RESEARCH STATEMENT My Research Lies in the General

RESEARCH STATEMENT

LENHARD NG

My research lies in the general mathematical area of geometry and topology, and more specifically in the fields of low-dimensional topology and symplectic geometry. Recently I have been collaborating intensively with physicists as well, as we have discovered that my work is closely related to current developments in string theory and mirror symmetry, and I anticipate that a fair amount of my future work will point in this direction. Low-dimensional topology is the study of three- and four-dimensional geometric spaces, as well as knots in three-dimensional spaces. This field has rapidly developed in the past three decades, in part because of powerful new approaches to the subject through algebra, analysis, and physics (especially gauge theory). Since then, ideas have continued to pour into the subject from other areas of mathematics. One of the currently hottest approaches to low-dimensional topology, and an area that I have devoted much of my research to, comes from symplectic geometry. Symplectic geometry, a venerable subject with origins in physics and specifically classical mechanics, can be thought of as a branch of differential geometry and involves much interplay between geometry and analysis. It studies symplectic manifolds, which are even-dimensional spaces with a closed nondegenerate differential 2-form. There is an odd-dimensional ana- logue, contact geometry, and the two are very closely interrelated. I work in what is typically called symplectic topology, the study of global properties of symplectic and contact manifolds. Even more than low-dimensional topology, symplectic topology has undergone a revolution in the past three decades, roughly dating to Gromov’s introduction in the 1980’s of holomorphic-curve techniques in symplectic and algebraic geometry. Although symplectic topology is now a vibrant and exciting field in its own right, it has produced a number of spectacular recent applications to low-dimensional topology. One particular celebrated example is Ozsváthand Szabó’sdevelopment of Heegaard Floer theory, which uses symplectic techniques to construct powerful invariants of smooth three- and four-manifolds, knots, and more. A key reason for the success of symplectic approaches to topological problems is their combinatorial nature, allowing for easy computations. My own work emphasizes this combinatorial character while retaining strong ties to the geometry underlying symplectic topology. A major line of research for me has been the development of a new package of knot invariants in low-dimensional topology, broadly called “knot contact homology,” which is quite powerful and relatively easy to compute, and which we now understand to have unexpected and deep ties to string theory and mirror symmetry. I have also worked extensively on questions more purely within symplectic topology, especially involving the study of Legendrian knots; this also has recently extended to areas outside of the subject, in particular to algebraic geometry and mathematical mirror symmetry. The remainder of this research statement discusses these various aspects of my work, as well as directions for future work.

Date: September 2015. 1 a. Knot Contact Homology In the past decade and a half, an exciting area of research in symplectic topology has concerned connections to low-dimensional topology via the study of cotangent bundles. A familiar construction from classical mechanics associates a symplectic structure to the cotangent bundle of any smooth manifold. An old idea of Arnol‘d is to study the topology of smooth manifolds by examining the symplectic geometry of their cotangent bundles. Question 1. Does the symplectic structure on a cotangent bundle T ∗M encode the smooth structure on the underlying manifold M? That is, do distinct (non-diffeomorphic) smooth manifolds necessarily have different (non-symplectomorphic) symplectic cotangent bundles? This question has recently been studied extensively by many researchers including myself, and current indications are that the answer, surprisingly, may be “yes.” If this is the case, or even if the symplectic structure on cotangent bundles encodes a large amount of smooth information, then one can profitably translate problems in smooth topology into problems in symplectic geometry. The recent advent of powerful holomorphic-curve techniques in symplectic geometry has turned this classical approach into a very interesting and lively subject (see, e.g., [AS06, Abo12, CL09, SW06, Vit]). My work in the subject begins with an extension of the cotangent construction: to a knot in usual three-space R3, one can naturally associate a Lagrangian submanifold, the knot’s conormal bundle, of the cotangent