Tag Archives: mobius band

reflector circle at a punctured torus

Sea T_o un toro 2-dimensional donde hemos removido un disco cerrado,
sea S=T_o\cup_{\partial}\bar{A} donde \bar{A} es aro S^1\times I con S^1\times 0 es la frontera
de una “vecindad” de una curva cerrada simple reflectora, y S^1\times 1 como la curva reflectora. Tal “aro reflector”, \bar{A} tiene como^ orbifold – grupo fundamental a \pi_1(\bar{A})=\Bbb{Z}\times{\Bbb{Z}}_2. Entonces el producto amalgamado es:

reflector circle

reflector circle

Esto es divertido por que es bien sabido que la superficie cerrada de género tres no orientable,  N_3 tiene presentación parecida a esta última.

Observemos que las respectivas abelianizaciones son \mathbb{Z}^2+\mathbb{Z}_2

Entonces es ¿cierto o no qué el concepto de curva reflectora dado por P.Scott no sea el mismo que el de curva reflectora en una superficie no orientable?

Recuerde que, poner una curva reflectora a una superficie orientable es para hacer una superficie no orientable de tipo T\#\cdots\#T\#{\Bbb{R}}P^2 de género impar, donde T es el toro 2D.

^ footnote{Ref[P. Scott en 424p. “The Geometries of 3-Manifolds“, 1983]}

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Filed under algebra, cucei math, group theory, low dimensional topology, topology

Moxi: the cartesian product of the mobius-strip and an interval

MoxI is a solid Klein bottle, the genus one nonorientable handlebody. Here we see it with some typical curves on


look a rough view

another earlier


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geodesics in the band with better numerical resolution

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perturbated core curve

the core curve of the mobius strip perturbated by 0.1\cos(11\ t) 

But if we use 0.4\cos(12.5t)\sin(3t)\cos(t)



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four geodesics in the band

on the right of this post you can see 4 (approximately) geodesics in the mobius strip, one is transversal to the core curve and the others (three)  begin at that transversal… One almost can see that all geodesics in the surface have curvature but without 3d torsion… would it be a theorem?  

Rendered with Mathematica-6.

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can you see a solid torus?

dcoato5The image on the right is a solid torus with four lateral annulus in colours: orange, green, yellow and blue.

Solid tori are important elementary type of 3-manifolds. Also called orientable genus one handlebodies and can be fibered -in the sense of Seifert- by longitudinal circles, and in many different ways.

Among three dimensional technologeeks, they are customed to see a solid torus that can be fibered as the M\ddot{o}\stackrel{\sim}\times I, the twisted interval bundle over the Möbius strip.  In constrast M\ddot{o}\times I is the genus one non orientable handlebody: the solid Klein bottle. Isn’t difficult to prove that they are the only two I-bundles over M\ddot{o}

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a non orientable 3d-manifold with boundary

Mö x I

Mö x I

The picture on the right is representation and a description of its parts of  a 3d chunk, it is the trivial I-bundle over the möbius-strip, M\ddot{o}\times I. It is  useful to determine which 3d spaces are non orientable.

A 3d-space is non orientable if it has a simple closed curve whose 3d regular neighborhood is homeomorphic to the model of the picture.

If we denote by C\ddot{o} the core of the möbius strip, you can deduce for the curve C\ddot{o}\times\{\frac{1}{2}\} (the black pearled one) that its tangent bundle can’t be embedded in the tangent bundle of \mathbb{R}^3 or in any other tangent bundle of an orientable one.

Remember N_2 is the Klein bottle. Let’s the image talks by itself.

Ah, il nome di questo grande pezzo è la bottiglia solida di Klein.


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