Mathematical writings

This is an archive of work authored (or coauthored) by me. A guide to availability types:

arXiv: papers on the arXiv preprint server, via the UC Davis front.
Journal/Book: direct link to the paper in the journal/book in which it appeared.
.tex: original LaTeX source.
.ps.gz: GNUzipped PostScript. If these don't uncompress automatically, type gunzip filename.ps.gz to produce a PostScript file.

Published papers and preprints
For a succinct list of my arXiv papers, please see here. Many of these papers are part of the AIM preprint series.

arXiv Grid diagrams, braids, and contact geometry
With Dylan Thurston.
Proceedings of Gökova Geometry-Topology Conference 2008, 120--136 (Gökova Geometry-Topology Conference (GGT), Gökova, 2009).
Also available as arXiv:0812.3665.

arXiv Rational Symplectic Field Theory for Legendrian knots
June 2008, arXiv:0806.4598.

arXiv A family of transversely nonsimple knots
With Tirasan Khandhawit.
Algebr. Geom. Topol., to appear.
Also available as arXiv:0806.1887.

Journal
arXiv A skein approach to Bennequin type inequalities
Int. Math. Res. Not. 2008, Art. ID rnn116, 18 pp.
September 2007, arXiv:0709.2141.

Journal Transverse knots distinguished by knot Floer homology
With Peter Ozsváth and Dylan Thurston.
J. Symplectic Geom. 6 (2008), no. 4, 461--490.
Also available as math.GT/0703446.

Here is a mirror of the C program cited in the paper: TransverseHFK.c. To compile and run it, type cc TransverseHFK.c followed by a.out at the command prompt. To try the program on a different grid diagram, first edit the file appropriately.

arXiv On arc index and maximal Thurston-Bennequin number
December 2006, math.GT/0612356.

Journal
arXiv A Legendrian Thurston-Bennequin bound from Khovanov homology
Algebr. Geom. Topol. 5 (2005), 1637--1653.
Also available as math.GT/0508649.
When I have time, I will post here diagrams of maximal Thurston-Bennequin fronts for all prime knots with 10 crossings (except the mirror of 10₁₃₂, which is the one open case). Corresponding diagrams for prime knots with 9 or fewer crossings can be found in Melvin and Shrestha, The nonuniqueness of Chekanov polynomials of Legendrian knots, Geom. Topol. 9 (2005), 1221--1252.

Journal The correspondence between augmentations and rulings for Legendrian knots
With Josh Sabloff.
Pacific J. Math. 224 (2006), no. 1, 141--150.
Also available as math.SG/0503168.

Book 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.
Also available as math.GT/0503162.

Book 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.
Also available as math.SG/0412330.
For another viewpoint on the same material, see the survey paper by Tobias Ekholm and John Etnyre.

arXiv Framed knot contact homology
Duke Math. J. 141 (2008), no. 2, 365--406.
Also available as math.GT/0407071.

Here are the Mathematica packages which compute the invariants in the paper. They are essentially souped-up versions of the packages from “Knot and braid invariants from contact homology I” below, with a few things removed. Currently they require a braid input; if there's interest, I can post updated versions which allow calculation in terms of a knot diagram.

framedDGA.m, which gives the full invariants in the noncommutative category. In order to run this package, the user first needs to install the noncommutative algebra package NCAlgebra/NCGB, available from http://www.math.ucsd.edu/~ncalg/.
framedDGAcomm.m, which gives the commutative version of the invariants and does not require any additional packages.

arXiv Legendrian solid-torus links
With Lisa Traynor.
J. Symplectic Geom. 2 (2005), no. 3, 411--443.
Also available as math.SG/0407068.

Journal Knot and braid invariants from contact homology II
Geom. Topol. 9 (2005), 1603--1637.
Also available as math.GT/0303343.

Journal Knot and braid invariants from contact homology I
Geom. Topol. 9 (2005), 247--297.
Also available as math.GT/0302099.

Here are the Mathematica notebooks and packages, in various flavors, which compute the invariants used in the paper.

DGA.nb, with the full versions of the invariants (except linearized homology; see below). In order to run this notebook, the user first needs to install the noncommutative algebra package NCAlgebra/NCGB, available from http://www.math.ucsd.edu/~ncalg/. Instructions on how to use the code are included in the notebook.
DGAabelian.m, which calculates the invariants in the commutative setting, and does not require NCAlgebra/NCGB. For instructions, input the package and type “?Instructions”.
DGAlin.m, which calculates linearized knot contact homology. For instructions, input the package and type “?Instructions”. This package requires the use of another package, IntegerSmithNormalForm.m, which can be downloaded from the Mathematica Information Center here.

