Kauffman Spin and Vassiliev Singularity

Follow up of Knot Theory and Polynomials

Prerequisite: algebra 2

Some more about the nature of knots.

Kauffman Spin

Kauffman assigned spins to each intersection on a knot, either "up"or "down". In Sossinsky's book Knots: Mathematics with a Twist, he uses sticks instead pointing to checkered A or B regions. Notice that if you travel along a positive crossing of a knot, A will always be to the right of B relative to the direction you are traveling, then change places with B after the intersection.

In this example of the figure eight knot, A and B regions are assigned like blacks and whites on a chessboard. Note that the outside is an A region. In this particular example there are three A spins and one B spin, as indicated by the sticks. If you smooth out the sticks it will either join or seal up shapes in the knot. In this particular example there are two closed shapes, so there are two gamma regions.

Plug in these numbers to the equation below and you will have only obtained a partial sum. There are four crossings in the figure eight knot, so there are sixteen possible assignments of A and B spins (given by 2^n, where n is the total number of crossings). For the fifteen other possible spin assignments, count A spins, B spins, gamma regions, obtain all partial sums, then calculate the grand sum. The result should be a polynomial for the knot K.

Reminiscent of electron spins. I like.

Vassiliev Singularity

Vassiliev had another idea of knot crossings. He treated crossings as singularities. A point, as you would with two intersecting lines on a graph. As a knot changes crossing through singularity, it becomes another knot and so passes to another domain on the map. All knots in the same domain are the same knot, and knots on boundaries have singularity crossings.

I. The value of the unknot is 0.

II. All routes produce the same results.

III. Crossing a boundary along its arrow is +1, and crossing a boundary against its arrow is -1.

Starting with O going to C, to F, to H, the corresponding values are 0 +1 -1 -1, the sum of which is -1. Taking the alternate route from O straight to H is 0 -1, which is also -1.

Neat! Except I am not sure how the map is determined. *Scratches head*.

This new rule is similar to Conway's Skein Relation again. This takes even more brain bending to get around. The second row shows unknots, which rule I determines has a value of 0. The two pairs of knots in the third row are zeroes, which total zero.

In the fourth row, the trefoil is broken down, then further broken down in the fifth row. There are three unknots and one trefoil, giving a value of 1. Knot crossings definitely do not form singularities in real life, but this idea does produce fascinating results.

I feel like I can never look at objects the same way again. Some part of me will tell me to mould it, and give it a little twist. There is more in the book that I am still trying to get my head around..