous deformation: crossing number, writhe, Alexander polynomial.

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📐 Invariants

Crossing #

Writhe

Alexander Δ(t)

t−1+t⁻¹

Unknotting #

ℹ️ Theory

Reidemeister moves: any two diagrams of the same knot are related by a sequence of 3 moves (R1: twist, R2: poke, R3: slide).

Alexander polynomial Δ(t): a Laurent polynomial invariant — if Δ(t) differs, knots are distinct. Trefoil: t − 1 + t⁻¹. Figure-eight: −t + 3 − t⁻¹.

Writhe: sum of signed crossings (+1 for right-hand, −1 for left-hand). Depends on diagram orientation.