Browsing by Subject "math.GT"
Now showing items 117 of 17

A complete knot invariant from contact homology
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 ... 
A topological introduction to knot contact homology
(Bolyai Society Mathematical Studies, 20140101)This is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects. 
Augmentations are Sheaves
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 in [BC], but is ... 
Harmonic Forms, Price Inequalities, and BenjaminiSchramm Convergence
We study Betti numbers of sequences of Riemannian manifolds which BenjaminiSchramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, ... 
Higher genus knot contact homology and recursion for colored HOMFLYPT polynomials
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 GromovWitten invariants and ChernSimons gauge theory, we ... 
Knot contact homology, string topology, and the cord algebra
(Journal de l’École polytechnique — Mathématiques, 2017)The conormal Lagrangian LKof a knot K in R3is the submanifold of the cotangent bundle T∗R3consisting of covectors along K that annihilate tangent vectors to K. By intersecting with the unit cotangent bundle S∗R3, one obtains ... 
Legendrian contact homology in R^3
This is an introduction to Legendrian contact homology and the ChekanovEliashberg differential graded algebra, with a focus on the setting of Legendrian knots in $\mathbb{R}^3$. 
Legendrian contact homology in the boundary of a subcritical Weinstein 4manifold
(Journal of Differential Geometry, 201509)We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1× S2or any connected sum #k(S1×S2), viewed as the contact boundary of the Weinstein manifold obtained by ... 
Null surgery on knots in Lspaces
Let $K$ be a knot in an Lspace $Y$ with a Dehn surgery to a surface bundle over $S^1$. We prove that $K$ is rationally fibered, that is, the knot complement admits a fibration over $S^1$. As part of the proof, we show that ... 
On Lspace knots obtained from unknotting arcs in alternating diagrams
Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an Lspace obtained by half integral surgery on a knot $K_D$. We denote the set of all such ... 
On the Stein framing number of a knot
For an integer $n$, write $X_n(K)$ for the 4manifold obtained by attaching a 2handle to the 4ball along the knot $K\subset S^3$ with framing $n$. It is known that if $n< \overline{\text{tb}}(K)$, then $X_n(K)$ admits ... 
On transverse invariants from Khovanov homology
(Quantum Topology, 2015)© European Mathematical Society. In [31], O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two re_nements of Plamenevskaya’s invariant, one valued in ... 
Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points
(20170601)We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of nonpositive sectional ... 
Representations, sheaves, and Legendrian $(2,m)$ torus links
We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$dimensional representations of the ChekanovEliashberg differential graded algebra of the link. This representation category ... 
The prism manifold realization problem
The spherical manifold realization problem asks which spherical threemanifolds arise from surgeries on knots in $S^3$. In recent years, the realization problem for C, T, O, and Itype spherical manifolds has been solved, ... 
The prism manifold realization problem II
We continue our study of the realization problem for prism manifolds. Every prism manifold can be parametrized by a pair of relatively prime integers $p>1$ and $q$. We determine a complete list of prism manifolds $P(p, q)$ ... 
The prism manifold realization problem III
Every prism manifold can be parametrized by a pair of relatively prime integers $p>1$ and $q$. In our earlier papers, we determined a complete list of prism manifolds $P(p, q)$ that can be realized by positive integral ...