Knot Invariants & Jones Polynomial

Trefoil, figure-8, unknot — Reidemeister moves and the Jones polynomial

Skein relation: t⁻¹·V(L₊) − t·V(L₋) + (t^(−1/2)−t^(1/2))·V(L₀) = 0
Crossings: 0 | Crossing number: 0 | Writhe: 0

Jones Polynomial V(t)

V(t) = 1 (unknot is trivially knotted)

Knot Invariants

Crossing number: 0 | Genus: 0 | Unknotting number: 0

Reidemeister Moves

R1: Add/remove a loop (changes writhe by ±1)
R2: Slide two strands past each other
R3: Triangle move — slide strand over crossing
Jones polynomial is invariant under all three!

Deep Connections

Jones polynomial = Chern-Simons path integral (Witten 1989)
Khovanov homology categorifies Jones: V(t) = Euler characteristic
Open question: Does Jones polynomial detect the unknot?