In the same way that a figure-eight knot is equal (after a suitable rotation) to its mirror knot, I am looking for simple tangles made of three ropes that are equal to their mirror tangle.

The tangles should have two further properties:

The six ends of the ropes should align with the x, y and z axes. Ropes can also start along one axis and end along another.

The region where the three ropes tangle up should be as simple as possible, but at the same time, the three ropes should be tangled or braided up, so as to keep the tangle of the three ropes bound up together.

What would the simplest tangles look like?

Remark: I have found one simple candidate tangle. Two of the ropes look like the two parentheses ) and ( shifted over each other, with a third rope going vertically through the "hole" created by the first two ropes. This "tangle", however, is not really bound up together. Therefore I pose the above question.

I have found one complex candidate tangle. The three ropes are braided like a girl's hair braid, with a total of six braiding motions.

Are there other possible tangles, maybe with intermediate complexity?

equalto the original tangle, or just isotopic? The former would be called something like "strongly mirror-symmetric" while the latter would be called "weakly mirror symmetric". These generally are not the same things -- not even for knots. $\endgroup$ – Ryan Budney Apr 11 '16 at 6:48