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

• nuclearhead
In summary, the conversation discusses various topological particles that can exist in different dimensions. In 2D space, there are two types of particles - a cross cap and an oriented handle, which add negative curvature to the surface. In 3D space, there can be more particles such as a 3D cross-cap, a toroidal hole, and a toroidal wormhole. The conversation also mentions the possibility of chiral topological particles and the connection to string theory. It also raises questions about the existence of anti-topological particles in 3D space and the potential use of Dehn Surgery to identify identical particles.
nuclearhead
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!

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.

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## 1. What is a topological particle?

A topological particle is a theoretical concept in physics that refers to a localized energy or excitation in a system that exhibits topological properties. These particles can occur in various systems, such as quantum materials, condensed matter, and even in some biological systems.

## 2. What is the significance of studying local topological particles?

Studying local topological particles can provide insights into the behavior and properties of complex systems. These particles can also help us understand the underlying topological nature of various physical phenomena, and potentially lead to the development of new technologies.

## 3. How are local topological particles different from other types of particles?

Local topological particles are distinct from other types of particles, such as elementary particles, in that they are not fundamental building blocks of matter. Instead, they arise from the collective behavior of a system and are characterized by their topological properties, which do not change under certain transformations.

## 4. Can local topological particles be observed experimentally?

Yes, there have been several experimental demonstrations of local topological particles in various systems, such as in artificial spin ice, photonic crystals, and topological insulators. However, their detection and manipulation still pose significant challenges, and further research is needed to fully understand their properties and potential applications.

## 5. What are some potential applications of local topological particles?

Local topological particles have potential applications in various fields, including quantum computing, telecommunications, and energy harvesting. They can also be used to study and control exotic physical phenomena, such as topological phase transitions and topological superconductivity.

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