About

Lenhard Lee Ng is a Professor of Mathematics at Duke University, holding a position since 2016 within the Trinity College of Arts & Sciences. His research primarily focuses on symplectic topology and low-dimensional topology, with particular interest in structures within symplectic and contact geometry, such as Weinstein manifolds, contact manifolds, Legendrian and transverse knots. He employs holomorphic-curve techniques to study these structures and is especially interested in extracting topological information about knots through cotangent bundles and exploring their relations to topological string theory. Ng has also contributed to Heegaard Floer theory, quantum topology, and sheaf theory, especially as they relate to Legendrian and transverse knots. His work includes constructing algebraic invariants for Legendrian knots, analyzing torsion in contact homology, and developing invariants of Legendrian links, supported by multiple grants from the National Science Foundation. Ng earned his Ph.D. from the Massachusetts Institute of Technology in 2001 and is recognized for his contributions to the field through publications and leadership in mathematical research.

Research topics

• Mathematics
• Pure mathematics
• Biology
• Combinatorics
• Geography
• Genetics
• Archaeology

Selected publications

• Torsion in Linearized Contact Homology for Legendrian Knots

  The Michigan Mathematical Journal · 2025-01-01 · 1 citations

  articleSenior author

  We present examples of Legendrian knots in R3 that have linearized Legendrian contact homology over Z containing torsion. As a consequence, we show that there exist augmentations of Legendrian knots over Z that are not induced by exact Lagrangian fillings, even though their mod 2 reductions are.

  PublisherDOI

• An L∞$L_\infty$ structure for Legendrian contact homology

  Journal of Topology · 2025-07-31

  article1st authorCorresponding

  Abstract For any Legendrian knot or link in , we construct an algebra that can be viewed as an extension of the Chekanov–Eliashberg differential graded algebra. The structure incorporates information from rational symplectic field theory and can be formulated combinatorially. One consequence is the construction of a Poisson bracket on commutative Legendrian contact homology, and we show that the resulting Poisson algebra is an invariant of Legendrian links under isotopy.

  PublisherDOI

• An L-infinity structure for Legendrian contact homology

  arXiv (Cornell University) · 2023-11-24

  preprintOpen access1st authorCorresponding

  For any Legendrian knot or link in $\mathbb{R}^3$, we construct an $L_\infty$ algebra that can be viewed as an extension of the Chekanov-Eliashberg differential graded algebra. The $L_\infty$ structure incorporates information from rational Symplectic Field Theory and can be formulated combinatorially. One consequence is the construction of a Poisson bracket on commutative Legendrian contact homology, and we show that the resulting Poisson algebra is an invariant of Legendrian links under isotopy.

  PublisherOA PDFDOI

• Torsion in linearized contact homology for Legendrian knots

  arXiv (Cornell University) · 2023-08-25

  preprintOpen accessSenior author

  We present examples of Legendrian knots in $\mathbb{R}^3$ that have linearized Legendrian contact homology over $\mathbb{Z}$ containing torsion. As a consequence, we show that there exist augmentations of Legendrian knots over $\mathbb{Z}$ that are not induced by exact Lagrangian fillings, even though their mod $2$ reductions are.

  PublisherOA PDFDOI

• Braid loops with infinite monodromy on the Legendrian contact DGA

  Journal of Topology · 2022 · 23 citations

  Senior authorCorresponding
  • Mathematics
  • Pure mathematics
  • Geography

  We present the first examples of elements in the fundamental group of the space of Legendrian links in ( S 3 , ξ st ) $(\mathbb {S}^3,\xi _{\text{st}})$ 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 new 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, and in particular give the first Floer-theoretic proof that Legendrian ( n , m ) $(n,m)$ torus links have infinitely many Lagrangian fillings if n ⩾ 3 , m ⩾ 6 $n\geqslant 3,m\geqslant 6$ or ( n , m ) = ( 4 , 4 ) , ( 4 , 5 ) $(n,m)=(4,4),(4,5)$ . 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 in the same smooth isotopy class, but distinct Hamiltonian isotopy classes. 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.

  PublisherDOI

• Braid Loops with infinite monodromy on the Legendrian contact DGA

  arXiv (Cornell University) · 2021-01-07

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Legendrian contact homology in $\mathbb{R}^3$

  Surveys in Differential Geometry · 2020 · 16 citations

  Senior authorCorresponding
  • Mathematics
  • Combinatorics
  • Genetics

  This is an introduction to Legendrian contact homology and the Chekanov-Eliashberg differential graded algebra, with a focus on the setting of Legendrian knots in $\mathbb{R}^3$.

  PublisherDOI

• Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials

  Advances in Theoretical and Mathematical Physics · 2020-01-01 · 13 citations

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Representations, sheaves, and Legendrian $(2,m)$ torus links

  HAL (Le Centre pour la Communication Scientifique Directe) · 2019-08-01 · 6 citations

  articleCorresponding

  v2: 50 pages, added discussion in the introduction about geometric motivation

  Publisher

• Higher genus knot contact homology and recursion for colored HOMFLY-PT\n polynomials

  arXiv (Cornell University) · 2018-03-11

  preprintOpen accessSenior author

  We sketch a construction of Legendrian Symplectic Field Theory (SFT) for\nconormal tori of knots and links. Using large $N$ duality and Witten's\nconnection between open Gromov-Witten invariants and Chern-Simons gauge theory,\nwe relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of\nthe link. We present an argument that the HOMFLY-PT wave function is determined\nfrom SFT by induction on Euler characteristic, and also show how to, more\ndirectly, extract its recursion relation by elimination theory applied to\nfinitely many noncommutative equations. The latter can be viewed as the higher\ngenus counterpart of the relation between the augmentation variety and\nGromov-Witten disk potentials established by Aganagic, Vafa, and the authors,\nand, from this perspective, our results can be seen as an SFT approach to\nquantizing the augmentation variety.\n

  PublisherOA PDFDOI

Recent grants

• Holomorphic Invariants in Symplectic Topology

  NSF · $356k · 2017–2021

• Holomorphic Invariants of Knots and Contact Manifolds

  NSF · $360k · 2020–2024

• Holomorphic Curves and Low-Dimensional Topology

  NSF · $129k · 2007–2010

• CAREER: Symplectic Field Theory and Low-Dimensional Topology

  NSF · $401k · 2009–2015

• Knots and contact topology through holomorphic curves

  NSF · $437k · 2014–2018

Frequent coauthors

• Tobias Ekholm

  18 shared

• Dylan P. Thurston

  Indiana University Bloomington

  5 shared

• Steven Sivek

  Imperial College London

  5 shared

• John B. Etnyre

  Georgia Institute of Technology

  5 shared

• Joshua M. Sabloff

  4 shared

• Joshua Sabloff

  Haverford College

  4 shared

• Vivek Shende

  University of Southern Denmark

  3 shared

• Peter Ozsváth

  Princeton University

  3 shared

Awards & honors

• Fellow of the American Mathematical Society (2018)