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

  1. Jun 13, 2015 #1
    I know the solution for R2. 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)

    A non-oriented handle (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, in R3 what kind of topological "particles" could you get?

    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!
  2. jcsd
  3. Jun 13, 2015 #2
    I've thought of some more.

    If you cut out a toroidal knot, and create a fibration of the knot with circles (e.g. decompose the torus into circles) then you can identify opposite points in these circles.
    The circles will loop N times around the major circle and M times around the minor circle so there should be lots of topological particles for each toroidal knot labelled by (N,M) where N and M are integers.

    But I don't know how to check how many of these are identical. I suppose you'd need the Poincare theorem for that(?)

    Edit: Oh seem I just reinvented Dehn Surgery. haha.
    Last edited: Jun 13, 2015
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