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.
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.
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
Henry Segerman said this on March 28, 2011 at 10:07 am |
That’s really neat. I wonder how hard it would be to make one out of a thick rope or cable.
Ken Baker said this on March 28, 2011 at 4:16 pm |
I think the tricky bit would be fusing the ends nicely, and I’m not sure you’d get it rigid enough.
Henry Segerman said this on March 28, 2011 at 8:30 pm
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.
Ken Baker said this on March 28, 2011 at 9:59 pm |
[…] 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, […]
Weekly Picks « Mathblogging.org — the Blog said this on April 7, 2011 at 2:58 am |
[…] Ken Baker: Once again, Whiteheadtangletangle…tangle, p q is q p […]
Third Xamuel.com Linkfest said this on April 10, 2011 at 3:58 pm |
isn’t it a complex infinity shape?
valeria said this on October 13, 2011 at 4:43 am |
I wonder what the electrostatic field would look like for a charged metallic trefoil knot.
Bill Livingston said this on January 30, 2013 at 4:42 am |