Reflective bandings

Let’s riff again on that Berge tangle.  The three-fold symmetry of the tangle in S^2 \times I about one arc allows the three fillings to be equivalent.

BTsymm

If we double the tangle with the red filling (that is, glue it to its reflection across the outer sphere) we obtain the two component unlink.

BT3XR BT3RR BT3RX

Getting the unlink from doubling is due to these filled tangles being rational tangles, but nevermind that for now. Instead, ponder this curiosity. Let’s keep the mirrored red filling up top and compare the blue and green down below.

BT3RB BT3RG

They’re the same except the small change between the blue and green filling.  (Taken together, the blue and green arcs bound a twisted rectangle, a “band”).  But they’re even closer in another way.  Mirror, then rotate:

BT3RBmirror BT3RBmirrorrotate
Mirror the blue guy across the horizontal plane, then rotate about the central vertical axis by a third.  And now it’s the same as the green guy.

This generalizes quite nicely.

You might regard the above example as nearly 3-fold. Let’s do 5-fold.

BT5RB-angle BT5RG-angle

BT5RBmirror-angle BT5RBmirrorrotate-angle

And 7-fold

BT7RB-angle BT7RG-angle

BT7RBmirror-angle BT7RBmirrorrotate-angle

Note that the red up top got spread apart passing from the 3-fold to the more-fold.

And of course the little twists could be replaced with other tangles.

General9RB General9RG

General9RBmirror General9RBmirrorrotate

The (double branched cover version of the) 3-fold version was noted by Bleiler-Hodgson-Weeks in Cosmetic Surgery on Knots.  Kazuhiro Ichihara and In Dae Jong recently showed in Cosmetic Banding on Knots and Links that (the double branched covers of) these more-fold generalizations can give counterexamples to one of BHW’s conjectures.

All these examples are bandings from one mess of knotted loops to its mirror image.  Is mirroring necessary? (Yep, there are some cheap shots.  Anything else?)

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~ by Ken Baker on June 27, 2016.

3 Responses to “Reflective bandings”

  1. Nice tangles! What are tools are you using to create the images?

    • Thanks Theo! All these are modeled with Rhino and rendered with the Rhino Renderer.

      • Cool! If you can export the models as .OBJ or .STL files then I can show you own to display them in 3D so you can pan, zoom and rotate in your posts with Three.js

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