3d-3d correspondence for knot complements with finite and large N

A bstract For G = SU( N ) at finite and large N , with a totally symmetric representation, we realize the homological block F K for a knot complement S 3 \ K , given in the form of the inverted Habiro series, as a half-index of a 3d 𝒩 = 2 theory, which is expected to provide a UV description of T [ M 3 ]. From the half-index expression, it is also possible to realize the colored HOMFLY-PT polynomial by summing the residues at a certain set of poles. Through the half-index realization, we describe a method for obtaining the G = SU( N ) homological block and its a -deformed version for S 3 \ K in the form of the inverted Habiro series from a given Habiro series expression for the colored HOMFLY-PT polynomial. We also discuss some properties of partition functions for arbitrary N .