bundle. (This construction has been studied in string theory, notably by Ooguri–Vafa [OV00], as well as symplectic topology.) As part of the Symplectic Field Theory package of Eliashberg, Givental, and Hofer [EGH00], one can assemble a count of certain holomorphic curves with boundary on this Lagrangian to give a Floer-theoretic structure, Legendrian contact homology [Eli98]. In particular, in the case of the conormal bundle to a knot, one obtains a new algebraic invariant of knots called knot contact homology. My collaborators and I have developed an extensive theory around this invariant, including its properties and its relationship to other parts of low-dimensional topology. This began with a series of my papers [Ng05a, Ng05b, Ng08a, Ng11] that introduced a combinatorial model for knot contact homology. Theorem 2 ([Ng08a]). To any topological knot, one can define a combinatorial chain complex whose homology gives a knot invariant, knot contact homology. This invariant is quite strong; in particular, it contains the Alexander polynomial and A-polynomial of the knot, and it can detect the unknot (distinguish the unknot from other knots). As an indication of the strength of knot contact homology, the property of unknot detection is not true for many classical knot invariants (it is famously an open question for the Jones polynomial). The fact that knot contact homology detects the unknot has been cited in the field as rather compelling evidence that the answer to Question 1 above may indeed be “yes.” Indeed, the following rather strong corroborating conjecture is currently open. Conjecture 3. Knot contact homology is a complete knot invariant. Very recently, Gordon and Lidman [GL15] have shown that knot contact homology detects torus knots and various topological properties of knots. Their work, which builds on my joint paper [CELN15] (see Section a.1 below), provides some further evidence for Conjecture 3. 2 Besides consequences for knot theory, my work cited above can be seen as part of a large- scale program to “combinatorialize” Floer theory. General Floer theory began in the 1980’s and has many current manifestations in topology (e.g., Heegaard Floer theory, Seiberg– Witten and monopole Floer theory) as well as symplectic geometry (e.g., Lagrangian inter- section Floer theory and Symplectic Field Theory), with numerous significant applications to each subject. In the past, a major impediment to large-scale calculations of Floer theories has been the daunting nature of the analysis necessary to solve the partial differential equa- tions underlying Floer theory. Recent work, however, has “reduced” these analytical issues in some circumstances to combinatorics. Joint work with Ekholm, Etnyre, and Sullivan [EENS13b] performed this function for knot contact homology. Instead of the algebraic formulation, the “proper” geometric definition for knot contact homology, via Symplectic Field Theory and Legendrian contact homology, involves a count of holomorphic curves in a six-dimensional symplectic manifold. We under- took an involved study of degenerations of holomorphic curves to obtain the following result, which comprises one of the principal instances in which one can systematically reduce the analysis behind Symplectic Field Theory to combinatorics. Theorem 4 ([EENS13b]). The combinatorial complex for knot contact homology agrees with the geometric Legendrian contact homology associated to a knot by the cotangent construction. This result successfully completed the foundations of knot contact homology, at least in R3. Symplectic topology says that a similar invariant should exist much more generally, for knots in arbitrary manifolds, and it is a very interesting open problem to develop knot 3 contact homology for knots in other 3-manifolds (perhaps beginning with RP , where there are close ties to physics, and where Ekholm and I have some preliminary progress) as well as in higher dimensions. a.1. Knot contact homology and string topology. An interesting open question is to determine the relation between knot contact homology and other known knot invariants. In this direction, several authors have separately noted that knot contact homology has some features that are reminiscent of a construction in algebraic topology, the string topology formalism of Chas and Sullivan [CS99]. Here the starting point is my work [Ng05b] giving a topological interpretation, the “cord algebra,” for a part of knot contact homology. This is an algebra generated by paths beginning and ending on K and subject to certain skein relations, and the relations resemble operations in string topology. In joint work in progress with Cieliebak, Ekholm, and Latschev [CELN15], we give a new interpretation for knot contact homology in terms of string topology. String