Book		Frontiers in Geometry and Topology Proc. Sympos. Pure Math., 109. Coedited with Paul Feehan and Peter Ozsváth.
			This volume contains the proceedings of the summer school and research conference “Frontiers in Geometry and Topology”, celebrating the sixtieth birthday of Tomasz Mrowka, which was held from August 1–12, 2022, at the Abdus Salam International Centre for Theoretical Physics (ICTP). The summer school featured ten lecturers and the research conference featured twenty-three speakers covering a range of topics. A common thread, reflecting Mrowka's own work, was the rich interplay among the fields of analysis, geometry, and topology. Articles in this volume cover topics including knot theory; the topology of three and four-dimensional manifolds; instanton, monopole, and Heegaard Floer homologies; Khovanov homology; and pseudoholomorphic curve theory.
	arXiv: 2311.14614	PDF	An L-infinity structure for Legendrian contact homology November 2023.
			Abstract: For any Legendrian knot or link in R3, we construct an L∞ algebra that can be viewed as an extension of the Chekanov-Eliashberg differential graded algebra. The L∞ 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.
	arXiv: 2308.13482	PDF	Torsion in linearized contact homology for Legendrian knots Joint with Robert Lipshitz. Michigan Math. J., to appear.
			Abstract: 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.
Journal	arXiv: 2101.02318	PDF	Braid loops with infinite monodromy on the Legendrian contact DGA Joint with Roger Casals. J. Topol. 15 (2022), no. 4, 1927-2016.	More files
			Abstract: 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.
Book	arXiv: 1811.10966	PDF	Legendrian contact homology in R3 Joint with John Etnyre. In Surveys in Differential Geometry vol. XXV, Surveys in 3-Manifold Topology and Geometry, edited by Ian Agol and David Gabai.
			Abstract: 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 R3.
Journal	arXiv: 1805.03603	PDF	Representations, sheaves, and Legendrian (2,m) torus links Joint with Baptiste Chantraine and Steven Sivek. J. London Math. Soc. (2) 100 (2019), no. 1, 41-82.
			Abstract: We study an A∞ category associated to Legendrian links in R3 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.
Journal	arXiv: 1803.04011	PDF	Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials Joint with Tobias Ekholm. Adv. Theor. Math. Phys. 24 (2020), no. 8, 2067-2145.
			Abstract: 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.
Journal	arXiv: 1606.07050	PDF	A complete knot invariant from contact homology Joint with Tobias Ekholm and Vivek Shende. Invent. Math. 211 (2018), no. 3, 1149-1200.
			Abstract: We 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].
Journal	arXiv: 1601.02167	PDF	Knot contact homology, string topology, and the cord algebra Joint with Kai Cieliebak, Tobias Ekholm, and Janko Latschev. J. Éc. polytech. Math. 4 (2017), 661-780.
			Abstract: The conormal Lagrangian LK of a knot K in R3 is the submanifold of the cotangent bundle T*R3 consisting 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 ΛK is a knot invariant of K, known as knot contact homology. We define a version of string topology for strings in R3 ∪ LK and 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*R3 with boundary on R3 ∪ LK.
Journal	arXiv: 1511.06724	PDF	The cardinality of the augmentation category of a Legendrian link Joint with Dan Rutherford, Vivek Shende, and Steven Sivek. Math. Res. Lett. 24 (2017), no. 6, 1845-1874.
			Abstract: We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R3. This `homotopy 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.
Journal	arXiv: 1502.04939	PDF	Augmentations are sheaves Joint with Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow. Geom. Topol. 24 (2020), no. 5, 2149-2286.
			Abstract: We 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 by Bourgeois and Chantraine, 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 Shende, Treumann, and Zaslow, 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.
Journal	arXiv: 1411.1364	PDF	Obstructions to Lagrangian concordance Joint with Chris Cornwell and Algebr. Geom. Topol. 16 (2016), no. 2, 797-824.
			Abstract: We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in R3. 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.
Journal	arXiv: 1307.8436	PDF	Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold Joint with Tobias Ekholm. J. Differential Geom. 101 (2015), no. 1, 67-157.
			Abstract: We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1 x S2 or any connected sum #k(S1 x 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, 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.
