Now showing items 1-18 of 18

• #### 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, 2014-01-01)
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 Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is ...
• #### Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence ﻿

We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm 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 HOMFLY-PT 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 Gromov-Witten invariants and Chern-Simons 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 Chekanov-Eliashberg 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 4-manifold ﻿

(Journal of Differential Geometry, 2015-09)
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 L-spaces ﻿

Let $K$ be a knot in an L-space $Y$ 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 3-braids and L-space knots ﻿

We classify closed 3-braids which are L-space knots.
• #### On L-space 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 L-space 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 4-manifold obtained by attaching a 2-handle to the 4-ball 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 ﻿

(2017-06-01)
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 non-positive 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 Chekanov-Eliashberg differential graded algebra of the link. This representation category ...
• #### The prism manifold realization problem ﻿

The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in $S^3$. In recent years, the realization problem for C, T, O, and I-type 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 ...