# Browsing by Author "Ng, L"

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Item Open Access A complete knot invariant from contact homologyEkholm, T; Ng, L; Shende, VWe construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant. Furthermore, we relate the holomorphic and sheaf approaches via calculations of partially wrapped Floer homology in the spirit of [BEE12].Item Open Access A topological introduction to knot contact homology(Bolyai Society Mathematical Studies, 2014-01-01) Ng, LThis is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects.Item Open Access Augmentations are SheavesNg, L; Rutherford, D; Shende, V; Sivek, S; Zaslow, EWe show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.Item Open Access Braid loops with infinite monodromy on the Legendrian contact DGA(Journal of Topology, 2022-12) Casals, R; Ng, LItem Open Access Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials(Advances in Theoretical and Mathematical Physics, 2020-01-01) Ekholm, T; Ng, LWe 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 varietyItem Open Access Knot contact homology(Geometry & Topology, 2013-04-30) Ekholm, T; Etnyre, JB; Ng, L; Sullivan, MGThe conormal lift of a link K in ℝ3is a Legendrian submanifold ∧Kin the unit cotangent bundle U*ℝ3of ℝ3with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of K, is defined as the Legendrian homology of ∧K, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization R × U*ℝ3with Lagrangian boundary condition R × ∧K. We perform an explicit and complete computation of the Legendrian homology of ∧Kfor arbitrary links K in terms of a braid presentation of K, confirming a conjecture that this invariant agrees with a previously defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot, which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.Item Open Access Knot contact homology, string topology, and the cord algebra(Journal de l’École polytechnique — Mathématiques, 2017) Cieliebak, K; Ekholm, T; Latschev, J; Ng, LThe conormal Lagrangian LKof a knot K in R3is the submanifold of the cotangent bundle T∗R3consisting of covectors along K that annihilate tangent vectors to K. By intersecting with the unit cotangent bundle S∗R3, one obtains the unit conormal ΛK, and the Legendrian contact homology of ΛKis a knot invariant of K, known as knot contact homology. We define a version of string topology for strings in R3∪ LKand prove that this is isomorphic in degree 0 to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree 0 knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in T∗R3with boundary on R3∪ LK.Item Open Access Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold(Journal of Differential Geometry, 2015-09) Ekholm, T; Ng, LWe give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1× S2or any connected sum #k(S1×S2), viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology [5], this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformations, from both the combinatorial and the analytical perspectives.Item Open Access ON ARC INDEX AND MAXIMAL THURSTON–BENNEQUIN NUMBER(Journal of Knot Theory and Its Ramifications, 2012-04) Ng, LWe discuss the relation between arc index, maximal ThurstonBennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal ThurstonBennequin number for all knots with at most 11 crossings. For some of these knots, the calculation requires a consideration of cables which also allows us to compute the maximal self-linking number for all knots with at most 11 crossings. © 2012 World Scientific Publishing Company.Item Open Access On transverse invariants from Khovanov homology(Quantum Topology, 2015) Lipshitz, R; Ng, L; Sarkar, S© European Mathematical Society. In [31], O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two re_nements of Plamenevskaya’s invariant, one valued in Bar-Natan’s deformation (from [2]) of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum (from [20]). We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskaya’s class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both re_nements are determined by the classical invariants of K.Item Open Access Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra(Algebraic & Geometric Topology, 2013-08-01) Ng, L; Rutherford, DWe develop a close relation between satellites of Legendrian knots in ℝ3and the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in ℝ3and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological type and Thurston-Bennequin number of the knot.Item Open Access The cardinality of the augmentation category of a Legendrian link(Mathematical Research Letters, 2017) Ng, L; Rutherford, D; Shende, V; Sivek, SWe introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R3. This ℓhomotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.Item Unknown Topological strings, D-model, and knot contact homology(Advances in Theoretical and Mathematical Physics, 2014) Aganagic, M; Ekholm, T; Ng, L; Vafa, C© 2014 International Press. We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov- Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model". In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Qdeformation of the extension of A-polynomials associated with the link complement.