Pietro Longhi

HOMFLYPT polynomials of knots in the 3-sphere in symmetric representations satisfy recursion relations. Their geometric origin is holomorphic curves at infinity on knot conormals that determine a D -module with characteristic variety the Legendrian knot conormal augmention variety and with the recursion relations as operator polynomial generators [1, 2]. We consider skein lifts of recursions and D -modules corresponding to skein valued open curve counts [3] that encode HOMFLYPT polynomials colored by arbitrary partitions. We define a worldsheet skein module which is the universal target for skein curve counts and a corresponding D -module. We then consider the concrete example of the Legendrian conormal of the Hopf link. We show that the worldsheet skein D -module for the Hopf link conormal is generated by three operator polynomials that annihilate the skein valued partition function for any choice of Lagrangian filling and recursively determine it uniquely. We find Lagrangian fillings for any point in the augmentation variety and show that their skein valued partition functions admit quiver-like expansions where all holomorphic curves are generated by a small number of basic holomorphic disks and annuli and their multiple covers.

A bstract: A certain class of A -branes in mirrors of toric Calabi-Yau threefolds can be described through the framework of foliations. This allows to develop an explicit description of their moduli spaces based on a cell decomposition, with strata of various dimensions glued together in a way that is dictated by partial degenerations of the underlying special Lagrangian. Examples of A -branes associated with ‘wild’ BPS states are considered in detail. The torus fixed points in their moduli spaces provide a decomposition of m -herds spectral networks into a number |Ω| of basic connected objects, where Ω is the corresponding rank-zero Donaldson-Thomas (DT) invariant. A relation between the surgery parameters of the special Lagrangian and the baryonic semi-invariants of the representation theory of m -Kronecker quivers is also discussed, providing a local map between moduli spaces of branes related by homological mirror symmetry.

Using skein valued holomorphic curve counting techniques, we give a flow loop formula for the skein valued partition function of the Lagrangian knot complement of a fibered knot (of the $A$-model open topological strings with Lagrangian $A$-branes wrapping the complement) in the cotangent bundle of the three-sphere and in the resolved conifold. For torus knots we show that the partition function in the cotangent bundle localizes on two or three holomorphic annuli and give a corresponding generalized quiver structure for the partition function in the resolved conifold. We connect the formula to the augmentation curve, the representation variety of the knot contact homology algebra of the knot, generated by Reeb chords of its Legendrian conormal and with differential given by holomorphic disks interpolating between words of Reeb chords. The curve admits a quantization as a $q$-difference equation for the generating function of symmetrically colored HOMFLYPT-polynomials of the knot or, geometrically, for the $U(1)$-partition function of the knot conormal. For $(2,2p+1)$-torus knots we show that, after a change of variables, the partition function of the knot complement also satisfies this $q$-difference equation. This gives another geometrically defined coordinate chart for the $D$-module defined by the quantized augmentation polynomial.

We propose a mirror derivation of the quiver description of open topological strings known as the knots-quivers correspondence, based on enumerative invariants of augmentation curves encoded by exponential networks. Quivers are obtained by studying M2 branes wrapping holomorphic disks with Lagrangian boundary conditions on an M5 brane, through their identification with a distinguished sector of BPS kinky vortices in the 3d-3d dual QFT. Our proposal suggests that holomorphic disks with Lagrangian boundary conditions are mirror to calibrated 1-chains on the associated augmentation curve, whose intersections encode the linking of boundaries.

We count holomorphic curves in complex 3-space with boundaries on three special Lagrangian solid tori. The count is valued in the HOMFLYPT skein module of the union of the tori. Using 1-parameter families of curves at infinity, we derive three skein valued operator equations which must annihilate the count, and which dequantize to a mirror of the geometry. We show algebraically that the resulting equations determine the count uniquely, and that the result agrees with the topological vertex from topological string theory.