Journal	arXiv: 1304.5778	PDF	Topological strings, D-model, and knot contact homology Joint with Mina Aganagic, Tobias Ekholm, and Cumrun Vafa. Adv. Theor. Math. Phys. 18 (2014), no. 4, 827-956. Click on “More files”, to the right of this text, for the associated Mathematica notebook with augmentation varieties for links.	More files
			Abstract: 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 Q-deformation of the extension of A-polynomials associated with the link complement.
Journal	arXiv: 1303.6371	PDF	On transverse invariants from Khovanov homology Joint with Robert Lipshitz and Sucharit Sarkar. Quantum Topol. 6 (2015), no. 3, 475-513.
			Abstract: O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. 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 refinements are determined by the classical invariants of K.
Book	arXiv: 1210.4803	PDF	A topological introduction to knot contact homology In Contact and Symplectic Topology, Bolyai Soc. Math. Stud. 26 (Springer, Berlin, 2014).
			Abstract: This is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects.
Journal	arXiv: 1206.2259	PDF	Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra Joint with Dan Rutherford. Algebr. Geom. Topol. 13 (2013), no. 5, 3047-3097.
			Abstract: We study satellites of Legendrian knots in R3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R3 and 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.
Journal	arXiv: 1109.1542	PDF	Knot contact homology Joint with Tobias Ekholm, John Etnyre, and Michael Sullivan. Geom. Topol. 17 (2013), 975-1112.
			Abstract: The conormal lift of a link K in R3 is a Legendrian submanifold ΛK in the unit cotangent bundle U*R3 of R3 with 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 ΛK, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization R x U*R3 with Lagrangian boundary condition R x ΛK. We perform an explicit and complete computation of the Legendrian homology of ΛK for 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.
Journal	arXiv: 1010.3997	PDF	An atlas of Legendrian knots Joint with Wutichai Chongchitmate. Exp. Math. 22 (2013), no. 1, 26-37.	More files
			Abstract: We present an atlas of Legendrian knots in standard contact three-space. This gives a conjectural Legendrian classification for all knots with arc index at most 9, including alternating knots through 7 crossings and nonalternating knots through 9 crossings. Our method involves a computer search of grid diagrams and applies to transverse knots as well. The atlas incorporates a number of new, small examples of phenomena such as transverse nonsimplicity and non-maximal non-destabilizable Legendrian knots, and gives rise to new infinite families of transversely nonsimple knots.
Journal	arXiv: 1010.0451	PDF	Combinatorial knot contact homology and transverse knots Adv. Math. 227 (2011), no. 6, 2189-2219.	More files
			Abstract: We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A-polynomial. We provide a number of computations of transverse homology that demonstrate its effectiveness in distinguishing transverse knots, including knots that cannot be distinguished by the Heegaard Floer transverse invariants or other previous invariants.
Journal	arXiv: 1010.0450	PDF	Filtrations on the knot contact homology of transverse knots Joint with Tobias Ekholm, John Etnyre, and Michael Sullivan. Math. Ann. 355 (2013), no. 4, 1561-1591.
			Abstract: We construct a new invariant of transverse links in the standard contact structure on R3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle S*R3 of R3, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.
Journal	arXiv: 1002.2400	PDF	Legendrian and transverse twist knots Joint with John Etnyre and Vera Vértesi. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 969-995.
			Abstract: In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the m(52) knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least n different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot K-2n with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that K-2n has exactly \lceil n^2/2\rceil Legendrian representatives with maximal Thurston-Bennequin number, and \lceil n/2 \rceil transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology.
Book	arXiv: 0812.3665	PDF	Grid diagrams, braids, and contact geometry Joint with Dylan Thurston. Proceedings of Gökova Geometry-Topology Conference 2008, 120-136 (Gökova Geometry-Topology Conference (GGT), Gökova, 2009).
			Abstract: We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.
Journal	arXiv: 0806.4598	PDF	Rational Symplectic Field Theory for Legendrian knots Invent. Math. 182 (2010), no. 3, 451-512.
			Abstract: We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.
Journal	arXiv: 0806.1887	PDF	A family of transversely nonsimple knots Joint with Tirasan Khandhawit. Algebr. Geom. Topol. 10 (2010), no. 1, 293-314.
			Abstract: We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with 10132. We also discuss the combinatorial relationship between grid diagrams, braids, and Legendrian and transverse knots in standard contact R3.
