curious little things

This page contains images related to the paper `New perspectives on self-linking' with Dev Sinha, Kevin Scannell and Jim Conant. The flavor of our results is that one can construct invariants of knots by counting `geometric coincidences' of points on the knot. More precisely, given a straight line that intersects a knot in exactly four points, there is a sign one can associate to it, +1 or -1. If you add up these signs over all the quadrisecants, you get a number. That number does not change as you play with the knot -- a pair of opposite sign quadrisecants can be created or destroyed, but that's is all that can happen. For technical reasons, this is only true for `long knots' as in the two pictures below. If the knot is closed, the conservation of quadrisecants is a little more elaborate. In the following two pictures, the positive sign quadrisecants are in yellow, the ones negative ones in blue.

Trefoil with alternating quadrisecants.

Figure-8 knot with alternating quadrisecants.

I've also created some rather complicated crossing-change type animations, where a 1-parameter family of immersions goes from the unknot to various torus knots. Here is one for the trefoil, here is one for the 7-3 torus knot, and here is one for a 7-5 torus knot.

This is a link to a finger-move isotopy of the figure-8 knot.

A (p,q) torus knot can be thought of as sitting in the 3-sphere as the solution to the equations x^p+y^q = 0, where you think of the 3-sphere as the unit sphere in complex 2-space with coordinates (x,y). The following animations were produced by following a path in the complement of the (p,q) torus knot, using that point as a stereographic projection point and then computing all the quadrisecants for the projected knot in the corresponding tangent space. This is a trefoil example. This is the example of a 3-4 torus knot.

This animation stays within the isotopy class of a trefoil knot, but introduces local curvature.

Update: In my paper A family of embedding spaces I extend the above result to compute the first non-trivial Vassiliev invariant of the space of long knots in R^n as a quadrisecant count over a (2n-6)-dimensional family of knots. Provided n is larger than 3, the first non-trivial Vassiliev invariant completely describes the first non-trivial homotopy group of the space of knots.

Sextuples -- the type-3 invariant:

When attempting to generalize the quadrisecants result to the type-3 invariant, for some time we were considering the cocircular sextuples `coincidence' as a possible candidate. I haven't worked on this in a while now, but here are some pictures of some early experiements.