Crossing number (knot theory)

[1] There has been very little progress on understanding the behavior of crossing number under rudimentary operations on knots.

A big open question asks if the crossing number is additive when taking knot sums.

It is also expected that a satellite of a knot K should have larger crossing number than K, but this has not been proven.

Basically, the higher the crossing number, the faster the relative velocity.

For composite knots, this does not appear to be the case, although experimental conditions can drastically change the results.

Trefoil knot without 3-fold symmetry with crossings labeled.
A table of all prime knots with seven crossing numbers or fewer (not including mirror images).
Square knot (cr(6)) = trefoil (cr(3)) + trefoil reflection (cr(3)).