Related papers: Knots in collapsible and non-collapsible balls
As a first step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a…
There are 352.2 million prime knots in the 3-sphere with at most 19 crossings. We study which of these knots are slice, in both the smooth and topological categories. While no algorithm is known for deciding whether a given knot is slice in…
It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised…
An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in the boundary torus of $M$. For such a triangulation, the gluing and completeness…
For any n\ge 2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. Berge and Kawauchi gave…
We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander…
We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…
We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.
It is shown that a hyperbolic knot in the 3-sphere admits at most nine integral surgeries yielding 3-manifolds which are reducible or whose fundamental groups are not infinite word-hyperbolic.
Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this…
We exhibit an algorithm to determine the bridge number of a hyperbolic knot in the 3-sphere. The proof uses adaptations of almost normal surface theory for compact surfaces with boundary in ideally triangulated knot exteriors.
We show that, for hyperbolic fibred knots in the three-sphere, the volume and the genus are unrelated. Furthermore, for such knots, the volume is unrelated to strong quasipositivity and Seifert form.
A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known…
A triangulation of a $3$-manifold can be shown to be homeomorphic to the $3$-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one…
The concordance group of knots in the three-sphere contains an infinite subgroup generated by elements of order two, each one of which is represented by a knot K with the property that for every n > 0, the n-fold cyclic cover of S^3…
Let n be any integer greater than two. We prove that there exists a projection P having the following properties. (1) P is not the projection of any unknotted knot. (2) The singular point set of P consists of double points. (3) P is the…
In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces…
The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable…
Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this…
We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of…