.ps.gz Problems in low dimensional contact geometry
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
December 2001 (though originally from April 2001), math.GT/0112105.
Portions of this paper are now incorporated into “Legendrian solid-torus links” above.

.ps.gz 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.

Journal
arXiv Invariants of Legendrian knots and coherent orientations
With John Etnyre and Josh Sabloff.
J. Symplectic Geom. 1 (2002), no. 2, 321--367.
Also available as math.SG/0101145, which incorporates some corrections to the published version. A list of just these corrections is available here in PostScript form.

arXiv Computable Legendrian invariants
Topology 42 (2003), no. 1, 55--82.
Also available as math.GT/0011265, but I should caution that the published version is more up-to-date than the preprint.

Journal Maximal Thurston-Bennequin number of two-bridge links
Algebr. Geom. Topol. 1 (2001), 427--434.
Also available as math.GT/0008242.
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.

arXiv Legendrian mirrors and Legendrian isotopy
August 2000, math.GT/0008210. A slightly different approach to a result also proven in “Computable Legendrian invariants,” above.

.tex, .ps.gz Almost complex structures and determinant line bundles
Preprint, 1998.

.tex, .ps.gz The rook on the half-chessboard, or how not to diagonalize a matrix
With Kiran Kedlaya.
Amer. Math. Monthly 105 (1998), no. 9, 819--824.

.tex, .ps.gz 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.
With Michelle Schultz.
From the 1994 Duluth REU. This paper gives me Erdös number 3: Michelle Schultz, Gary Chartrand, Paul Erdös.

Expository papers and miscellany

.ps.gz Invariants of Legendrian knots and links
Slides for my talk on May 31, 2001, at the Georgia International Topology Conference.

.tex, .ps.gz Notes on the K-theory of complex grassmannians
A paper for 18.979 (K-theory and the index theorem) at MIT, spring 1997.

.tex, .ps.gz Manifolds of positive scalar curvature
A paper for 18.966 (geometry of manifolds) at MIT, spring 1997.

.tex, .ps.gz Classical and modern formulations of curvature
A paper for Math 230a (differential geometry) at Harvard, fall 1994.

.tex, .ps.gz Putnam practice problems
From a Putnam practice session I organized at Harvard in fall 1995, mainly taken from books and other paraphernalia I stole from Ravi Vakil.

