The First Post — Jones Polynomial

The Jones polynomial via Wilson loop is essentially obtained from Cayley-Hamilton’s equation. We will work with SU(2)\(_k\) gauge group. Let the trivial loop in 2+1 spacetime of a majorana zero mode be \(d\) then

\[ d = \int \mathcal{D}a e^{iS_\mathrm{CS}[a]}W_{\gamma}[a]. \]

Now for two majorana zero modes they could braid. Before they braid we pick a time slice and set it to \(t=0\). Then

\[ \psi = \int \mathcal{D}a e^{iS[a]}W_{\gamma_-}W_{\gamma^\prime_-} \exp\left(\int_{-\infty}^0 dt \int d^2x \mathcal{L}_\mathrm{CS}\right). \]

After the time \(t=0\) it braids to be, for example \(\sigma_i\psi\). By drawing we find that

\[ \langle \psi | \psi\rangle = d^2, \quad \langle \psi | \sigma_i |\psi\rangle = \langle \psi | \sigma_i^{-1} |\psi\rangle = d. \]

Let the quasiparticle be in \(j=1/2\) irrep then \(\sigma\) is a 2-by-2 matrix, so it satisfies the Caylay-Hamilton’s equation

\[ \sigma - \mathrm{tr}\sigma + \det \sigma \cdot \sigma^{-1} = 0. \]

The eigenvalues are \(R_1^{\sigma \sigma} = -q^{3/2}\) and \(R_\psi^{\sigma \sigma} = q^{-1/2}\) where \(q=-e^{-\pi i /2(k+2)}\) then finally we obtain

\[ q^{-1/2} \sigma_i - q^{1/2} \sigma_i^{-1} = q - q^{-1} \]

which is the skein relation of the Jones polynomial.