A CATALOGUE OF ORIENTABLE 3-MANIFOLDS TRIANGULATED BY 30 COLORED TETRAHEDRA

2008 ◽  
Vol 17 (05) ◽  
pp. 579-599 ◽  
Author(s):  
MARIA RITA CASALI ◽  
PAOLA CRISTOFORI
Keyword(s):  
Automatic Generation ◽  
Computational Approach ◽  
Equivalence Classes ◽  
Colored Graphs ◽  
One To One ◽  
Genus Two ◽  
Jsj Decompositions

The present paper follows the computational approach to 3-manifold classification via edge-colored graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 colored tetrahedra), in [2] (with respect to non-orientable 3-manifolds up to 26 colored tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 colored tetrahedra): in fact, by automatic generation and analysis of suitable edge-colored graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting colored triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to-one correspondence with the homeomorphism classes of the represented manifolds.

Nearnesses and T0-Extensions of Topological Spaces

1983 ◽  
Vol 26 (4) ◽  
pp. 430-437 ◽  
Author(s):  
Alice M. Dean
Keyword(s):  
Topological Space ◽  
Equivalence Classes ◽  
Topological Spaces ◽  
One To One

AbstractIn [3], Reed establishes a bijection between the (equivalence classes of) principal T1-extensions of a topological space X and the compatible, cluster-generated, Lodato nearnesses on X. We extend Reed's result to the T0 case by obtaining a one-to-one correspondence between the principal T0-extensions of a space X and the collections of sets (called “t-grill sets”) which generate a certain class of nearnesses which we call “t-bunch generated” nearnesses. This correspondence specializes to principal T0-compactifications. Finally, we show that there is a bijection between these t-grill sets and the filter systems of Thron [5], and that the corresponding extensions are equivalent.

scholarly journals Classification of fused links

2016 ◽  
Vol 25 (14) ◽  
pp. 1650076 ◽  
Author(s):  
Timur Nasybullov
Keyword(s):  
Symmetric Group ◽  
Equivalence Classes ◽  
One To One ◽  
Text Component

We construct the complete invariant for fused links. It is proved that the set of equivalence classes of [Formula: see text]-component fused links is in one-to-one correspondence with the set of elements of the abelization [Formula: see text] up to conjugation by elements from the symmetric group [Formula: see text].

Detecting Non-Triviality of Virtual Links

2003 ◽  
Vol 12 (06) ◽  
pp. 781-803 ◽  
Author(s):  
Teruhisa Kadokami
Keyword(s):  
Equivalence Classes ◽  
Geometric Method ◽  
Virtual Link ◽  
Virtual Knot ◽  
Stable Equivalence ◽  
Link Diagrams ◽  
Virtual Links ◽  
Algebraic Invariants ◽  
One To One

J. S. Carter, S. Kamada and M. Saito showed that there is one to one correspondence between the virtual Reidemeister equivalence classes of virtual link diagrams and the stable equivalence classes of link diagrams on compact oriented surfaces. Using the result, we show how to obtain the supporting genus of a projected virtual link by a geometric method. From this result, we show that a certain virtual knot which cannot be judged to be non-trivial by known algebraic invariants is non-trivial, and we suggest to classify the equivalence classes of projected virtual links by using the supporting genus.

scholarly journals Low dimensional homotopy for monoids II: groups

1999 ◽  
Vol 41 (1) ◽  
pp. 1-11 ◽  
Author(s):  
STEPHEN J. PRIDE
Keyword(s):  
Free Group ◽  
Equivalence Classes ◽  
Normal Closure ◽  
Group Presentation ◽  
One To One ◽  
Low Dimensional ◽  
Reduced Words

Consider a group presentation: $$\hat{[Pscr ]}\tfrm{=<\tfbf{x};}\tfbf{r}\tfrm{>}$$. Here x is a set and r is a set of non-empty, cyclically reduced words on the alphabet x ∪ x−1 (where x−1 is a set in one-to-one correspondence x[harr ]x−1 with x). We assume throughout that $\hat{[Pscr ]}$ is finite. Let $\hat{F}$ be the free group on x (thus $\hat{F}$ consists of free equivalence classes [W] of word on x∪x−1), and let N be the normal closure of {[R] : R∈r} in $\hat{F}$. Then the group G=G($\hat{[Pscr ]}$) defined by $\hat{[Pscr ]}$ is $\hat{F}\tfrm{/}N$. We will write W1 =GW2 if [W1]N=[W2]N.

