
arXiv 
PDF

A complete knot invariant from contact homology
Joint with Tobias Ekholm and Vivek Shende,
June 2016.





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,
holomorphiccurve 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].



arXiv 
PDF 
Knot contact homology,
string topology, and the cord algebra
Joint with Kai Cieliebak, Tobias Ekholm, and Janko Latschev, January
2016.





Abstract:
The conormal Lagrangian L_{K} of a knot K in R^{3}
is the
submanifold of the cotangent bundle T^{*}R^{3} consisting of
covectors
along K that annihilate
tangent vectors to K. By
intersecting with
the unit cotangent bundle S^{*}R^{3}, one obtains the unit
conormal $\Lambda_{K}$, and the Legendrian contact homology of
$\Lambda_{K}$ is a knot invariant of K,
known as
knot contact homology. We
define a version of string topology for strings in $\R^{3}\cup L_K$
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^{*}R^{3} with boundary on
$\R^{3}\cup
L_K$.



arXiv 
PDF






Abstract:
We introduce a notion of cardinality for the augmentation category
associated to a Legendrian knot or link in standard contact R^3. 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.



arXiv 
PDF 





Abstract:
We show that the set of augmentations of the ChekanovEliashberg
algebra of a Legendrian link
underlies the structure of a unital Ainfinity category. This differs
from the nonunital 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 xline, 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 
PDF 





Abstract:
We investigate the question of the existence of a Lagrangian
concordance
between two Legendrian knots in R^{3}. 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 nonreversible
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 
PDF 
Legendrian contact
homology in the boundary of a subcritical Weinstein 4manifold
Joint with Tobias Ekholm.
J. Differential Geom. 101
(2015), no. 1, 67157.





Abstract:
We give a combinatorial description of the Legendrian contact homology
algebra associated to a Legendrian link in S^{1} x S^{2}
or any
connected sum #^{k}(S^{1} x S^{2}), viewed as
the contact boundary of
the Weinstein manifold obtained by attaching 1handles to the
4ball. In view of the surgery formula for symplectic homology, this
gives a combinatorial description of the
symplectic homology of any 4dimensional 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 
PDF 
Topological strings, Dmodel,
and knot contact homology
Joint with Mina Aganagic, Tobias Ekholm, and Cumrun Vafa.
Adv. Theor. Math. Phys. 18
(2014), no. 4, 827956.
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 GromovWitten disk
amplitudes of a Lagrangian associated to a knot and augmentations of
its contact homology algebra. This also implies the equality between
the Qdeformed Apolynomial 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
CalabiYau. 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 Dmodules, which we call the "Dmodel". 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 Dmodules. 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 Apolynomials associated with the link complement.


Journal

arXiv 
PDF 





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 BarNatan'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
smallcrossing transverse knots K, both refinements are determined by
the classical invariants of K.


Book

arXiv 
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 
PDF 
Satellites of Legendrian
knots and representations of the ChekanovEliashberg algebra
Joint with Dan Rutherford.
Algebr. Geom.
Topol. 13 (2013), no. 5, 30473097.





Abstract:
We study satellites of Legendrian knots in R^{3} and their
relation to the ChekanovEliashberg differential graded algebra of the
knot.
In particular, we generalize the wellknown correspondence
between rulings of a Legendrian knot in R^{3} and augmentations
of its DGA by showing that the DGA has finitedimensional
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 finitedimensional representations for the DGA of a Legendrian
knot depends only on the topological type and ThurstonBennequin number
of the knot.


Journal

arXiv 
PDF 
Knot contact homology
Joint with Tobias Ekholm, John
Etnyre, and Michael Sullivan.
Geom. Topol. 17 (2013), 9751112.





Abstract:
The conormal lift of a link K in R^{3} is a Legendrian
submanifold
Λ_{K} in the unit cotangent bundle U^{*}R^{3}
of R^{3}
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 Λ_{K}, the homology of a differential
graded algebra generated by Reeb chords whose differential
counts holomorphic disks in the symplectization R x U^{*}R^{3}
with Lagrangian boundary condition
R x Λ_{K}.
We perform an explicit and complete computation of the Legendrian
homology of Λ_{K} for arbitrary links K in terms of a braid
presentation of K, confirming a conjecture that this invariant
agrees with a previouslydefined 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 
PDF 
An atlas of Legendrian knots
Joint with Wutichai Chongchitmate.
Exp. Math. 22 (2013), no. 1, 2637.

More files




Abstract:
We present an atlas of Legendrian knots in standard contact
threespace. 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 nonmaximal
nondestabilizable Legendrian knots, and gives rise to new infinite
families of transversely nonsimple knots.


Journal 
arXiv 
PDF 
Combinatorial knot contact
homology and transverse knots
Adv. Math.
227 (2011), no. 6, 21892219.

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 threevariable knot
polynomial related to the Apolynomial. 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 
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, 15611591.





