p q is q p
The (p,q)-torus knot is isotopic to the (q,p)-torus knot. Here we use the trefoil as an example. (Trefoil, always the trefoil…) It’s both a (2,3)-torus knot and a (3,2)-torus knot.
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To see that these are equivalent, shift your point of view between inside and outside the torus on which the knot lies. After all, this Heegaard torus separates the 3-sphere into two solid tori. People don’t always find this approach satisfying.
Alternatively, we step between these two solid tori.
Let’s push these curves around to groom their alignment.
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Once groomed, it becomes more simple to see the annulus that traces their isotopy.
More pics are here.
This 3d print seems particularly appropriate to this subject:
http://www.shapeways.com/model/11915/knotted_gear.html?gid=mg
That’s really neat. I wonder how hard it would be to make one out of a thick rope or cable.
I think the tricky bit would be fusing the ends nicely, and I’m not sure you’d get it rigid enough.
Yeah, I was thinking about splicing the ends of a rope together while using some other rigging to hold everything else in place. But having enough leeway to actually do the splicing would probably make the entire thing too loose. I’d reckon you’d want sufficiently large p,q to get a longer section of rope to be almost linear to work with.
I guess in that knotted gear the two components are actually rigid. Maybe some other medium would be suitable.
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isn’t it a complex infinity shape?
I wonder what the electrostatic field would look like for a charged metallic trefoil knot.