Browsing by Author "Ng, Lenhard"

Now showing 1 - 4 of 4
  • Thumbnail Image
    ItemOpen Access
    Braid Loops with infinite monodromy on the Legendrian contact DGA
    (2021-01-06) Casals, Roger; Ng, Lenhard
    We present the first examples of elements in the fundamental group of the space of Legendrian links in the standard contact 3-sphere whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These families include the first known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, in particular giving the first Floer-theoretic proof that Legendrian (n,m) torus links have infinitely many Lagrangian fillings, if n is greater than 3 and m greater than 6, or (n,m)=(4,4),(4,5). In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.
  • Thumbnail Image
    ItemOpen Access
    Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials
    Ekholm, Tobias; Ng, Lenhard
    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 by Aganagic, Vafa, and the authors, and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety.
  • Thumbnail Image
    ItemOpen Access
    Obstructions to Lagrangian concordance
    (Algebraic & Geometric Topology, 2016-04-26) Cornwell, Christopher; Ng, Lenhard; Sivek, Steven
    We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in $\mathbb{R}^3$. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with non-reversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slice.
  • Thumbnail Image
    ItemOpen Access
    Representations, sheaves, and Legendrian $(2,m)$ torus links
    Chantraine, Baptiste; Ng, Lenhard; Sivek, Steven
    We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a category of sheaves of microlocal rank $n$ constructed by Shende, Treumann, and Zaslow. We establish the cohomological version of this conjecture for a family of Legendrian $(2,m)$ torus links.