Related papers: Upper bounds for the complexity of torus knot comp…

We establish a framework for extending invariants of sutured manifolds to invariants of pairs of sutured manifolds who differ by attaching a basic slice along a torus boundary component. In the particular case of (bordered-)sutured Floer…

We give a necessary condition for a torus knot to be untied by a single twisting. By using this result, we give infinitely many torus knots that cannot be untied by a single twisting.

It is a basic question in contact geometry to classify all non-isotopic tight contact structures on a given 3-manifold. If the manifold has a boundary, we need also specify the dividing set on the boundary. In this paper, we answer the…

In this paper, we give some relations in the mapping class groups of oriented closed surfaces in the form that a product of a small number of right hand Dehn twists is equal to a single commutator. Consequently, we find upper bounds for the…

Let $M$ be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if $M$ is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a…

We answer a question posed by Fielder in [1] concerning two notions of crossing number for algebraic knots $K$ under Hopf fibration, one topological, denoted $h(K)$, the other coming from the realization of such knots around complex…

We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higher-dimensional fibred knots.

A new lower bound on the complexity of a 3-manifold is given using the Z2-Thurston norm. This bound is shown to be sharp, and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of…

Let $M$ be a compact orientable Seifered fibered 3-manifold without a boundary, and $\alpha$ an $S^1$-invariant contact form on $M$. In a suitable adapted Riemannian metric to $\alpha$, we provide a bound for the volume $\text{Vol}(M)$ and…

Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -- this could be described as Seifert fibering)There are group-theoretic obstructions to the…

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…

We prove Farber's conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our…

Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geq\alpha_0…

We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact…

We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised…

Infinite families of 3-dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $\text{SL}_2(\mathbb{F})$-characters…

If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an…

We show that the problem of deciding whether a knot in a fixed closed orientable 3-dimensional manifold bounds a surface of genus at most $g$ is in co-NP. This answers a question of Agol, Hass, and Thurston in 2002. Previously, this was…

We prove an upper bound of higher topological complexity $TC_n(X)$ using higher $\mathcal{D}$-topological complexity $TC_n^{\mathcal{D}}(X)$ of a space $X$. An intermediate invariant $\widetilde{TC}_n(X)$ is used in the proof. We interpret…

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds…