Abstract
The expectation value of Wilson loop operators in three-dimensional $\mathrm{SO}(N)$ Chern-Simons gauge theory gives a known knot invariant: the Kauffman polynomial. Here this result is derived, at the first order, via a simple variational method. With the same procedure the skein relation for Sp(N) are also obtained. Jones polynomial arises as special cases: Sp(2), $\mathrm{SO}(\ensuremath{-}2)$, and $\mathrm{SL}(2,\mathbb{R})$. These results are confirmed and extended up to the second order, by means of perturbation theory, which moreover let us establish a duality relation between $\mathrm{SO}(\ifmmode\pm\else\textpm\fi{}N)$ and $\mathrm{Sp}(\ensuremath{\mp}N)$ invariants. A correspondence between the first orders in perturbation theory of $\mathrm{SO}(\ensuremath{-}2)$, Sp(2) or SU(2) Chern-Simons quantum holonomy's traces and the partition function of the $Q=4$ Potts model is built.
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