Kauffman knot invariant from $\mathrm{SO}(N)$ or $\mathrm{Sp}(N)$ Chern-Simons theory and the Potts model

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), $\mathrm{SO}(-2)$, 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 $\mathrm{SO}(\pm N)$ and $\mathrm{Sp}(\mp N)$ invariants. A correspondence between the first orders in perturbation theory of $\mathrm{SO}(-2)$, Sp(2) or SU(2) Chern-Simons quantum holonomy’s traces and the partition function of the $Q=4$ Potts model is built.

Figure 1

Different crossing configurations involved in the skein relations. Dealing with unoriented links, arrows can be ignored because they carry no sensitive information.

Figure 2

Abstract diagrammatic representation of Fierz identity for $\mathrm{SO}(N)$.

Figure 3

Diagrammatic closure of the $\mathrm{SO}(N)$ skein relation (3.3).

Figure 4

Fierz identity for $\mathrm{Sp}(N)$, dots represent points where orientations of the line change.

Figure 5

Diagrammatic representation of Fierz identity for Sp(2).

Figure 6

Skein relation (2.1)-(i) applied to the upper $\mathcal{H}\mathcal{L}$ crossing.

Figure 7

$K(G)$ $\leftrightarrow$ shading of $K(G)$ $\leftrightarrow$ emerging of lattice graph $G$ inside $K$ $\leftrightarrow$ $G(K)$.