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Notes on tiled incompressible tori

Leonid Plachta (2012)

Open Mathematics

Let Θ denote the class of essential tori in a closed braid complement which admit a standard tiling in the sense of Birman and Menasco [Birman J.S., Menasco W.W., Special positions for essential tori in link complements, Topology, 1994, 33(3), 525–556]. Moreover, let R denote the class of thin tiled tori in the sense of Ng [Ng K.Y., Essential tori in link complements, J. Knot Theory Ramifications, 1998, 7(2), 205–216]. We define the subclass B ⊂ Θ of typical tiled tori and show that R ⊂ B. We also...

On the structure of closed 3-manifolds

Jan Mycielski (2003)

Fundamenta Mathematicae

We will show that for every irreducible closed 3-manifold M, other than the real projective space P³, there exists a piecewise linear map f: S → M where S is a non-orientable closed 2-manifold of Euler characteristic χ ≡ 2 (mod 3) such that | f - 1 ( x ) | ≤ 2 for all x ∈ M, the closure of the set x ∈ M : | f - 1 ( x ) | = 2 is a cubic graph G such that S - f - 1 ( G ) consists of 1/3(2-χ) + 2 simply connected regions, M - f(S) consists of two disjoint open 3-cells such that f(S) is the boundary of each of them, and f has some additional interesting properties....

Quotients compacts des groupes ultramétriques de rang un

Fanny Kassel (2010)

Annales de l’institut Fourier

Soit G l’ensemble des points d’un groupe algébrique semi-simple connexe de rang relatif un sur un corps local ultramétrique. Nous décrivons tous les sous-groupes discrets de type fini sans torsion de  G × G qui agissent proprement et cocompactement sur  G par multiplication à gauche et à droite. Nous montrons qu’après une petite déformation dans  G × G un tel sous-groupe agit encore librement, proprement discontinûment et cocompactement sur  G .

Sharp edge-homotopy on spatial graphs.

Ryo Nikkuni (2005)

Revista Matemática Complutense

A sharp-move is known as an unknotting operation for knots. A self sharp-move is a sharp-move on a spatial graph where all strings in the move belong to the same spatial edge. We say that two spatial embeddings of a graph are sharp edge-homotopic if they are transformed into each other by self sharp-moves and ambient isotopies. We investigate how is the sharp edge-homotopy strong and classify all spatial theta curves completely up to sharp edge-homotopy. Moreover we mention a relationship between...

Symmetries of embedded complete bipartite graphs

Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn (2014)

Fundamenta Mathematicae

We characterize which automorphisms of an arbitrary complete bipartite graph K n , m can be induced by a homeomorphism of some embedding of the graph in S³.

Symmetries of spatial graphs and Simon invariants

Ryo Nikkuni, Kouki Taniyama (2009)

Fundamenta Mathematicae

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3 + 3 vertices in detail, and determine...

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