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.

23-torusknot 32-torusknot
23-torusknot-tilt 32-torusknot-tilt

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.

23-32-together

23-32-together4

Let’s push these curves around to groom their alignment.

torusknots-bothsides2

torusknots-bothsides1

32-torusknot-thin1 23-torusknot-thin

Once groomed, it becomes more simple to see the annulus that traces their isotopy.

torusknots-bothsidesannulus1-big

torusknots-bothsidesannulus2-big

torusknots-bothsidesannulus5-big

torusknots-bothsidesannulus7-big

torusknots-bothsidesmanycurves

torusknots-bothsidesmanycurves5

More pics are here.

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~ by Ken Baker on March 28, 2011.

8 Responses to “p q is q p”

  1. This 3d print seems particularly appropriate to this subject:

    http://www.shapeways.com/model/11915/knotted_gear.html?gid=mg

  2. 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.

  3. [...] was incredibly rich: sketches of topology had some amazing eye candy for us, republic of math discussed (and ranted a little) about the future of mathematical textbooks, [...]

  4. [...] Ken Baker: Once again, Whiteheadtangletangle…tangle, p q is q p [...]

  5. isn’t it a complex infinity shape?

  6. I wonder what the electrostatic field would look like for a charged metallic trefoil knot.

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