Braid group Bₙ: generators σ₁,…,σₙ₋₁
σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁ (Reidemeister III)
σᵢσⱼ = σⱼσᵢ, |i−j| ≥ 2
Markov's theorem: braids ↔ knots
under stabilization & conjugacy
Jones: V_K(t) via Burau / Temperley-Lieb
Skein: −t⁻¹V₊ + (t⁻½−t½)V₀ + tV₋ = 0
V(unknot) = 1
Writhe w = Σ signs of crossings
V_K(t) = (−t)^(−3w) · f(K)
σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁ (Reidemeister III)
σᵢσⱼ = σⱼσᵢ, |i−j| ≥ 2
Markov's theorem: braids ↔ knots
under stabilization & conjugacy
Jones: V_K(t) via Burau / Temperley-Lieb
Skein: −t⁻¹V₊ + (t⁻½−t½)V₀ + tV₋ = 0
V(unknot) = 1
Writhe w = Σ signs of crossings
V_K(t) = (−t)^(−3w) · f(K)
3
Jones Polynomial V_K(t):
Enter braid word above...
Type a braid word using integers: positive n = σₙ (over-crossing), negative = σₙ⁻¹ (under-crossing). Strands numbered 1 to N-1. The braid closure turns the braid into a knot/link.