Chern-Simons Theory — Link Invariants

SCS = k/4π ∫ Tr(A∧dA + ⅔A∧A∧A)  |  Witten 1989
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Chern-Simons theory is a 3D topological quantum field theory whose partition function and Wilson loop expectation values compute topological link invariants. For gauge group SU(2) at level k, the path integral Z = ∫DA e^{iS_CS[A]} with Wilson loops W_R[C] = Tr_R P exp(i∮_C A) computes the Jones polynomial (Witten 1989, Fields Medal): ⟨W_R[L]⟩ = V_L(q) with q = e^{2πi/(k+2)}. The skein relation Δ⁺V₊ − Δ⁻V₋ = (√q − 1/√q)V₀ determines the polynomial recursively. The visualization shows 3D projections of standard links as animated space curves, with their crossing data, writhe (w), and computed link invariants.