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The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander polynomial of a knot in terms of the augmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. The expression is derived from a description of the Alexander polynomial as a count of Floer strips and holomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangian with the topology of the knot complement and the zero-section, and from a description of the boundary of the moduli space of such annuli with one positive puncture.

Consider a pair (X, L) of a Weinstein manifold X with an exact Lagrangian submani-fold L, with ideal contact boundary (Y, A), where Y is a contact manifold and A c Y is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, CE*(A), with coefficients in chains of the based loop space of A, and study its relation to the Floer cohomology CF*(L) of L. Using the augmentation induced by L, CE*(A) can be expressed as the Adams cobar construction 2 applied to a Legendrian coalgebra, LC*(A). We define a twisting cochain t: LC*(A)-* B(CF*(L))# via holomorphic curve counts, where B denotes the bar construction and # the graded linear dual. We show under simple-connectedness assumptions that the corresponding Koszul complex is acyclic, which then implies that CE*(A) and CF*(L) are Koszul dual. In particular, t induces a quasi-isomorphism between CE*(A) and 2CF*(L), the cobar of the Floer homology of L.This generalizes the classical Koszul duality result between C*(L) and C -*(2L) for L a simply connected manifold, where 2L is the based loop space of L, and provides the geometric ingredient explaining the computations given by Etgu and Lekili (2017) in the case when X is a plumbing of cotangent bundles of 2-spheres (where an additional weight grading ensured Koszulity of t). We use the duality result to show that under certain connectivity and local-finiteness assumptions, CE*(A) is quasi-isomorphic to C -*(2L) for any Lagrangian filling L of A.Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that CE*(A) is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk C in the Weinstein domain obtained by attaching T*(A x [0, oo)) to X along A (or, in the terminology of Sylvan (2019), the wrapped Floer cohomology of C in X with wrapping stopped by A). Along the way, we give a definition of wrapped Floer cohomology via holomorphic buildings that avoids the use of Hamiltonian perturbations, which might be of independent interest.

We introduce generalized quiver partition functions of a knot K and conjecture a relation to generating functions of symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincare polynomials. We interpret quiver nodes as certain basic holomorphic disks in the resolved conifold, with boundary on the knot conormal L-K, a positive multiple of a unique closed geodesic, and with their (infinitesimal) boundary linking density measured by the adjacency matrix of the generalized quiver. The basic holomorphic disks that are quiver nodes appear in a certain U(1)-symmetric configuration. We propose an extension of the quiver partition function to arbitrary, not U(1)-symmetric, configurations as a function with values in chain complexes. The chain complex differential is trivial at the U(1)-symmetric configuration, under deformations the complex changes, but its homology remains invariant. We also study recursion relations for the partition functions connected to knot homologies. We show that, after a suitable change of variables, any (generalized) quiver partition function satisfies the recursion relation of a single toric brane in C-3.

We generalize the F-K invariant, i.e. (Z) over cap for the complement of a knot Kin the 3-sphere, the knots-quivers correspondence, and A-polynomials of knots, and find several interconnections between them. We associate an F-K invariant to any branch of the A-polynomial of K and we work out explicit expressions for several simple knots. We show that these F-K invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using R-matrices. We generalize the quantum a-deformed A-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter a, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide t-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d N = 2 theory T[M-3] and to the data of the associated modular tensor category MTC[M-3].

The Chekanov–Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams [22]. It also leads to a proof of a conjectured isomorphism [17, 25] between partially wrapped Floer cohomology and Chekanov–Eliashberg dg-algebras with coefficients in chains on the based loop space.

We define a refined Gromov-Witten disk potential of monotone immersed Lagrangian surfaces in a symplectic 4-manifold that are self-transverse as an element in a capped version of the Chekanov- Eliashb erg dg-algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point. We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5-sphere.

Reducing a 6d fivebrane theory on a 3-manifold Y gives a q-series 3-manifold invariant (Z ) over cap (Y). We analyse the large-N behaviour of F-K = (Z ) over cap (M-K), where M-K is the complement of a knot K in the 3-sphere, and explore the relationship between an a-deformed (a = q(N)) version of F-K and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of F-K in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for a-deformed F-K for (2, 2p + 1) torus knots and an order-by-order construction for other cases. They both suggest a further -deformation based on superpolynomials, which can be used to obtain a 1-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how F-K transforms under natural geometric operations on K indicates relations to quantum modularity in a new setting.

We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large N duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We present an argument that the HOMFLY-PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the higher genus counterpart of the relation between the augmentation variety and Gromov-Witten disk potentials established in [1] by Aganagic, Vafa, and the authors, and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety.

The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver descriptions of the geometry. Embedding into M-theory, a large class of dualities of 3d N = 2 theories associated to quivers is obtained. The multi-cover skein relation admits a compact formulation in terms of quantum torus algebras associated to the quiver and in this language the relations are similar to wall-crossing identities of Kontsevich and Soibelman.

The recently conjectured knots-quivers correspondence (Kucharski et al. in Phys RevD96(12):121902, 2017. arXiv:1707.02991, Adv Theor Math Phys 23(7):1849-1902, 2019. arXiv:1707.04017) relates gauge theoretic invariants of a knot K in the 3sphere to the representation theory of a quiver Q(K) associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of Ooguri-Vafa large N duality (Ooguri and Vafa in Nucl Phys B 577:419-438, 2000), that relates knot invariants to counts of holomorphic curves with boundary on LK, the conormal Lagrangian of the knot in the resolved conifold, and corresponding Mtheory considerations. From the physics side, we show that the quiver encodes a 3d N = 2 theory T [Q(K)] whose low energy dynamics arises on the worldvolume of anM5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of Q(K). From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on LK is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver QK and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to Q(K) and (doubly) refined version of LMOV invariants (Ooguri and Vafa 2000; Labastida and Marino in Commun Math Phys 217(2):423-449, 2001. arXiv:hep-th/0004196; Labastida et al. in JHEP 11:007, 2000. arXiv:hep-th/0010102; Aganagic and Vafa in Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots. arXiv:1204.4709; Fuji et al. in Nucl Phys B 867:506-546, 2013. arXiv:1205.1515).