That pretzel knot again

Remember P(-2,3,7), that pretzel knot? (Yeah, big edit there. Yikes!)

Here it is again.
P-237onfiber-knotflat
Doesn’t quite look like it, I know. But I drew it that way because this is a trefoil with its fiber:
P-237onfiber-trefoil
and that pretzel knot sits nicely on the fiber:
P-237onfiber-knotonflatfiber

Because it sits on the fiber of a genus one fibered knot (and thus belongs to a family of Berge’s doubly primitive knots), it has a lens space Dehn surgery along the framing the fiber induces. Let’s get a glimpse of this.

We’re gonna make a genus 2 Heegaard surface from two fibers with the knot sitting on one of them.

Billow out the surface, and carry the knot along.

P-237onfiber-billow P-237onfiber-billowwithknot

Then billow it out in the other direction too and we’ll have a nice genus 2 Heegaard surface with the knot on it.

P-237onfiber-HSknot

P-237onfiber-HSknot3

Rather than looking for the primitivizing disks, we’ll find compressing disks for the handlebodies on each side of the Heegaard surface that are disjoint from the knot. See, the Dehn surgery on the knot transforms the handlebodies into two solid tori and these two disks will become meridional disks.

We’ll start by finding an essential arc on the blue surface that is disjoint from the knot. (If we wanted primitivizing disks, we would start with an essential arc that crossed the knot once.)

P-237onfiber-knotarcfiber

Now we’ll sweep this arc through the fibers to the red surface, though the two handlebodies. The easy way is through the billowing we did.

P-237onfiber-arcstraightwdisk

This gives one meridional curve.

P-237onfiber-HSproduct

To get the other… well, going through the fibers outside the billowing, we apply the monodromy of the trefoil (a Dehn twist along two curves) to the arc on the red surface. Together with the arc on the blue surface we obtain this loop:
P-237onfiber-arcmonodromy
If you start by pulling the bottom red bit down and unhook it from the blue twisting, you might be able to convince yourself that this is indeed an unknotted loop. With a bit more work you could even convince yourself that it actually bounds a disk disjoint from the surface. Drawing that disk would’ve made quite a mess. Here’s the curve as it sits on the surface.
P-237onfiber-HSproductmonodromy

Now we can look at both these curves on the Heegaard surface together.
P-237onfiber-lensspacedisks2
Count that they intersect 19 times. You see 18 in the middle here. The last comes from the red arc they have in common, but that can be perturbed to give a single transverse intersection.
P-237onfiber-lensspacedisks4

One may count that the twisting is 7 — plus or minus, and perhaps the inverse mod 19 depending on how you’re counting. Starting off with the common red arc of the two meridians as 0, then count the intersections in order along the light blue meridian and again in order along the dark blue meridian. The torsion is the number that you multiply by the light blue count of an intersection to get the dark blue count of the same intersection, mod 19. A portion of this counting is shown here:
P-237onfiber-counting
There’s an intersection that the light blue counts as 2 and the dark blue counts as 5. So if 2q=5 mod 19 then we may take q=12. Since 7=-12 mod 19 we’ve got the lens space claimed (up to homeomorphism).

Here are those two meridians again with the knot.

P-237onfiber-HSallcurves

P-237onfiber-HSallcurves2

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~ by Ken Baker on September 2, 2010.

4 Responses to “That pretzel knot again”

  1. Nice! I would be very interested in knowing more about how you produced these pictures. I guess you used Rhino but can you describe your workflow a little bit? How do you get the nice surfaces and how do you get the curves to sit nicely on the surfaces? Is this all Nurbs? which kind of tools do you use? I don’t have access to Rhino but I’d love to be able to do this with blender.

    • I’ve been using Rhino more and more these days, though I feel I should embark upon learning Blender. This is indeed all done with NURBS. I don’t know how it’s handled in Blender, but in addition to sweeping/extruding/lofting to create surfaces Rhino has many transformational commands that allow for twisting and bending. What I try to avoid doing is explicitly messing with and tweaking control points. There are some times that that’s useful, but most of the time it’s a headache.

      For this set there were two main ways I put the curves on the surface (which I then put small tubes around).
      Since the surfaces are patchworks of (trimmed) rectangles mapped into 3D, you can easily extract curves that were horizontal or vertical prior to the mapping. That’s basically what I did for the parts of the curves along the two handles. For the curves in the middle, I had Rhino find some nice curves between the endpoints of the curves on the handles and then projected them onto the surface.

      • Thanks your your answer. Unfortunately it’s not a very good time to start learning Blender. There is a new major version in beta test (blender 2.5). It changes quite a lot of things, including most of the user interface. This version is well under way but still poorly documented so I would recommend waiting a few months if you have no good reason to start. In addition you must be warned that blender is really polygon based. It has rudimentary nurb support and nice projects for more but I guess you will be very disappointed if you hope to get back your nice nurb tools (this is why I feared your answer to my question).

      • That’s a good reason for holding off on Blender for now. I’ve not done much with mesh/polygon modeling aside from SketchUp. Though with the proper transformational tools I’d reckon most of my approaches to modeling with nurbs would transfer. Guess I’ll find out eventually.

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