My Math Forum 1-D string: 1- or 3-D knot?

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 May 28th, 2019, 06:58 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 422 Thanks: 27 Math Focus: Number theory 1-D string: 1- or 3-D knot? Is a knot in a 1-dimensional string itself 1- or 3-dimensional?
 May 29th, 2019, 06:15 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 619 Thanks: 391 Math Focus: Dynamical systems, analytic function theory, numerics 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. Thanks from topsquark and Loren
 May 29th, 2019, 09:49 PM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 422 Thanks: 27 Math Focus: Number theory 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.
 May 29th, 2019, 09:55 PM #4 Senior Member   Joined: Sep 2016 From: USA Posts: 619 Thanks: 391 Math Focus: Dynamical systems, analytic function theory, numerics 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 Thanks from topsquark and Loren
 May 30th, 2019, 03:30 AM #5 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,943 Thanks: 1132 Math Focus: Elementary mathematics and beyond
May 30th, 2019, 09:56 AM   #6
Senior Member

Joined: May 2015
From: Arlington, VA

Posts: 422
Thanks: 27

Math Focus: Number theory
Quote:
 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

 Tags knot, makes, string

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