Back to Lenny Ng's homepage

Published papers and preprints For a succinct list of my arXiv papers, please see here. Many of these papers are part of the AIM preprint series.
arXiv	Grid diagrams, braids, and contact geometry With Dylan Thurston. Proceedings of Gökova Geometry-Topology Conference 2008, 120--136 (Gökova Geometry-Topology Conference (GGT), Gökova, 2009). Also available as arXiv:0812.3665.
arXiv	Rational Symplectic Field Theory for Legendrian knots June 2008, arXiv:0806.4598.
arXiv	A family of transversely nonsimple knots With Tirasan Khandhawit. Algebr. Geom. Topol., to appear. Also available as arXiv:0806.1887.
Journal arXiv	A skein approach to Bennequin type inequalities Int. Math. Res. Not. 2008, Art. ID rnn116, 18 pp. September 2007, arXiv:0709.2141.
Journal	Transverse knots distinguished by knot Floer homology With Peter Ozsváth and Dylan Thurston. J. Symplectic Geom. 6 (2008), no. 4, 461--490. Also available as math.GT/0703446. Here is a mirror of the C program cited in the paper: TransverseHFK.c. To compile and run it, type `cc TransverseHFK.c` followed by `a.out` at the command prompt. To try the program on a different grid diagram, first edit the file appropriately.
arXiv	On arc index and maximal Thurston-Bennequin number December 2006, math.GT/0612356.
Journal arXiv	A Legendrian Thurston-Bennequin bound from Khovanov homology Algebr. Geom. Topol. 5 (2005), 1637--1653. Also available as math.GT/0508649. When I have time, I will post here diagrams of maximal Thurston-Bennequin fronts for all prime knots with 10 crossings (except the mirror of 10₁₃₂, which is the one open case). Corresponding diagrams for prime knots with 9 or fewer crossings can be found in Melvin and Shrestha, The nonuniqueness of Chekanov polynomials of Legendrian knots, Geom. Topol. 9 (2005), 1221--1252.
Journal	The correspondence between augmentations and rulings for Legendrian knots With Josh Sabloff. Pacific J. Math. 224 (2006), no. 1, 141--150. Also available as math.SG/0503168.
Book	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. Also available as math.GT/0503162.
Book	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. Also available as math.SG/0412330. For another viewpoint on the same material, see the survey paper by Tobias Ekholm and John Etnyre.
arXiv	Framed knot contact homology Duke Math. J. 141 (2008), no. 2, 365--406. Also available as math.GT/0407071. Here are the Mathematica packages which compute the invariants in the paper. They are essentially souped-up versions of the packages from “Knot and braid invariants from contact homology I” below, with a few things removed. Currently they require a braid input; if there's interest, I can post updated versions which allow calculation in terms of a knot diagram. framedDGA.m, which gives the full invariants in the noncommutative category. In order to run this package, the user first needs to install the noncommutative algebra package NCAlgebra/NCGB, available from http://www.math.ucsd.edu/~ncalg/. framedDGAcomm.m, which gives the commutative version of the invariants and does not require any additional packages.
arXiv	Legendrian solid-torus links With Lisa Traynor. J. Symplectic Geom. 2 (2005), no. 3, 411--443. Also available as math.SG/0407068.
Journal	Knot and braid invariants from contact homology II Geom. Topol. 9 (2005), 1603--1637. Also available as math.GT/0303343.
Journal	Knot and braid invariants from contact homology I Geom. Topol. 9 (2005), 247--297. Also available as math.GT/0302099. Here are the Mathematica notebooks and packages, in various flavors, which compute the invariants used in the paper. DGA.nb, with the full versions of the invariants (except linearized homology; see below). In order to run this notebook, the user first needs to install the noncommutative algebra package NCAlgebra/NCGB, available from http://www.math.ucsd.edu/~ncalg/. Instructions on how to use the code are included in the notebook. DGAabelian.m, which calculates the invariants in the commutative setting, and does not require NCAlgebra/NCGB. For instructions, input the package and type “?Instructions”. DGAlin.m, which calculates linearized knot contact homology. For instructions, input the package and type “?Instructions”. This package requires the use of another package, IntegerSmithNormalForm.m, which can be downloaded from the Mathematica Information Center here.
.ps.gz	Problems in low dimensional contact geometry 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 December 2001 (though originally from April 2001), math.GT/0112105. Portions of this paper are now incorporated into “Legendrian solid-torus links” above.
.ps.gz	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.
Journal arXiv	Invariants of Legendrian knots and coherent orientations With John Etnyre and Josh Sabloff. J. Symplectic Geom. 1 (2002), no. 2, 321--367. Also available as math.SG/0101145, which incorporates some corrections to the published version. A list of just these corrections is available here in PostScript form.
arXiv	Computable Legendrian invariants Topology 42 (2003), no. 1, 55--82. Also available as math.GT/0011265, but I should caution that the published version is more up-to-date than the preprint.
Journal	Maximal Thurston-Bennequin number of two-bridge links Algebr. Geom. Topol. 1 (2001), 427--434. Also available as math.GT/0008242. 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.
arXiv	Legendrian mirrors and Legendrian isotopy August 2000, math.GT/0008210. A slightly different approach to a result also proven in “Computable Legendrian invariants,” above.
.tex, .ps.gz	Almost complex structures and determinant line bundles Preprint, 1998.
.tex, .ps.gz	The rook on the half-chessboard, or how not to diagonalize a matrix With Kiran Kedlaya. Amer. Math. Monthly 105 (1998), no. 9, 819--824.
.tex, .ps.gz	*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. With Michelle Schultz. From the 1994 Duluth REU. This paper gives me Erdös number 3: Michelle Schultz, Gary Chartrand, Paul Erdös.
Expository papers and miscellany
.ps.gz	Invariants of Legendrian knots and links Slides for my talk on May 31, 2001, at the Georgia International Topology Conference.
.tex, .ps.gz	Notes on the K-theory of complex grassmannians A paper for 18.979 (K-theory and the index theorem) at MIT, spring 1997.
.tex, .ps.gz	Manifolds of positive scalar curvature A paper for 18.966 (geometry of manifolds) at MIT, spring 1997.
.tex, .ps.gz	Classical and modern formulations of curvature A paper for Math 230a (differential geometry) at Harvard, fall 1994.
.tex, .ps.gz	Putnam practice problems From a Putnam practice session I organized at Harvard in fall 1995, mainly taken from books and other paraphernalia I stole from Ravi Vakil.