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:

  1. 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.

  2. 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?

  • $\begingroup$ Are you asking for the mirror of the tangle to be equal to 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
  • $\begingroup$ Weakly mirror symmetric is sufficient. $\endgroup$ – Charles Dupont Apr 13 '16 at 3:41

Let one rope be straight along the x axis, another along the y axis, but slightly deformed, so that they just touch each other. Then take a third rope starting from -z infinity, wind it clockwise around the x axis rope, anticlockwise around the y axis rope, and continue to +z infinity. (Both times just one turn.)

I have no proof, but this could be the first nontrivial answer.

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