3-MANIFOLDS With Planar Presentations and the Width of Satellite Knots (original) (raw)

Intrinsic linking and knotting of graphs in arbitrary 3–manifolds

Algebraic & Geometric Topology, 2006

We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S 3 . Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S 3 . 05C10, 57M25

Three manifolds as geometric branched coverings of the three sphere

Boletin De La Sociedad Matematica Mexicana, 2007

One method for obtaining every closed orientable 3-manifold is as branched covering of the 3-sphere over a link. There is a classical topological result showing that the minimun possible number of sheets in the covering is three. In this paper we obtain a geometric version of this result. The interest is given by the growing importance of geometry in 3-manifolds theory.

Representing links in 3-manifolds¶ by branched coverings of S 3

manuscripta mathematica, 1998

We introduce a planar coloured-diagram representation of links in 3-manifolds given as branched coverings of the 3-sphere. We also prove an equivalence theorem based on local moves and the existence of a universal configuration for such representation. As an application we give unified proofs of two different results on existence of fibered links in 3-manifolds.

Knots, butterflies and 3-manifolds

… del Seminario de …, 2004

1. INTRODUCTION Recently, 1. Izmestiev and M. Joswig ([6] and [5]) have shown that for any trian-gulation .6. in a manifold N, there is an associated group II(.6.), called the group of projectivities of .6.. This group has sorne similarities with the fundamental group, even though it is not a ...

Thin position for knots and 3-manifolds

Topology and its Applications, 2008

We prove that for 2-bridge knots and 3-bridge knots in thin position the double branched cover inherits a manifold decomposition in thin position. We also argue that one should not expect this sort of correspondence to hold in general.

Knots and topologically transitive flows on 3-manifolds

Topology, 2004

Suppose that ' is a nonsingular (ÿxed point free) ow on a smooth three-dimensional manifold M. Suppose the orbit though a point p ∈ M is dense in M. Let D be an imbedded disk in M containing p which is transverse to the ow. Suppose that q ∈ D is a point in the forward orbit of p. Under certain assumptions on M , which include the case M = S 3 , we prove that if q is su ciently close to p then the orbit segment from p to q together with a compact segment in D from p to q forms a nontrivial prime knot in M .

Knots, Surfaces, 3-manifols

2021

CMO Workshop 21w5094 delved into the inner workings of 3-manifolds and how knots and surfaces can be positioned in 3-manifolds. Specifically, participants conferred on recent research about structures, especially geometric structures, exhibited by 3-manifolds and how distinct such structures interact and inform us about each other. An unexpected recurring theme came via covering spaces: Inspired by Etienne Ghys’ ICM plenary lecture, Tali Pinsky discussed Anosov flows, paying particular attention to the complement of the trefoil knot. The trefoil knot complement admits a degree 6 cover that illustrates Anosov flows and shows how dynamics and topology interact. On a different note, Emily Hamilton discussed hyperbolic 3-manifolds with infinitely many virtual geometric triangulations. The structures involved, geometric triangulations, live in covers of the actual 3-manifold and can only exist when certain algebraic conditions (subgroup separability) on the fundamental group of the 3-man...

Closed oriented 3-manifolds as 3-fold branched coverings of S 3 of special type

Pacific Journal of Mathematics, 1976

It has been shown by Hilden and Montesinos independently that any closed oriented 3-manif old Mis a 3-fold irregular branched covering of S 3 , p:M^>S B. The purpose of this paper is to show that the branch covering space map can be chosen in such a way that the set of points at which p fails to be a local homeomorphism is the boundary of a disc in M. One application of this result is a new proof that a closed oriented 3-manifold is parallelizable.