I know the solution for R(adsbygoogle = window.adsbygoogle || []).push({}); _{2}. That is a for an infinite plane you can have one of 2 things (from the classification of 2D surfaces):

1) cross cap (cut a circle out of the plane and identify opposite points).

2) a oriented handle (cut two circles out and identify points on one with reflected points on the other - like a wormhole)

Anon-orientedhandle (cut two circles out and identify points on one with equivalent points on the other) is equivalent to two cross-caps.

Each of these "particles" adds negative curvature to the surface.

So that got me thinking, inRwhat kind of topological "particles" could you get?_{3}

I think there will be more since you can cut out spherical holes or toroidal holes (which could be knotted). You could get 3 dimensional equivalents of (1) and (2) but can you get anything else interesting? And will they all add negative curvature?

The ones I can think of are:

1) Cut out a spherical hole and identify opposite points (a 3D cross-cap - whatever that is called!!)

2) Cut out two spherical holes and identify reflected points (like a wormhole)

3) Cut out a torus (perhaps knotted) and identify opposite points at each cross-section.

4) Cut out a torus (perhaps knotted) and identify opposite points but reflected

5) Cut out two tori and identify points - (Like a toroidal wormhole - not sure if this can be composed of others)

I know the wormholes (2) are solutions of General Relativity. Are any of the others? Does that mean that these things exist or not? Are non-orientable topological defects allowed in General Relativity? If so, would they act like fermions?

Also, can there be any chiral topological particles? Maybe made out of a trefoil knotted torus or something simpler?

Would something like (3) act like a string from string theory or something else? What is their curvature? I imagine it is zero. Hence they might be solutions to empty space in GR.

In 2D space there is no-such thing as an anti-topological particle, since two cross-caps don't cancel each other out, they produce a non-oriented handle. (Being negatively curved they just add together). Are there any such things as anti-toplogical particles in 3D? (i.e. if both particle and anti-particle exist on the same plane it is equivalent to R3).

Sorry, lots of questions! This was just on my mind today!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# What kind of local topological "particles" can you get in R3?

**Physics Forums | Science Articles, Homework Help, Discussion**