Chris Cornwell

Research
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Research
 Teaching
Publications
KCH representations, augmentations, and A-polynomials
, J. Symplectic Geometry (to appear). 		arXiv. files
Augmentation rank of satellites with braid pattern
(with D. Hemminger), Communications in Analysis and Geometry 24 (2016), 939-967. 		arXiv
Obstructions to Lagrangian concordance
(with L. Ng and S. Sivek), Algebraic & Geometric Topology 16 (2016), 797-824. 		arXiv
Knot contact homology and representations of knot groups
, J. Topology 7 (2014), 1221--1242. 		arXiv
A polynomial invariant for links in lens spaces
, J. Knot Theory and its Ramifications 21 (2012), #1250060 (31 pages). 		arXiv
Bennequin type inequalities in lens spaces
, International Mathematics Research Notices 2012, 1890-1916. 		arXiv
Counting fundamental paths in certain Garside semigroups
, J. Knot Theory and its Ramifications 17 (2008), 191-211. 		abstract
Preprints
Character varieties of knot complements and branched double-covers via the cord ring.
 arXiv.
Berge duals and universally tight contact structures.
 arXiv.
A strong correspondence principle for smooth, monotone environments (with F. Christensen).
Topics
  Knot contact homology
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Much of my current research involves a graded non-commutative algebra called Knot Contact Homology (KCH), which is defined from the Legendrian DGA of certain tori in the cotangent bundle. KCH is an invariant of a knot or link, and it appears to carry a great deal of information. One effective way of extracting the information is through augmentations. In particular, augmentations allow one to draw relationships to numerous other constructions studied in low-dimensional topology.
A large-scale aim of my research is to discover just how much we can see from KCH. In the two papers below I describe a correspondence between representations of the knot group and augmentations of KCH. The correspondence provides a new way to study augmentations and provides an indicator for how KCH and augmentations might be related to certain quantum knot invariants. I also indicate a new approach to study an open problem on Kirby's list, which asks whether the meridional rank and bridge number are equal. My approach uses a notion of augmentation rank, which is well-defined by the correspondence.
I have mentored two undergraduate students, each of which began studying with me through the PRUV summer program at Duke. Daniel Vitek studied augmentations of torus knots and David Hemminger studied how augmentation rank behaves under an operation called cabling.
David Hemminger and I have prepared a preprint detailing our results, now submitted for publication.
  Legendrian & transverse links
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Embeddings in a contact structure can be special in some sense, as were the tori mentioned before. These embeddings are called Legendrian. Legendrians in the standard contact 3-space have the type of some underlying link, with some tangency restrictions. As in smooth knot theory, one can study Legendrians up to (Lagrangian) concordance in standard 4-space.
There are some interesting phenomena, for example, the relation of Lagrangian concordance is not symmetric as in the smooth case (though it is reflexive and transitive). In joint work with Lenhard Ng and Steven Sivek, we investigated the (still open) possibility that the relation is anti-symmetric.
  Lens spaces
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My thesis work considered invariants of knots in lens spaces, a certain infinite family of 3-manifolds, and invariants of Legendrian knots in lens spaces with a contact structure that pulls back to the standard contact structure in the 3-sphere.
Before my thesis, polynomial invariants computable from skein moves had, with one exception, not been constructed in a closed manifold other than the 3-sphere. In the paper below, I showed how to construct a HOMFLY-PT polynomial of links in any lens space (this is a two-variable polynomial). A specialization then provides a Jones polynomial for links in lens space.
In the 80's, independent work of Morton and Franks-Williams had shown that a degree of the HOMFLY-PT polynomial in the 3-sphere provides a bound on certain Legendrian knot invariants. In fact, any invariant computable from skein moves that satisfy some criteria will provide such a bound, as shown by Ng. In this paper I generalized Ng's result to lens spaces with a contact structure defined to pull back to the standard contact structure. In particular this shows the inequality of Morton, Franks-Williams extends to this setting.
Dehn surgery on knots and links is one of the principal methods of constructing 3-manifolds. There is a near 30-year-old conjecture concerning which knots in the 3-sphere produce lens spaces through this construction. I have a program for solving this problem by understanding the Legendrian knot invariants I studied in the paper above.
An important question surrounding such an understanding is if in lens spaces certain fibered knots are associated, via a well-known correspondencd in contact geometry, to the contact structure I studied above. The following paper shows that many of these knots are not associated to this contact structure.
  Garside (semi-)groups
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The following paper was the result of research I began as an undergraduate. I began by indicating a combinatorial equivalence between some walks in the plane and paths in the Cayley graph of the 3-stranded braid group representing a particular fundamental element. I then extended the equivalence to a large class of Garside semigroups and proved that, similar to the identity on binomial coefficients that arises from Pascal's triangle, the Cayley graphs of these Garside groups admit a Pascal-like identity.

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