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We construct 3D $\mathcal{N}=2$ abelian gauge theories on $\mathbb{S}^2\times \mathbb{S}^1$ labeled by knot diagrams whose K-theoretic vortexpartition functions, each of which is a building block of twisted indices, givethe colored Jones polynomials of knots in $\mathbb{S}^3$. The colored Jonespolynomials are obtained as the Wilson loop expectation values along knots in$SU(2)$ Chern-Simons gauge theories on $\mathbb{S}^3$, and then ourconstruction provides an explicit correspondence between 3D $\mathcal{N}=2$abelian gauge theories and 3D $SU(2)$ Chern-Simons gauge theories. We verify,in particular, the applicability of our constructions to a class of tanglediagrams of 2-bridge knots with certain specific twists.

The colored Jones polynomials as vortex partition functions