Knot Theory and Polynomials

Followup of Braid Theory

Prerequisite: algebra 2

Mathematics is broad and wide. Nonmathematicians have probably never heard of topology, which is about the properties of deformed objects and spaces. Under topology is knot theory.

Receive an introduction from Numberphile:
https://www.youtube.com/watch?v=aqyyhhnGraw

And take up some basics from this site. I love the commutativity and associativity of knots. Simple yet mindblowing:
http://www.popmath.org.uk/exhib/knotexhib.html

I was at this awkward stage where the understandable information are too easy and the advanced information are too difficult. Then I found Knots: Mathematics with a Twist by Alexei Sossinsky. It tells of ways people applied rules to discover invariants, consistent properties that suggests an underlying truth. I bent my brains a little, but at least it is still topologically the same brain..

Conway's Skein Relation

A knot is topologically the same no matter how you stretch and tangle it. But once you cut and reattach sections, it is no longer the same knot. There are three possible ways to attach two sections, eerily reminiscent of topoisomerase and their work on DNA strands.

In order to attach numerical value to knots, three rules are established (pardon me for using N instead of L). The upsidedown ∆ denotes "the polynomial of" whatever is in the bracket.

I. Two knots are the same if their polynomials are the same.
II. The polynomial of the unknot is 1.
III. Conway's Skein Relation: [the polynomial of N+] minus [the polynomial of N-] is [x times the polynomial of N˚].

Rule III is better understood with this diagram, where the intersection/break is the only point of difference between these three knots.

If we plug in the unknot into Conway's Skein Relation, we get 1 - 1 = 0x, which is just 0. From rules II and III we found that the polynomial of a double ring is 0.

If you still have not figured out how Conway's Skein Relation works, notice how the parts outside the dashed circles are the same, and the insides are N+, N-, and N˚.

Conway's Skein Relation is as easy as algebra. If we plug in the double ring and unknot along with their polynomials, we get ∆(H+) - (0) = x(1). The polynomial of the hopf link is x.

We can bring it even further. You need to stretch your imagination a little more in this example. (1) - ∆(T) = x(x), do the algebra, and figure that the trefoil knot is -(x^2)+1.

Note that rule I applies all the way.. so far. The problem with Conway's polynomials is that a trefoil knot and its mirror image have the same polynomial. They are topologically different because one cannot be deformed into the other without breaking, and Conway's polynomials do not express that. More advances will be made to solve this problem.

Kauffman Bracket

The revised rule I is another form of Conway's Skein Relation. In rule II <KUO> denotes a knot added to an unknot. Rule III states again that the polynomial of an unknot is 1.

You can figure that <00> is (-a^2 - a^-2) by rule II. Let K be an unknot, which is linked to another unknot. (-a^2 - a^-2)<1> is just (-a^2 - a^-2). <00> = (-a^2 - a^-2).

Here are some more polynomial assigning that you can do according to Kauffman's revision. More algebra, but with a knotty twist. You may have noticed that the two knots on the same row as the hopf link are in fact unknots, but their polynomials are not the same! One has a positive degree while the other is negative.

The little formula at the base solves the problem. The w(K) (called a writhe) is equal to the number of positive crossing minus the number of negative crossings for a knot K.

Then Jones did another revision after Kauffman. I am not sure how recent updates work, or if there are even any.

There is something mysterious about knots. It may appear to be just a piece of string, yet it can be so spatially complex. It is a mesmerizing tangle of intraplay. True, simple, aesthetic. I like.

Damn. What will humans decide to number next?