topology is notoriously difficult to make rigorous, and our techniques involve delicate transversality arguments to formulate string topology in our setting, along with a study of moduli spaces of holomorphic curves with boundary that switches between two Lagrangians. The result is a topological knot invariant that we call string homology, which we prove is isomorphic to knot contact homology and the cord algebra in the appropriate degree. (There is previous work [BMSS12] that uses a slightly different version of string topology to recover a specialization of the cord algebra, but not the entire cord algebra.) As an offshoot of our work, we show that the cord algebra is completely determined by 3 the knot group π1(R − K), in a specific way. A corollary is a new, cleaner proof of my 3 old result from [Ng08a] that knot contact homology detects the unknot. This technique has been notably extended by Gordon and Lidman [GL15] to prove that knot contact homology detects other classes of knots as well. There is related work by my former postdoc Cornwell [Cor13], building on an observation from [Ng06], which establishes an exact correspondence between one-dimensional representations (“augmentations”) of the cord algebra and certain 3 GL(n, C)-representations of the knot group π1(R − K), generalizing my earlier work in [Ng08a] relating knot contact homology to the A-polynomial. a.2. Knot contact homology and physics. The papers [Ng08a, Ng11] used knot contact homology to construct an auxiliary invariant of knots, the augmentation polynomial, which is a three-variable polynomial associated to a topological knot K. Excitingly, the augmentation polynomial unexpectedly seems to appear in string theory, in work of Vafa and others on topological strings and mirror symmetry. More precisely, in 2012, in the context of large N string duality, Aganagic and Vafa [AV12] proposed a “generalized Strominger– Yau–Zaslow conjecture,” which associates to a knot K a Calabi–Yau manifold XK that is 1 a mirror manifold to the resolved conifold O(−1) ⊕ O(−1) → CP . By physical arguments [GSV05], the geometry of XK should contain a great deal of information about K, including such recently-developed knot invariants as the HOMFLY-PT polynomial and Khovanov homology. In turn, XK is itself determined by a three-variable polynomial, the “Q-deformed A-polynomial.” In joint work with Aganagic, Ekholm, and Vafa [AENV14], we proved that for a family of examples, this polynomial coincides with the augmentation polynomial from knot contact homology. Furthermore, we conjectured that this is true for all knots. Conjecture 5 ([AENV14]). The Q-deformed A-polynomial and the augmentation polynomial agree for all knots. This conjecture, which is motivated by arguments in physics, is currently open, but has subsequently been verified for more families of knots by several groups of physicists. In the meantime, even the conjectured relation between the math and physics sides has produced results for both sides. In [AENV14], we constructed, on the math side, new link invariants via representation theory in knot contact homology, and on the physics side, a new topological string related to D-modules. Conjecture 5 appears to be the beginning of what I hope will be a rich vein of research. Ekholm and I have proposed a strategy for a proof by considering partial Lagrangian fillings of the conormal bundle to a knot (see [AENV14, §6]). If true, Conjecture 5 would, among other things, provide a rigorous construction of a large class of mirror manifolds, which is of interest in homological mirror symmetry; allow for many calculations in topological string theory that are currently impossible to do; and establish a variant of the celebrated AJ conjecture in knot theory. Even as an open question, Conjecture 5 suggests a number of avenues for further work purely on the mathematical side. For instance, the physics side suggests that instead of the augmentation polynomial, the proper object of study in knot contact homology is a D-module. It is currently unknown how to obtain a D-module from knot contact homology, but Ekholm and I have made some progress. Such a noncommutative structure has not previously appeared in Legendrian contact homology, and its existence would suggest a much stronger algebraic structure to contact homology than is currently known. In a different 4 direction, work by physicists [FGS13] on a categorification of colored HOMFLY-PT polynomials suggests that the augmentation polynomial should in fact be a specialization of a four-variable polynomial that they call the super-A-polynomial. Finding this polynomial on the contact-geometry side could lead to a “categorified” version of knot contact homology whose Euler characteristic recovers the original theory.

