That pretzel knot

That pretzel knot P(-2,3,7) is a mischievous fella.

(Edit: Even more mischievous than I had reckoned for! Big thanks to Hyun-Jong Song for pointing out that I had the 18-fold cover wrong…. and the 19-fold cover too! The braids were correct in my notes, but they sure weren’t entered into KnotPlot correctly. Then I noticed the first picture of that alleged pretzel knot was actually of P(2,3,7) for cryin’ out loud!

I’ve redone the pictures correctly now, I hope. It’s somewhat reassuring that the relaxed versions of the covers now suggest the right symmetries for the lens space covers. For fun I’ve added some animations of these spinning. Braid words for the covers are at the end of the post as are pics of how they were derived.)

One of its famous tricks is that both 18 and 19 surgeries yield lens spaces. Since lens spaces are covered by the 3-sphere, the associated knots in these lens spaces lift to knots in the 3-sphere.

Starting from the grid number one descriptions of these associated knots in their lens spaces we can obtain grid diagrams (of grid numbers 18 and 19) for the lifts of these two knots. From grid diagrams we obtain braid descriptions that are more easily thrown into KnotPlot. We then let KnotPlot do its thing to obtain some “relaxed” pictures.

Here’s the 18-fold cover as the input closed braid with some views of its relaxation.

And here’s the 19-fold cover as the input closed braid with some views of its relaxation.

Note: I’m making no claims about orientations. Maybe either or both of the braids should have been mirrored.

If you want to check these out yourself, here are the braids of the two covers. I’ll write them both for input into KnotPlot and SnapPy.

18-fold cover:
In KnotPlot type into the command line
braid bccdCBabbcBADCbccdCBabbcBADCbccdCBab
then click on “close” towards the bottom of the “Cons” pane.

In SnapPy type
M=Manifold('braid[2,3,3,4,-3,-2,1,2,2,3,-2,-1,-4,-3,2,3,3,4,-3,-2,1,2,2,3,-2,-1,-4,-3,2
,3,3,4,-3,-2,1,2](0,0)(1,0)')
then try out things like M.volume() and M.symmetry_group()

Hmmm… had to put in carriage returns for the whole line to appear.

19-fold cover:
In KnotPlot type into the command line
braid cdefDCabcdBAEDbcdeCBFEcdefDCabcdBAEDbcdeCBFEcdefDCabcd
then click on “close” towards the bottom of the “Cons” pane.

In SnapPy type
N=Manifold('braid[3,4,5,6,-4,-3,1,2,3,4,-2,-1,-5,-4,2,3,4,5,-3,-2,-6,-5,3,4,5,6,-4,-3,1
,2,3,4,-2,-1,-5,-4,2,3,4,5,-3,-2,-6,-5,3,4,5,6,-4,-3,1,2,3,4](0,0)(1,0)')
then try out things like N.volume() and N.symmetry_group()

Finally, here are two pics that suggest how these braids were obtained from the grid number one presentations of the knots dual to the lens space surgeries of P(-2,3,7).

~ by Ken Baker on October 20, 2009.