Classical knot theory is about how to figure out whether

  • a knot is actually knotted or only seems to be knotted
  • one knot is the same as another one via some moves.

If you look at the image below

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you easily recognize that they are the same and they can be deformed continuously one into the another. The general situation is not that simple though. Can you immediately see whether the one below (or worse) is also just a circle?

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So, in the early 20th century Reidemeister discovered how to simplify the question and defined three, so-called, Reidemeister moves:

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They are actually really what one does in daily life to unknot a bundle. If you use the moves on a simple example you can see how it works:

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This idea solves to some extend the question whether one knot is the same as another. From a mathematical point of view this is somewhat unsatisfactory because one cannot analytically handle little pictures of knots. Instead of this it was discovered that one can uniquely associate polynomials with knots and the transition from a knot figure to a polynomial goes like this. Any crossing in a knot diagram is split into two parts:

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and the split generates a set of simplified knot figures where the splitting in one or the other direction is kept via the letters A and B:

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In the result you can see that you have a mixture of two things: the letters and the amount of loops (simple circles). So, one sums it all up into a polynomial

$$\langle K\rangle = \sum_f P_f\,x^{L_f -1}$$

where

  • $K$ is an arbitrary knot
  • the sum runs over the set of figures when decomposing $K$ into figures
  • $P_f$ is the product of the letters in the figure
  • $L_f$ is the number of loops.

For the example above this leads to

$$A^3x^{2-1} + A^2B x^{1-1} + A^2Bx^{1-1}+AB^2x^{2-1}+A^2Bx^{1-1}+AB^2x^{2-1}+AB^2x^{2-1}+B^3x^{3-1}\\ =A^3x + 3A^2B + 3AB^2x + B^3x^2.$$

With this in place there various fun things you can do and prove statements like the following

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Of course assigning a polynomial to a knot should be well-defined, meaning that it should be independent of how you shuffle it using the moves defined above. So one looks for values of A,B and x such that the polynomial is stable under the Reidemeister moves. As it stands the solution of this challenge does not work without assigning a direction to a knot (and/or a figure). The invariance requires one to assign a way of walking around on a knot $K$ and defining the writhe $w(K)$:

$$w(K) = \sum_{c\in K} \text{sign}(c)$$

where the sign of the crossing is defined like below

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With this in place the normalized bracket polynomial can be defined as

$$\mathcal{L}_K = (-A^3)^{-w(K)}\langle K \rangle$$

with

$$B = A^{-1},\, x = -A^2-A^{-2}$$

and solves the initial two questions above:

  • if two knots have the same Jones polynomial they are the same, up to some Reidemeister moves
  • if a knot has the same Jones polynomial as the unknot it’s not knotted.

The normalized bracket polynomial is a function of a variable $A$ which can be anything you like (a Grassmann number, a matrix, whatnot). If you make the substitution $A \rightarrow z^{-1/4}$ you get the Jones polynomial:

$$V_K(t) := \mathcal{L}_K(t^{-1/4}). $$

Much more can be said about knots and how they capture a lot of physics. The amazing Knots and physics is full of fun and deep ideas. Highly recommended.