scholarly journals Universal Families of Rational Tropical Curves

2013 ◽  
Vol 65 (1) ◽  
pp. 120-148 ◽  
Author(s):  
Georges Francois ◽  
Simon Hampe
Keyword(s):  
Moduli Space ◽  
Equivalence Classes ◽  
Tropical Curves ◽  
One To One ◽  
Tropical Varieties

AbstractWe introduce the notion of families of n-marked, smooth, rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of n-marked, abstract, rational, tropical curves Mn.

scholarly journals Rectangular diagrams of Legendrian graphs

2014 ◽  
Vol 23 (13) ◽  
pp. 1450074 ◽  
Author(s):  
Maxim Prasolov
Keyword(s):  
Equivalence Classes ◽  
Combinatorial Object ◽  
Edge Contraction ◽  
One To One

In this paper Legendrian graphs in (ℝ3, ξst) are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram — a purely combinatorial object. Moves of rectangular diagrams are introduced so that equivalence classes of Legendrian graphs and rectangular diagrams coincide. Using this result we prove that the classes of Legendrian graphs are in one-to-one correspondence with fence diagrams modulo fence moves introduced by Rudolph.

scholarly journals On the Combinatorial Structure of Arrangements of Oriented Pseudocircles

10.37236/1783 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Johann Linhart ◽  
Ronald Ortner
Keyword(s):  
Representation Theorem ◽  
Equivalence Classes ◽  
Oriented Matroids ◽  
Combinatorial Structure ◽  
Topological Representation ◽  
Combinatorial Properties ◽  
One To One ◽  
Orientable Surfaces

We introduce intersection schemes (a generalization of uniform oriented matroids of rank 3) to describe the combinatorial properties of arrangements of pseudocircles in the plane and on closed orientable surfaces. Similar to the Folkman-Lawrence topological representation theorem for oriented matroids we show that there is a one-to-one correspondence between intersection schemes and equivalence classes of arrangements of pseudocircles. Furthermore, we consider arrangements where the pseudocircles separate the surface into two components. For these strict arrangements there is a one-to-one correspondence to a quite natural subclass of consistent intersection schemes.

REMARKS ON THE IMPRIMITIVITY THEOREM FOR NONLOCALLY COMPACT POLISH GROUPS

2000 ◽  
Vol 03 (02) ◽  
pp. 247-262
Author(s):  
FRANCESCO FIDALEO
Keyword(s):  
Invariant Measure ◽  
Probability Measure ◽  
Invariant Probability Measure ◽  
Equivalence Classes ◽  
Fréchet Spaces ◽  
Locally Compact ◽  
Polish Groups ◽  
Polish Group ◽  
One To One

In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space [Formula: see text] of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem [Formula: see text] in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.

C*-ALGEBRAS OF TRIVIAL K-THEORY AND SEMILATTICES

1999 ◽  
Vol 10 (01) ◽  
pp. 93-128 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
Direct Limit ◽  
Tensor Products ◽  
Equivalence Classes ◽  
Direct Sums ◽  
C Algebras ◽  
Direct Limits ◽  
One To One ◽  
K Theory ◽  
Essential Extensions ◽  
Distributive Semilattices

We give a class of nuclear C*-algebras which contains [Formula: see text] and is closed under stable isomorphism, ideals, quotients, hereditary subalgebras, tensor products, direct sums, direct limits as well as extensions. We show that this class of C*-algebras is classified by their equivalence classes of projections and there is a one to one correspondence between (unital) C*-algebras in the class and countable distributive semilattices (with largest elements). One of the main results is that essential extensions of a C*-algebras which is a direct limit of finite direct sums of corners of [Formula: see text] by the same type of C*-algebras are still direct limits of finite direct sums of corners of [Formula: see text].

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