A two-variable series for knot complements

Abstract

The physical 3d N D 2 theory T OEY was previously used to predict the existence of some 3-manifold invariants y Z a .q/ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in S 3 , the analogue of the invariants y Z a .q/ should be a twovariable series F K .x; q/ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates F K .x; q/ to the invariants y Z a .q/ for Dehn surgeries on the knot. We provide explicit calculations of F K .x; q/ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figureeight knot, up to any desired order, and use this to understand y Z a .q/ for some hyperbolic 3-manifolds.

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

“…The choice of g(x) is determined by the choice of X. In other words, in the present setup the dictionary (1.1) reads [8]. A large class of examples that produce (3.1) with different g(x) comes from X with one isolated fixed point p ∈ X, such that T X| p ∼ = C 2n has weights (w 1 , −w 1 , w 2 , −w 2 , .…”

“…is basically a q-deformation of the inverse Alexander polynomial [8]. Here, the Alexander variable x encodes the dependence on Spin c structure, cf.…”

“…One can also motivate Conjecture 3 and MTC[X p ] by interpreting Z-invariants as a non-perturbative definition of the analytically continued Chern-Simons theory with complex gauge group G C . In this theory, the infinite-dimensional Hilbert space H(T 2 ) carries an action of the modular group MCG(T 2 ) = SL(2, Z) that plays an important role in surgery operations [8,84]. At special values of q = ζ p it contains a finite-dimensional subrepresentation associated with compact submanifolds in the character variety M flat (G C , T 2 ).…”

“…This space (or, its ALF version X = Taub-NUT) is the Coulomb branch of 3d Seiberg-Witten gauge theory [7], so that the Rozansky-Witten invariants in this case are equal to the Seiberg-Witten invariants of M 3 . The latter depend on the choice of additional structure, namely the choice of Spin c structure on M 3 , which itself has a non-trivial behavior under cutting and gluing [8]. However, apart from keeping track of this additional structure, the Reidemeister-Turaev-Milnor torsion of M 3 computed by the Seiberg-Witten theory [9,10] has a very simple behavior under cutting and gluing: it is basically multiplicative [11,12].…”

“…A seemingly different class of new invariants, introduced in [16,28], associates a q-series with integer coefficients to a 3-manifold M 3 equipped with a choice of an abelian flat connection (or, more precisely, a choice of Spin c structure [8]). These q-series invariants, denoted Z b (M 3 ), provide a non-perturbative completion to all-loop perturbative expansions [29] in SL(2, C) Chern-Simons theory that behave well under cutting and gluing (surgery) operations.…”