Links with alternating diagrams on closed surfaces of positive genus

1995 ◽  
Vol 117 (1) ◽  
pp. 113-128 ◽  
Author(s):  
Chuichiro Hayashi
Keyword(s):  
Orientable Surface ◽  
Heegaard Splitting ◽  
Closed Orientable Surface ◽  
Closed Surfaces ◽  
Positive Genus

AbstractWe consider links in an orientable 3-manifold M which have an alternating diagram on a closed orientable surface F of positive genus. We see that if the diagram is ‘complex’ enough and if F gives a Heegaard splitting of M, then such a link L is prime and M—L does not contain an essential torus.

scholarly journals ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

2020 ◽  
pp. 1-10
Author(s):  
MARK GRANT ◽  
AGATA SIENICKA
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Closed Surface ◽  
Orientable Surface ◽  
Mapping Class ◽  
Closed Orientable Surface ◽  
Outer Action ◽  
Positive Genus ◽  
Closed Sphere ◽  
Group Modulo

Abstract The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

scholarly journals On the entropy norm on the group of diffeomorphisms of closed oriented surface

2018 ◽  
Vol 12 (01) ◽  
pp. 105-111
Author(s):  
Michael Brandenbursky ◽  
Arpan Kabiraj
Keyword(s):  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Oriented Surface ◽  
Group Of Diffeomorphisms ◽  
Positive Genus ◽  
Closed Oriented Surface

We prove that the entropy norm on the group of diffeomorphisms of a closed orientable surface of positive genus is unbounded.

scholarly journals DISTORTION OF SPHERES AND SURFACES IN SPACE

2020 ◽  
Vol 71 (3) ◽  
pp. 981-988
Author(s):  
Sebastian Baader ◽  
Luca Studer ◽  
Roger Züst
Keyword(s):  
Lower Bound ◽  
Unit Disc ◽  
Closed Surfaces ◽  
Positive Genus ◽  
Sharp Lower Bound ◽  
Uniformly Bounded

Abstract It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb{R}^3$. In this note we show that distortion minimizers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $\pi /2$ is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.

scholarly journals Ranks of mapping tori via the curve complex

2019 ◽  
Vol 2019 (748) ◽  
pp. 153-172 ◽  
Author(s):  
Ian Biringer ◽  
Juan Souto
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Curve Complex ◽  
Mapping Torus ◽  
Closed Orientable Surface ◽  
Mapping Tori

Abstract We show that if ϕ is a homeomorphism of a closed, orientable surface of genus g, and ϕ has large translation distance in the curve complex, then the fundamental group of the mapping torus {M_{\phi}} has rank {2g+1} .

scholarly journals Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons

2017 ◽  
Vol 29 (06) ◽  
pp. 1750018 ◽  
Author(s):  
Sven Bachmann
Keyword(s):  
Ground State ◽  
Entanglement Entropy ◽  
Orientable Surface ◽  
Cell Complex ◽  
Local Order ◽  
Topological Order ◽  
Cohomology Groups ◽  
Closed Orientable Surface ◽  
Ground State Degeneracy ◽  
Comprehensive Study

In this comprehensive study of Kitaev’s abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterize the elementary anyonic excitations. The homology and cohomology groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterizations of topological order.

UNKNOTTING ORIENTABLE SURFACES IN THE 4-SPHERE

1995 ◽  
Vol 04 (02) ◽  
pp. 213-224 ◽  
Author(s):  
JONATHAN A. HILLMAN ◽  
AKIO KAWAUCHI
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Flat Embedding ◽  
Orientable Surfaces

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an infinite cyclic fundamental group.

scholarly journals Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces

2015 ◽  
Vol 26 (09) ◽  
pp. 1550066 ◽  
Author(s):  
Michael Brandenbursky
Keyword(s):  
Braid Group ◽  
Infinite Family ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Invariant Metrics ◽  
Closed Orientable Surface ◽  
Infinite Dimensional ◽  
Hamiltonian Diffeomorphisms ◽  
Injective Homomorphisms ◽  
Area Preserving

Let Σg be a closed orientable surface of genus g and let Diff 0(Σg, area ) be the identity component of the group of area-preserving diffeomorphisms of Σg. In this paper, we present the extension of Gambaudo–Ghys construction to the case of a closed hyperbolic surface Σg, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Σg defines a nontrivial homogeneous quasi-morphism on the group Diff 0(Σg, area ). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff 0(Σg, area ) is infinite-dimensional. Let Ham (Σg) be the group of Hamiltonian diffeomorphisms of Σg. As an application of the above construction we construct two injective homomorphisms Zm → Ham (Σg), which are bi-Lipschitz with respect to the word metric on Zm and the autonomous and fragmentation metrics on Ham (Σg). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham (Σg).

scholarly journals Singular pleated surfaces and CP1–structures

1995 ◽  
Vol 37 (2) ◽  
pp. 179-190 ◽  
Author(s):  
Ser Peow Tan
Keyword(s):  
Teichmüller Space ◽  
Universal Cover ◽  
Orientable Surface ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Distinguished Point ◽  
Totally Geodesic ◽  
Closed Orientable Surface ◽  
Hyperbolic Structures ◽  
Usual Interpretation

Let Fg be a closed orientable surface of genus g > 1 and let be the Teichmuller space of Fg, i.e., the space of marked hyperbolic structures on Fg We shall also denote by the space of marked hyperbolic structures on Fgwith one distinguished point; by this, we mean a distinguished point on the universal cover gof Fg. This space is isomorphic to the space of marked complete hyperbolic structures on a genus g surface with 1 cusp which is the usual interpretation of . Choose a decomposition of Fginto pairs of pants by a collection of non–intersecting, totally geodesic simple closed curves. The Fenchel–Nielsen coordinates for relative to this decomposition are given by the lengths of the curves as well as twist parameters defined on each curve. Varying the length and twist parameters gives deformations of the marked hyperbolic structures.

ON FRAMED LINK PRESENTATIONS OF SURFACE BUNDLES

1998 ◽  
Vol 07 (08) ◽  
pp. 1087-1092
Author(s):  
KAZUHIRO ICHIHARA
Keyword(s):  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Surface Bundles ◽  
Fibered Link ◽  
Surface Bundle ◽  
Framed Link

In this paper, we show that every closed orientable surface bundle over the circle is represented by a fibered link in the 3-sphere with framings induced by the fibration of the complement.

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