May 28th, 2019, 07:58 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory  1D string: 1 or 3D knot?
Is a knot in a 1dimensional string itself 1 or 3dimensional?

May 29th, 2019, 07:15 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 680 Thanks: 454 Math Focus: Dynamical systems, analytic function theory, numerics 
What exactly do you mean by knot? In general, a "1dimensional string" is still 1dimensional regardless of the space you embed it in. It is also still 1dimensional 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. 
May 29th, 2019, 10:49 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory 
I once read that knots were unique in that they were only threedimensional, and that quality made them special to topology. However, a knotted onedimensional string seems to violate this. Great selfcheck, SDK. I go for two dimensions, although the circle may be a projection of a 3D cylinder, etc. 
May 29th, 2019, 10:55 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 680 Thanks: 454 Math Focus: Dynamical systems, analytic function theory, numerics 
A circle is in fact 1dimensional. 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, nonrigorous, 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 
May 30th, 2019, 04:30 AM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,974 Thanks: 1156 Math Focus: Elementary mathematics and beyond  
May 30th, 2019, 10:56 AM  #6  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory  Quote:
https://en.wikipedia.org/wiki/Manifold  

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