相关论文: Distance and bridge position

We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimal genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic…

For any pair of integers $m$ and $n$ such that $3<m<n$, we provide an infinite family of links, where each link in the family has a locally minimal $n$-bridge position and a globally minimal $m$-bridge position. We accomplish this by…

We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum…

We show that except for $n = 2$ if a bridge surface for a knot is an index $n$ topologically minimal surface, then after a perturbation it is still topologically minimal with index at most $n+1$.

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show…

Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a…

Let $M=W\cup_T V$ be an amalgamation of two compact 3-manifolds along a torus, where $W$ is the exterior of a knot in a homology sphere. Let $N$ be the manifold obtained by replacing $W$ with a solid torus such that the boundary of a…

Information Theoretic analysis of the periods of a hyperelliptic curve provides more information about the well--known but abstract relationship between the branch points and the periods. Here one constructs a canonical homology basis for a…

Given $M_\varphi$, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy $\varphi$ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the…

We present a one-to-one correspondence between equivalence classes of embeddings of a manifold (into a larger manifold of the same dimension) and equivalence classes of certain distances on the manifold. This correspondence allows us to use…

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the…

We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic…

A $(1,1)$-knot in the 3-sphere is a knot that admits a 1-bridge presentation with respect to a Heegaard torus in $\mathbb{S}^{3}$. A new parameterization of $(1,1)$-knots distinct from the classical ones is introduced. This parameterization…

We consider the family of Heegaard splittings of genus $g$ at least three which are defined via a glueing map that is the $n$-th power of the Dehn twist along a curve that satisfies a natural topological assumption, namely pared…

We give a distance estimate for the metric on the disk complex and show that it is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error…

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 $\mathbf{u}$…

It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in the…

A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of…

We analyze the orbifolds that can be obtained as quotients of hyperbolic 3-manifolds admitting a Heegaard splitting of genus two by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the…

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions.…