Tobias Ekholm

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.

We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi–Yau 3-folds. We show that the expected codimension one phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows that counting holomorphic curves by the class of their boundaries in the skein module of the Lagrangian gives a deformation invariant result. This is a mathematically rigorous incarnation of Witten’s assertion that boundaries of open topological strings create line defects in Chern–Simons theory [42]. Using this theory, we rigorously establish the following prediction of Ooguri and Vafa: the coefficients of the HOMFLYPT polynomial of a link in the three-sphere count the holomorphic curves in the resolved conifold, with boundary on (a push-off of) the link conormal.

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.

Given a link in the three-sphere, its Lagrangian conormal can be transplanted to the “resolved conifold,” which is a certain noncompact Calabi–Yau threefold. Here we show that, as predicted by Ooguri and Vafa using string theoretic arguments, the count of all holomorphic curves in the resolved conifold ending on this Lagrangian is, appropriately understood, the collection of the HOMFLYPT invariants of all colorings of the link. This generalizes our previous work, Skeins on branes, arXiv:1901.08027, which identified a curve count that captures the uncolored case. The main ingredient in the present work is a skein-valued multiple cover formula for an isolated embedded annulus.

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.

This is an overview paper that describes Eliashberg's Legendrian surgery approach to wrapped Floer cohomology and use it to derive the basic relations between various holomorphic curve theories with additional algebraic constructions. We also give a brief discussion of further results that use the surgery perspective, e.g., for holomorphic curve invariants of singular Legendrians and Lagrangians.

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 [arXiv:1304.5778, arXiv:1803.04011]. We consider skein lifts of recursions and $D$-modules corresponding to skein valued open curve counts [arXiv:1901.08027] 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.