b. Algebraic Structures in Symplectic Topology Symplectic Field Theory (SFT), mentioned in the previous section, was introduced in the late 1990’s and comprises one of the most significant recent applications of holomorphic-curve techniques to symplectic geometry. By counting certain holomorphic curves in symplectic manifolds, one obtains powerful new invariants of symplectic and contact manifolds, with links to such subjects as integrable systems and algebraic geometry (where the analogous construction yields Gromov–Witten invariants); the resulting algebraic structure forms the SFT package. Although the general picture of SFT was laid out by Eliashberg, Givental, and Hofer more than a decade ago, working out the details in almost all circumstances has proved to be surprisingly tricky. In particular, the analytical underpinnings of the theory are still hotly debated in the subject. However, in certain settings one can perform enough of the analysis to reduce the story to algebra and combinatorics. This allows one to concretely study new symplectic invariants and to give an illustration of how the general story might proceed. A celebrated example of this philosophy was carried out by Chekanov [Che02] and Eliash- berg [Eli98] in the late 1990’s, resulting in a combinatorial description of Legendrian contact homology for Legendrian knots in R3. This was a major milestone and contributed greatly to the development of Legendrian knot theory. Much of my work as a graduate student and postdoc was devoted to exploring ramifications of Chekanov and Eliashberg’s work in R3, and this has continued more recently in a joint paper [NR13] with Dan Rutherford, in which we show that matrix representations of the Chekanov–Eliashberg algebra of a Leg- endrian knot correspond to one-dimensional representations of the algebra for satellites of the knot. This is closely related to a joint paper with Cornwell and Sivek [CNS14] in which we use satellites and representations of the Chekanov–Eliashberg algebra to provide useful obstructions to the existence of Lagrangian concordances between Legendrian knots. I have extended the Chekanov–Eliashberg work in a couple of other directions. One is a formulation of a relative version of rational SFT for Legendrian knots. Despite the outline given by the general theory of SFT, coming up with even the precise algebraic setup for SFT in the presence of a Legendrian knot was (and is) a perplexing and important question in the subject for the past decade. (The American Institute of Mathematics sponsored a workshop in 2007 whose sole aim was to solve this problem.) Although the general formulation remains elusive, I was able to give the answer for genus 0 curves with boundary on a Legendrian knot in R3, extending the Chekanov–Eliashberg construction. Theorem 6 ([Ng10]). For any Legendrian knot in R3, one can construct a combinatorial fil- tered chain complex, encoding the genus 0 holomorphic curves counted in SFT. The homology of this complex is an invariant of the Legendrian knot. Another direction is joint work with Ekholm [EN15], in which we extended the Chekanov– Eliashberg work to a formula for Legendrian contact homology for Legendrian knots in 5 connected sums of S1 × S2. This consists of three steps: giving an algebraic definition of a differential graded algebra associated to a Legendrian knot in #k(S1 × S2); carrying out the analytical construction of Legendrian contact homology in this setting; and proving that the algebraic and analytic definitions agree. (Outside of R3, each of these single steps has only been carried out for sporadic cases of contact 3-manifolds.) We were especially interested in #k(S1 × S2) due to famous recent work of Bourgeois, Ekholm, and Eliashberg [BEE12] on another fundamental symplectic invariant, symplectic homology. Their work, combined with ours, gives a combinatorial formula for the symplectic homology of any Stein surface (Weinstein 4-manifold), via Legendrian knots that specify the attaching data for the Stein surface. Such an explicit formula is currently very hard to come by in any amount of generality. As a sample application, we can recover the following result, previously proven by Seidel–Smith [SS05] and McLean [McL09] through very different means.

Theorem 7 ([SS05, McL09]). There are exotic Stein structures on R8. c. Contact Knot Theory Within the realm of contact geometry, there is much current activity studying the inter- action between contact structures and knots. In a 3-dimensional manifold with a contact structure, there are two types of knots of particular interest: Legendrian knots (discussed previously), which are everywhere tangent to the contact 2-plane field, and transverse knots, which are everywhere transverse. Here I will highlight some recent work on transverse knots. Compared to Legendrian knots, transverse knots are much less understood. An important question in contact topology and braid theory is the classification problem for transverse knots: Question 8. Classify transverse knots of a given topological knot type. This problem has seen limited progress compared to the analogous question for Legendrian knots, in large part because it has proved difficult to come up with so-called “effective” invariants of transverse knots, which can distinguish between different transverse knots. One candidate was an invariant in Heegaard Floer homology devised by Ozsváth,Szabó, and Thurston [OST08]. Joint work with Ozsváthand Thurston [NOT08] and follow-up work with my students Tirasan Khandhawit [KN10] and Wutichai Chongchitmate [CN13] showed that the Heegaard Floer invariant was indeed effective, the first demonstration of an effective transverse invariant. In joint