**Knots and their polynomials**

## The Jones Polynomial

The simplest invariant that distinguishes the two trefoils was
discovered quite recently, by Vaughan F. R. Jones in 1984. It can be
calculated by anyone with high-school algebra and a cool head. This
invariant is a generalized polynomial:
an expression like *t*^{-2} + 2*t*^{-1}
+ 3 - 2*t*^{2}, where both positive and negative powers
can appear. For our purposes you should think of the variable
*t* as a place-keeper: what matters are the coefficients that
go with the various powers.
The Jones polynomial is calculated from an oriented knot diagram, that is,
a 2-dimensional picture of a knot with overcrossings and undercrossings,
and with a consistent set of directions indicated along the string.
We will use the pictorial notation

[*t*]
for the Jones polynomial of an oriented knot diagram.

The theory behind the Jones polynomial shows that it only depends on
the knot as it exists in 3-dimensional space, and not on the particular
diagram chosen to represent it (twisting a knot around in space can
lead to very different diagrams). Also it does not depend on the orientation
chosen in the diagram, so we can speak of ``the Jones polynomial of the
knot.''

The algorithm for the calculation of the Jones polynomial
from an oriented knot
diagram is based on two principles.

1. The Jones polynomial assigns the value 1 to any oriented
diagram representing the unknot.

[*t*] = [*t*]= 1.
2. The skein relation: whenever three oriented diagrams differ only at
one crossing
point, their Jones polynomials are related by the equation

*t*^{-1}[*t*] - *t* [*t*]
=
(*t*^{1/2} - *t*^{-1/2})[*t*].
In words, this means that when three oriented diagrams are the same except
at one crossing, *t*^{-1} times the Jones polynomial of
the one with the upper-left-to-lower-right undercrossing minus
*t* times the Jones polynomial of
the one with the upper-left-to-lower-right overcrossing is equal to
(*t*^{1/2}-*t*^{-1/2}) times
the Jones polynomial of
the one with two downward-oriented uncrossed strands.

We will calculate a Jones polynomial by repeated application of this
relation.

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