That pretzel knot

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

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.

The Flickr set has more pics of these.

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

Also I nudged the relaxations a bit to help coerce it along… slight chance that a crossing change occurred. Here are the input files I used for the KnotPlot relaxations.

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Possibly related posts: (automatically generated)

• The Berge Conjecture for one-tunnel knots
• Knot complements covering knot complements
• Polynomial Knots I

~ by Ken Baker on October 20, 2009.