work with Etnyre and Vértesi[ENV13], we used knot Floer homology and convex- surface techniques to classify a large family of transverse knots, the so-called “twist” knots. Classifying transverse knots is currently rather sparsely understood, and this constitutes arguably the first classification for a “transversely nonsimple” knot type. I have worked more generally on the relation between Legendrian knots and various recently-introduced topological knot invariants, including knot Floer homology as well as Khovanov homology and Khovanov–Rozansky homology. In contact geometry, there is a natural knot invariant, the Thurston–Bennequin number, which is known to be related to various knot polynomials. In [Ng08b], I gave a general approach that unifies the disparate proofs of many of these relations and also proves relations to Khovanov homology (generalizing my work in [Ng05c]) and Khovanov–Rozansky homology. 6 To return to transverse knots, an offshoot of my work on knot contact homology is a new invariant of transverse knots called transverse homology. This comes from a filtration on knot contact homology naturally induced by a transverse knot, and was defined combinatorially in [Ng11], and concurrently in a geometric setting in joint work with Ekholm, Etnyre, and Sullivan [EENS13a]. Theorem 9 ([Ng11, EENS13a]). For a transverse knot, there is a filtration on knot contact homology that can be combinatorially computed. This results in an effective invariant of transverse knots that can be used to distinguish transverse knots of the same topological type and self-linking number. Along with the Heegaard Floer invariant, transverse homology is the second (and essentially only other) transverse invariant that has been proven to be effective, and indeed in examples it seems to be the most refined and powerful tool that we currently have for distinguishing between transverse knots; see [Ng11] for a comparison between the two invariants. It may be interesting to mention that the Heegaard Floer transverse invariant has a pre- cursor in Khovanov homology, introduced by Plamenevskaya [Pla06]. Surprisingly, it is still unknown whether the Plamenevskaya transverse invariant is effective. In joint work with Lip- shitz and Sarkar [LNS15], we proved that the Plamenevskaya invariant does not distinguish between transverse knots related by negative flypes, which the Heegaard Floer invariant and transverse homology do, and this suggests that perhaps the Plamenevskaya invariant is not effective. This work built on a previous joint paper with Thurston [NT09] establishing grid diagrams as a simultaneous means to depict Legendrian knots, transverse knots, and braids.

d. Legendrian Knots and Constructible Sheaves A very active area of research in recent years has connected symplectic topology and algebraic geometry through sheaf theory and homological mirror symmetry. One manifestation of this connection is the celebrated Nadler–Zaslow correspondence [NZ09], which relates sheaves on a smooth manifold M with a derived Fukaya category of Lagrangian submanifolds in the symplectic manifold T ∗M (cf. the discussion in Section a). In 2014, Shende, Treumann, and Zaslow [STZ14] introduced Legendrians and contact manifolds into the picture, defining a category STZ∗(Λ) of constructible sheaves associated to a Legendrian knot Λ. They noticed that this category shares many properties with Legendrian contact homology, and conjectured a relation between the two, via work of Bourgeois and Chantraine [BC14]. In recent joint work with Rutherford, Shende, Sivek, Treumann, and Zaslow [NRS+15], we constructed a category, the augmentation category Aug∗(Λ) of a Legendrian knot, and proved: + ∗ ∼ ∗ Theorem 10 ([NRS 15]). There is an A∞ equivalence of categories STZ (Λ) = Aug (Λ). This establishes a strong new relation between contact geometry and sheaf theory. On the sheaf side, it enables easy computation of the STZ category; on the contact side, it produces a strong new Legendrian invariant, closely tied to holomorphic curves and generating-family theory. Our work suggests a way to use Legendrian contact homology to construct the Fukaya category of Lagrangians with asymptotic boundary conditions: one might describe our project as developing “mirror symmetry for the 4-dimensional ball,” with the Fukaya category on one side and the augmentation category on the other. 7 I believe that this work opens up new areas that should be rather fruitful in the near future, with a number of natural generalizations of Theorem 10 to pursue. For instance, in place of augmentations, which are one-dimensional representations of a differential graded algebra, one can consider higher rank representations on the contact side, in the spirit of [NR13]. On the sheaf side, the STZ category involves so-called rank 1 constructible sheaves, but one could instead consider higher rank sheaves. We believe that there should be a higher- rank version of the isomorphism in Theorem 10. The isomorphism is much less clear in this general setting but would be a significant step in using contact geometry to understand arbitrary constructible sheaves. Even more intriguingly, one can attempt to generalize our result to higher-dimensional Legendrian submanifolds in higher-dimensional contact manifolds. One case of particular interest is when the Legendrian submanifold is the conormal bundle to a knot in R3: here the category on the contact side is given by knot contact homology. It is completely unclear what the category on the sheaf side should be, but we anticipate that it will provide a mathematical approach to the topological-string work of Aganagic, Vafa, et al., thus tying together two of my ongoing research projects.