Journal	arXiv: 0709.2141	PDF	A skein approach to Bennequin type inequalities Int. Math. Res. Not. 2008, Art. ID rnn116, 18 pp.
			Abstract: We give a simple unified proof for several disparate bounds on Thurston-Bennequin number for Legendrian knots and self-linking number for transverse knots in R^3, and provide a template for possible future bounds. As an application, we give sufficient conditions for some of these bounds to be sharp.
Journal	arXiv: 0703.5446	PDF	Transverse knots distinguished by knot Floer homology Joint with Peter Ozsváth and Dylan Thurston. J. Symplectic Geom. 6 (2008), no. 4, 461-490.	More files
			Abstract: We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.
Journal	arXiv	PDF	On arc index and maximal Thurston-Bennequin number J. Knot Theory Ramifications 21 (2012), no. 4, 1250031, 11 pp.
			Abstract: We discuss the relation between arc index, maximal Thurston-Bennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal Thurston-Bennequin 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.
Journal	arXiv	PDF	A Legendrian Thurston-Bennequin bound from Khovanov homology Algebr. Geom. Topol. 5 (2005), 1637-1653.
			Abstract: We establish an upper bound for the Thurston-Bennequin number of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for all alternating links, and knots with nine or fewer crossings.
Journal	arXiv	PDF	The correspondence between augmentations and rulings for Legendrian knots Joint with Josh Sabloff. Pacific J. Math. 224 (2006), no. 1, 141-150.
			Abstract: We strengthen the link between holomorphic and generating-function invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot's contact homology to the complete ruling invariant of Chekanov and Pushkar.
Book	arXiv	PDF	Plane curves and contact geometry In Proceedings of 12th Gökova Geometry-Topology Conference, ed. S. Akbulut, T. Önder, and R. J. Stern (International Press, 2006), 165-174.
			Abstract: We apply contact homology to obtain new results in the problem of distinguishing immersed plane curves without dangerous self-tangencies.
Book	arXiv		Conormal bundles, contact homology, and knot invariants In The Interaction of finite-type and Gromov-Witten invariants at the Banff International Research Station (2003), Geom. Topol. Monogr. 8 (2006), 129-144.
			Abstract: We summarize recent work on a combinatorial knot invariant called knot contact homology. We also discuss the origins of this invariant in symplectic topology, via holomorphic curves and a conormal bundle naturally associated to the knot.
Journal	arXiv	PDF	Framed knot contact homology Duke Math. J. 141 (2008), no. 2, 365-406.	More files
			Abstract: We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in S3 and can be generalized to knots in arbitrary manifolds, distinguishes the unknot and can distinguish mutants. It contains the Alexander polynomial and naturally produces a two-variable polynomial knot invariant which is related to the A-polynomial.
Journal	arXiv	PDF	Legendrian solid-torus links Joint with Lisa Traynor. J. Symplectic Geom. 2 (2005), no. 3, 411-443.
			Abstract: Differential graded algebra invariants are constructed for Legendrian links in the 1-jet space of the circle. In parallel to the theory for R3, Poincare-Chekanov polynomials and characteristic algebras can be associated to such links. The theory is applied to distinguish various knots, as well as links that are closures of Legendrian versions of rational tangles. For a large number of two-component links, the Poincare-Chekanov polynomials agree with the polynomials defined through the theory of generating functions. Examples are given of knots and links which differ by an even number of horizontal flypes that have the same polynomials but distinct characteristic algebras. Results obtainable from a Legendrian satellite construction are compared to results obtainable from the DGA and generating function techniques.
Journal	arXiv	PDF	Knot and braid invariants from contact homology II Geom. Topol. 9 (2005), 1603-1637.
			Abstract: We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branched covers for general knots. In the appendix we show that the cord ring is determined by the fundamental group and peripheral structure of a knot and give applications.
Journal	arXiv	PDF	Knot and braid invariants from contact homology I Geom. Topol. 9 (2005), 247-297.	More files
			Abstract: We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative contact homology of certain Legendrian tori in five-dimensional contact manifolds. We present several computations and derive a relation between the knot invariant and the determinant.
