Jones polynomial (thing)

One of the goals of knot theory is to find invariants of knots. An invariant takes the same value on equivalent knots so this can be a way to show that two knots are different.

In 1984 the world of knot theory was rocked by an incredible invention of Vaughan Jones, the Jones polynomial, descibed in his paper A Polynomial Invariant for Knots via von Neumann Algebras.

Motivated by ideas from theoretical physics and operator algebras Jones found an invariant that could distinguish more knots than any previous invariant.

Here are some of the properties of the Jones polynomial taken from Jones' paper. For each knot K its Jones polynomial is a Laurent polynomial V_K(t) with integer coefficients (we have equality of polynomials for equivalent knots).

The unknot (i.e. an unknotted circle) has V_unknot(t)=1.

An interesting example is the (left-handed) trefoil knot

      /\ /\
     /  \  \
    /  / \  \
   /_ /______\
     /     \
    /_______\

It has Jones polynomial t^-1+t^-3-t^-4. The mirror image of this knot is the right-handed trefoil. Draw one! It has Jones polynomial t+t³-t⁴.

This illustrates a more general property found by Jones; the polynomial of the mirror image of a knot is obtained by replacing t by t^-1, that is:
V_K*(t)=V_K(t^-1)

The Jones polynomial can also be defined for (oriented) links (which are several knots linked together) (it turns out that orientation is not significant for knots but is for links). This slightly more general definition allows us to effectively compute the polynomial by breaking a link down into simpler components, using the skein relation. This says that
t^-1V_K -tV_K' = (t^1/2-t^-1/2)V_L where the links K,K',L are related by altering one crossing of the link like so:

 K               K'              L

 \               \              ----
- \ --          ------
   \               \
    \               \           ----

(The orientation in the diagram is such that for each strand we pass from left to right.)