8 References [Abo12] Mohammed Abouzaid. Framed bordism and Lagrangian embeddings of exotic spheres. Ann. of Math. (2), 175(1):71–185, 2012. [AENV14] Mina Aganagic, Tobias Ekholm, Lenhard Ng, and Cumrun Vafa. Topological strings, D-model, and knot contact homology. Adv. Theor. Math. Phys., 18(4):827–956, 2014. [AS06] Alberto Abbondandolo and Matthias Schwarz. On the Floer homology of cotangent bundles. Comm. Pure Appl. Math., 59(2):254–316, 2006. [AV12] Mina Aganagic and Cumrun Vafa. Large N duality, mirror symmetry, and a Q-deformed A- polynomial for knots. Preprint, arXiv:1204.4709, 2012. [BC14] FrédéricBourgeois and Baptiste Chantraine. Bilinearized Legendrian contact homology and the augmentation category. J. Symplectic Geom., 12(3):553–583, 2014. [BEE12] FrédéricBourgeois, Tobias Ekholm, and Yasha Eliashberg. Effect of Legendrian surgery. Geom. Topol., 16(1):301–389, 2012. With an appendix by Sheel Ganatra and Maksim Maydanskiy. [BMSS12] Somnath Basu, Jason McGibbon, Dennis Sullivan, and Michael Sullivan. Transverse string topology and the cord algebra. Preprint, arXiv:1210.5722, 2012. [CELN15] Kai Cieliebak, Tobias Ekholm, Janko Latschev, and Lenhard Ng. Relative contact homology, string topology, and the cord algebra. In preparation, 2015. [Che02] Yuri Chekanov. Differential algebra of Legendrian links. Invent. Math., 150(3):441–483, 2002. [CL09] Kai Cieliebak and Janko Latschev. The role of string topology in symplectic field theory. In New perspectives and challenges in symplectic field theory, volume 49 of CRM Proc. Lecture Notes, pages 113–146. Amer. Math. Soc., Providence, RI, 2009. [CN13] Wutichai Chongchitmate and Lenhard Ng. An atlas of Legendrian knots. Exp. Math., 22(1):26– 37, 2013. [CNS14] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., to appear, arXiv:1411.1364, 2014. [Cor13] Christopher Cornwell. KCH representations, augmentations, and A-polynomials. Preprint, arXiv:1310.7526, 2013. [CS99] Moira Chas and Dennis Sullivan. String topology. Preprint, arXiv:math.GT/9911159, 1999. [EENS13a] Tobias Ekholm, John Etnyre, Lenhard Ng, and Michael Sullivan. Filtrations on the knot contact homology of transverse knots. Math. Ann., 355(4):1561–1591, 2013. [EENS13b] Tobias Ekholm, John B Etnyre, Lenhard Ng, and Michael G Sullivan. Knot contact homology. Geom. Topol., 17(2):975–1112, 2013. [EGH00] Y. Eliashberg, A. Givental, and H. Hofer. Introduction to symplectic field theory. Geom. Funct. Anal., (Special Volume, Part II):560–673, 2000. GAFA 2000 (Tel Aviv, 1999). [Eli98] Yakov Eliashberg. Invariants in contact topology. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, pages 327–338 (electronic), 1998. [EN15] Tobias Ekholm and Lenhard Ng. Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold. J. Differential Geom., 101(1):67–157, 2015. [ENV13] John Etnyre, Lenhard Ng, and Vera Vértesi. Legendrian and transverse twist knots. J. Eur. Math. Soc. (JEMS), 15(3):969–995, 2013. [FGS13] Hiroyuki Fuji, Sergei Gukov, and Piotr Sulkowski. Super-A-polynomial for knots and BPS states. Nuclear Phys. B, 867(2):506–546, 2013. [GL15] Cameron Gordon and Tye Lidman. Knot contact homology detects cables, composite, and torus knots. Preprint, arXiv:1509:01642, 2015. [GSV05] Sergei Gukov, Albert Schwarz, and Cumrun Vafa. Khovanov-Rozansky homology and topological strings. Lett. Math. Phys., 74(1):53–74, 2005. [KN10] Tirasan Khandhawit and Lenhard Ng. A family of transversely nonsimple knots. Algebr. Geom. Topol., 10(1):293–314, 2010. [LNS15] Robert Lipshitz, Lenhard Ng, and Sucharit Sarkar. On transverse invariants from Khovanov homology. Quantum Topol., 6(3):475–513, 2015. [McL09] Mark McLean. Lefschetz fibrations and symplectic homology. Geom. Topol., 13(4):1877–1944, 2009.

