Tuesday, July 27, 2021

What is a Klein bottle?

Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.

Other related non-orientable objects include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

The Klein bottle was first described in 1882 by the German mathematician Felix Klein.

Construction of the Klein bottle would look like this:

(Note that this is an “abstract” gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle.)

The following square is a fundamental polygon of the Klein bottle. The idea is to ‘glue’ together the corresponding red and blue edges with the arrows matching.

To construct the Klein bottle, glue the blue arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends of the cylinder together so that the arrows on the circles match, one would pass one end through the side of the cylinder. This creates a circle of self-intersection – this is an immersion of the Klein bottle in three dimensions.

This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has no boundary, where the surface stops abruptly, and it is non-orientable, as reflected in the one-sidedness of the immersion.

The common physical model of a Klein bottle is a similar construction. The Science Museum in London has a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett.

Regular 3D embeddings of the Klein bottle fall into three regular homotopy classes (four if one paints them). The three are represented by:

1. The “traditional” Klein bottle

2. Left-handed figure-8 Klein bottle

3. Right-handed figure-8 Klein bottle

The traditional Klein bottle embedding is achiral. The figure-8 embedding is chiral. If the traditional Klein bottle is cut lengthwise it deconstructs into two, oppositely chiral Möbius strips.

If a left handed figure-8 Klein bottle is cut it deconstructs into two left handed Möbius strips, and similarly for the right handed figure-8 Klein bottle.

Painting the traditional Klein bottle two colors induces chirality on it, creating four homotopy classes.

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