		PDF	Problems in low dimensional contact geometry Joint with John Etnyre. In Topology and Geometry of Manifolds, Proc. Sympos. Pure Math. 71 (2003), 337-357, the proceedings of the 2001 Georgia International Topology Conference.
	arXiv		The Legendrian satellite construction The key portions of this paper are now incorporated into “Legendrian solid-torus links” above.
			Abstract: We examine the Legendrian analogue of the topological satellite construction for knots, and deduce some results for specific Legendrian knots and links in standard contact three-space and the solid torus. In particular, we show that the Chekanov-Eliashberg contact homology invariants of Legendrian Whitehead doubles of stabilized knots contain no nonclassical information.
		PDF	Invariants of Legendrian links My Ph.D. dissertation, April 2001. Many of the main results are included in “Computable Legendrian invariants,” “The Legendrian satellite construction,” and “Maximal Thurston-Bennequin number of two-bridge links,” found elsewhere on this page. One other result which might be of interest is a calculation of maximal Thurston-Bennequin number for many pretzel knots.
			Abstract: We introduce new, readily computable invariants of Legendrian knots and links in standard contact three-space, allowing us to answer many previously open questions in contact knot theory. The origin of these invariants is the powerful Chekanov-Eliashberg differential graded algebra, which we reformulate and generalize. We give applications to Legendrian knots and links in three-space and in the solid torus. A related question, the calculation of the maximal Thurston-Bennequin number for a link, is answered for some large classes of links.
Journal	arXiv		Invariants of Legendrian knots and coherent orientations With John Etnyre and Josh Sabloff. J. Symplectic Geom. 1 (2002), no. 2, 321-367. The latest arXiv version incorporates some corrections to the published version. A list of just these corrections is available here.
			Abstract: We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R3 and a relative version of Eliashberg and Hofer's Contact Homology. We use this translation to transport the idea of “coherent orientations” from the Contact Homology world to Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's differential graded algebra invariant to an algebra over Z[t,t-1] with a full Z grading.
Journal	arXiv		Computable Legendrian invariants Topology 42 (2003), no. 1, 55-82. Caution: the published version is more up-to-date than the arXiv version.
			Abstract: We establish tools to facilitate the computation and application of the Chekanov-Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots in standard contact three-space. More specifically, we reformulate the DGA in terms of front projection, and introduce the characteristic algebra, a new invariant derived from the DGA. We use our techniques to distinguish between several previously indistinguishable Legendrian knots and links.
Journal	arXiv		Maximal Thurston-Bennequin number of two-bridge links Algebr. Geom. Topol. 1 (2001), 427-434. There is a typographical error in the table at the end of the paper (p. 433): the row for 915 should be -1,-10 rather than -10,-1.
			Abstract: We compute the maximal Thurston-Bennequin number for a Legendrian two-bridge knot or oriented two-bridge link in standard contact R3, by showing that the upper bound given by the Kauffman polynomial is sharp. As an application, we present a table of maximal Thurston-Bennequin numbers for prime knots with nine or fewer crossings.
	arXiv		Legendrian mirrors and Legendrian isotopy A slightly different approach to a result also proven in “Computable Legendrian invariants,” above.
			Abstract: We resolve a question of Fuchs and Tabachnikov by showing that there is a Legendrian knot in standard contact three-space with zero Maslov number which is not Legendrian isotopic to its mirror. The proof uses the differential graded algebras of Chekanov.
		PDF	The rook on the half-chessboard, or how not to diagonalize a matrix Joint with Kiran Kedlaya. Amer. Math. Monthly 105 (1998), no. 9, 819-824.
		PDF	Heisenberg model, Bethe ansatz, and random walks My undergraduate senior thesis, spring 1996, written under the direction of Persi Diaconis.
Journal			Hamiltonian decomposition of lexicographic products of digraphs J. Combin. Theory Ser. B 73 (1998), no. 2, 119-129. From research conducted in the 1995 Duluth REU summer program.
Journal			Hamiltonian decomposition of complete regular multipartite digraphs Discrete Math. 177 (1997), no. 1-3, 279-285. From the 1995 Duluth REU.
Journal			k-ordered hamiltonian graphs J. Graph Theory 24 (1997), no. 1, 45-57. Joint with Michelle Schultz. From the 1994 Duluth REU. This paper gives me Erdös number 3: Michelle Schultz, Gary Chartrand, Paul Erdös.