Abstract:
We construct a new invariant of transverse links in the standard
contact structure on R^{3}. 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^{*}R^{3} of R^{3}, 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 
PDF 





Abstract:
In 1997, Chekanov gave the first example of a Legendrian nonsimple knot
type: the m(5_{2}) knot. Epstein, Fuchs, and Meyer extended his
result by showing that there are at least n different Legendrian
representatives with maximal ThurstonBennequin 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
ThurstonBennequin number, and \lceil n/2 \rceil transverse
representatives with maximal selflinking number. Our techniques
include convex surface theory, Legendrian ruling invariants, and
Heegaard Floer homology.


Book

arXiv 
PDF 





Abstract:
We use grid diagrams to present a unified picture of braids, Legendrian
knots, and transverse knots.


Journal 
arXiv

PDF 
Rational Symplectic Field
Theory for Legendrian knots
Invent.
Math. 182 (2010), no. 3, 451512.





Abstract:
We construct a combinatorial invariant of Legendrian knots in standard
contact threespace. 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

PDF 
A family of transversely
nonsimple knots
Joint with Tirasan Khandhawit.
Algebr. Geom.
Topol. 10 (2010), no. 1, 293314.





Abstract:
We apply knot Floer homology to exhibit an infinite family of
transversely nonsimple prime knots starting with 10_{132}. We
also discuss the combinatorial relationship between grid diagrams,
braids, and Legendrian and transverse knots in standard contact
R^{3}.


Journal

arXiv

PDF 





Abstract:
We give a simple unified proof for several disparate bounds on
ThurstonBennequin number for Legendrian knots and selflinking 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

PDF 

More
files




Abstract:
We exhibit pairs of transverse knots with the same selflinking 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 





Abstract:
We 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
selflinking number for all knots with at most 11 crossings.


Journal

arXiv 
PDF

A Legendrian
ThurstonBennequin bound from Khovanov
homology
Algebr. Geom.
Topol. 5 (2005), 16371653.





Abstract:
We establish an upper bound for the ThurstonBennequin 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, 141150.





Abstract:
We strengthen the link between holomorphic and generatingfunction
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






Abstract:
We apply contact homology to obtain new results in the problem of
distinguishing immersed plane curves without dangerous
selftangencies.


Book 
arXiv

PDF

Conormal bundles, contact
homology, and knot invariants
In The
Interaction of finitetype and GromovWitten
invariants at the Banff International Research Station (2003), Geom.
Topol. Monogr. 8 (2006), 129144.





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, 365406.

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 S^{3} 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 twovariable polynomial knot invariant which is related to
the Apolynomial.


Journal

arXiv 
PDF 





Abstract:
Differential graded algebra invariants are constructed for Legendrian
links in the 1jet space of the circle. In parallel to the theory for R^{3},
PoincareChekanov 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 twocomponent links, the
PoincareChekanov 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), 16031637.





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 higherdimensional knots. We calculate the knot invariant
for twobridge 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), 247297.

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 fivedimensional contact manifolds. We present several
computations and derive a relation between the knot invariant and the
determinant.




PDF




arXiv 

The Legendrian
satellite construction
The key portions of this paper are now incorporated into “Legendrian
solidtorus 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 threespace and the solid torus. In
particular, we show that the ChekanovEliashberg 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 ThurstonBennequin number of twobridge links,”
found elsewhere on this page. One other result which might be of
interest
is a calculation of maximal ThurstonBennequin number for many pretzel
knots.





Abstract:
We introduce new, readily computable invariants of Legendrian knots
and links in standard contact threespace, allowing us to answer many
previously open questions in contact knot theory. The origin of these
invariants is the powerful ChekanovEliashberg differential graded
algebra, which we reformulate and generalize.
We give applications to Legendrian
knots and links in threespace and in the solid torus.
A related question, the calculation of the maximal ThurstonBennequin
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, 321367.
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 R^{3}
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, 5582.
Caution: the published version is more uptodate than the arXiv
version.





Abstract:
We establish tools to facilitate the computation and application of the
ChekanovEliashberg differential graded algebra (DGA), a
Legendrianisotopy invariant of Legendrian knots in standard contact
threespace. 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 ThurstonBennequin
number of twobridge links
Algebr. Geom.
Topol. 1 (2001), 427434.
There is a typographical error in the table at the end of the paper (p.
433): the row for 9 _{15} should be 1,10 rather than 10,1.





Abstract:
We compute the maximal ThurstonBennequin number for a Legendrian
twobridge knot or oriented twobridge link in standard contact R^{3},
by showing that the upper bound given by the Kauffman polynomial is
sharp. As an application, we present a table of maximal
ThurstonBennequin 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 threespace 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 halfchessboard, or how not to diagonalize
a matrix
With Kiran Kedlaya.
Amer.
Math. Monthly 105 (1998), no. 9, 819824. 



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, 119129.
From research conducted in the 1995 Duluth REU summer program.


Journal 


Hamiltonian decomposition of complete regular
multipartite digraphs
Discrete
Math. 177 (1997), no. 13, 279285.
From the 1995 Duluth REU. 

Journal 


kordered hamiltonian graphs
J.
Graph
Theory 24 (1997), no. 1, 4557.
With Michelle Schultz.
From the 1994 Duluth REU. This paper gives me Erdös
number 3: Michelle Schultz, Gary Chartrand, Paul Erdös. 
