Kauffman Bracket — Knot Polynomials & Skein Relations

The Kauffman bracket ⟨K⟩ is a polynomial in A satisfying the skein relation ⟨L⟩ = A⟨L₀⟩ + A⁻¹⟨L∞⟩ and ⟨unknot⟩ = 1. After writhe correction, it yields the Jones polynomial V(t) at t=A⁻⁴. Skein relations reduce any knot diagram to sums of unknots, computing an invariant that detects chirality and distinguishes many knots that Alexander polynomial cannot.

Kauffman bracket:
⟨ ⟩ = A⟨ ⟩₀ + A⁻¹⟨ ⟩∞
⟨K⊔O⟩ = δ⟨K⟩ (δ=−A²−A⁻²)
⟨O⟩ = 1

Jones polynomial:
V_L(t) = (−A)^{−3w}⟨L⟩|_{A²=t^{−1/2}}
w = writhe = Σ signs of crossings

HOMFLY-PT:
PL(v,z): v⁻¹P(L+)−vP(L−)=zP(L₀)
Specialization: Jones at v=t, z=√t−1/√t

Colored Jones:
J_n(K;q) = quantum dimension n
Volume conjecture: 2π lim J_n/n = vol
Select a knot to see its Jones polynomial.