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May 28th, 2019, 06:58 PM   #1
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1-D string: 1- or 3-D knot?

Is a knot in a 1-dimensional string itself 1- or 3-dimensional?
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May 29th, 2019, 06:15 PM   #2
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What exactly do you mean by knot? In general, a "1-dimensional string" is still 1-dimensional regardless of the space you embed it in. It is also still 1-dimensional if you twist it into the only thing I can imagine you mean by a knot. Does this make sense?

As a self check you could ask yourself whether you think a circle is 1 or 2 dimensional.
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May 29th, 2019, 09:49 PM   #3
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I once read that knots were unique in that they were only three-dimensional, and that quality made them special to topology. However, a knotted one-dimensional string seems to violate this.

Great self-check, SDK. I go for two dimensions, although the circle may be a projection of a 3-D cylinder, etc.
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May 29th, 2019, 09:55 PM   #4
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A circle is in fact 1-dimensional. The difficulty here is separating the dimension of an object from the dimension of the space it is embedded in. Consider for example if a circle is embedded into 3D or 4D or any higher space, you certainly would not expect the dimension of the circle to change.

A crude, non-rigorous, but instructive way to see this is to note that the following function of a single variable parameterizes the circle:
\[f(t) = (\cos(t), \sin(t)) \]
so it must have dimension no greater than 1. This intuition can be made more precise as would typically be done in a first topology course. See for example:

https://en.wikipedia.org/wiki/Manifold
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May 30th, 2019, 03:30 AM   #5
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https://en.wikipedia.org/wiki/Knot_(mathematics)
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May 30th, 2019, 09:56 AM   #6
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One-dimensional manifolds include lines and circles, but not figure eights (because they have crossing points that are not locally homeomorphic to Euclidean 1-space).
Does that figure in?

https://en.wikipedia.org/wiki/Manifold
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