Let’s riff again on that Berge tangle. The three-fold symmetry of the tangle in about one arc allows the three fillings to be equivalent.
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.
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.
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:
This generalizes quite nicely.
You might regard the above example as nearly 3-fold. Let’s do 5-fold.
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.
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?)