The colored Jones polynomials as vortex partition functions

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We construct 3D <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 2 abelian gauge theories on <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathbbm{S} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></jats:alternatives></jats:inline-formula><jats:sup>2</jats:sup> × <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathbbm{S} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></jats:alternatives></jats:inline-formula><jats:sup>1</jats:sup> labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathbbm{S} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></jats:alternatives></jats:inline-formula><jats:sup>3</jats:sup>. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathbbm{S} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></jats:alternatives></jats:inline-formula><jats:sup>3</jats:sup>, and then our construction provides an explicit correspondence between 3D <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 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 tangle diagrams of 2-bridge knots with certain specific twists.</jats:p>