In a knot, the (two-dimensional) or 2d writhe is the sum of all positive crossings minus the sum of all negative crossings. The 2d writhe is always an integer. There is also, for each knot, a smallest possible 2d writhe.
The 3d writhe is the average of 2d writhe taken over all projection directions in 3d. The 3d writhe is a real, non-integer number.
A knot is ideal if one imagines it as made of a tube of slippery rope, and then one tightens the rope as much as possible.
Now the question:
Is the 3d writhe of ideal knots proportional to their smallest possible 2d writhe?
In other words, taking many knots with different complexities, is their 3d writhe proportional to the smallest possible 2d writhe?