9 [Ng05a] Lenhard Ng. Knot and braid invariants from contact homology. I. Geom. Topol., 9:247–297 (electronic), 2005. [Ng05b] Lenhard Ng. Knot and braid invariants from contact homology. II. Geom. Topol., 9:1603–1637 (electronic), 2005. With an appendix by the author and Siddhartha Gadgil. [Ng05c] Lenhard Ng. A Legendrian Thurston-Bennequin bound from Khovanov homology. Algebr. Geom. Topol., 5:1637–1653 (electronic), 2005. [Ng06] Lenhard Ng. Conormal bundles, contact homology, and knot invariants. Geom. Topol. Monogr., 8:129–144, 2006. [Ng08a] Lenhard Ng. Framed knot contact homology. Duke Math. J., 141(2):365–406, 2008. [Ng08b] Lenhard Ng. A skein approach to Bennequin-type inequalities. Int. Math. Res. Not. IMRN, pages Art. ID rnn116, 18 pp., 2008. [Ng10] Lenhard Ng. Rational symplectic field theory for Legendrian knots. Invent. Math., 182(3):451– 512, 2010. [Ng11] Lenhard Ng. Combinatorial knot contact homology and transverse knots. Adv. Math., 227(6):2189–2219, 2011. [NOT08] Lenhard Ng, Peter Ozsváth,and Dylan Thurston. Transverse knots distinguished by knot Floer homology. J. Symplectic Geom., 6(4):461–490, 2008. [NR13] Lenhard Ng and Dan Rutherford. Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra. Algebr. Geom. Topol., 13:3047–3097 (electronic), 2013. [NRS+15] Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow. Augmentations are sheaves. Preprint, arXiv:1502.04939, 2015. [NT09] Lenhard Ng and Dylan Thurston. Grid diagrams, braids, and contact geometry. In Proceedings of GökovaGeometry-Topology Conference 2008, pages 120–136. Gökova Geometry/Topology Conference (GGT), Gökova, 2009. [NZ09] David Nadler and Eric Zaslow. Constructible sheaves and the Fukaya category. J. Amer. Math. Soc., 22(1):233–286, 2009. [OST08] Peter Ozsváth,ZoltánSzabó,and Dylan Thurston. Legendrian knots, transverse knots and combinatorial Floer homology. Geom. Topol., 12(2):941–980, 2008. [OV00] Hirosi Ooguri and Cumrun Vafa. Knot invariants and topological strings. Nuclear Phys. B, 577(3):419–438, 2000. [Pla06] Olga Plamenevskaya. Transverse knots and Khovanov homology. Math. Res. Lett., 13(4):571– 586, 2006. [SS05] Paul Seidel and Ivan Smith. The symplectic topology of Ramanujam’s surface. Comment. Math. Helv., 80(4):859–881, 2005. [STZ14] Vivek Shende, David Treumann, and Eric Zaslow. Legendrian knots and constructible sheaves. Preprint, arXiv:1402.0490, 2014. [SW06] D. A. Salamon and J. Weber. Floer homology and the heat flow. Geom. Funct. Anal., 16(5):1050– 1138, 2006. [Vit] Claude Viterbo. Functors and computations in Floer homology with applications II. Geom. Funct. Anal., to appear; http://www.math.ens.fr/˜viterbo/FCFH.II.2003.pdf.