Related papers: Fibered Transverse Knots and the Bennequin Bound

Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given…

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the…

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic…

We develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic…

We perform a compare-and-contrast investigation between the equilibrium shapes of physical and ideal trefoil knots, both in closed and open configurations. Ideal knots are purely geometric abstractions for the tightest configuration tied in…

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact…

In general, the bridge index of a knot is less than or equal to its braid index. A natural question is when these two values coincide. Motivated by a conjecture of Krishna and Morton, we prove that the bridge index and the braid index…

We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a…

This paper introduces a new type of open book decomposition for a contact three-manifold with a specified characteristic foliation $\mathcal{F}_\xi$ on its boundary. These \textit{foliated open books} offer a finer tool for studying contact…

Let p and n be positive integers with p>1, and let E(p,n) be the oriented 3-manifold obtained by performing pn(p-1)-1 surgery on a positive torus knot of type (p, pn+1). We prove that E(2,n) does not carry tight contact structures for any…

We show that a group presented by a labelled oriented tree presentation in which the tree has diameter at most three is an HNN extension of a finitely presented group. From results of Silver, it then follows that the corresponding higher…

We prove that any weakly symplectically fillable contact manifold is tight. Furthermore we verify the strong Weinstein conjecture for contact manifolds that appear as the concave boundary of a directed symplectic cobordism whose positive…

Let $\phi : S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $\phi$ to K has exactly two (non-degenerate) critical points (and K is…

We use parameterized Morse theory on the pages of an open book decomposition to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to…

A ribbon is, intuitively, a smooth mapping of an annulus $S^1 \times I$ in 3-space having constant width $\varepsilon$. This can be formalized as a triple $(x,\varepsilon, \mathbf{u})$ where $x$ is smooth curve in 3-space and $\mathbf{u}$…

We use the skein theory of $\mathfrak{sl}_3$-webs to study the properties of the quantum $\mathfrak{sl}_3$-link polynomial of positive links. We give explicit formulae for the three leading terms of the polynomial on positive links in terms…

We show that a metric $f$-manifold $(M^{2n+s}, \phi, \xi_i, \eta_j, g)$ satisfying the property $[\xi_i, \xi_j]=0$ for all $i, j\in\{1, \ldots, s\}$ admits a metric connection $\nabla$ with skew-torsion $T$ preserving the structure if and…

We prove contact big fiber theorems, analogous to the symplectic big fiber theorem by Entov and Polterovich, using symplectic cohomology with support. Unlike in the symplectic case, the validity of the statements requires conditions on the…

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is…

Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat…