Symmetry enhancement and closing of knots in 3d/3d correspondence

We revisit Dimofte-Gaiotto-Gukov's construction of 3d gauge theories associated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d $N=(2,0)$ theory of $A_1$-type on a 3-manifold, we propose a topological criterion for $SU\left(2\right)/SO\left(3\right)$ flavor symmetry enhancement for the $u\left(1\right)$ symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the construction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising $SU\left(3\right)$ symmetry enhancement which is unexpected from the 6d viewpoint.