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Have you tried something like this (you probably have):

Take a diagram of a complicated hyperbolic knot and draw a diagram for the Whitehead double. Then change a crossing in the clasp.

I'd like to test this, but my SnapPea can't handle it.

Edit: You could also try Haken's unknots.

show/hide this revision's text 1

Have you tried something like this (you probably have):

Take a diagram of a complicated hyperbolic knot and draw a diagram for the Whitehead double. Then change a crossing in the clasp.

I'd like to test this, but my SnapPea can't handle it.