The Jones polynomial is an invariant of knots. This means that if two knots are equivalent in the sense that knot theory takes it to mean then they have the same Jones polynomial. You can compute the Jones polynomial by making crossing changes until the knot is trivial and keeping track of the changes. The Jones polynomial was defined as the trace of a braid corresponding to the knot in a certain representation of the braid group.

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 VK(t) with integer coefficients (we have equality of polynomials for equivalent knots).

The unknot (i.e. an unknotted circle) has Vunknot(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:

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-1VK -tVK' = (t1/2-t-1/2)VL
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.)

The Jones Polynomial is actually a special case of a more general polynomial invariant called the HOMFLY polynomial, the skein polynomial, or the LYMPH-TOFU polynomial, depending on who you ask. These three are equivalent polynomials with slight variation in notation.

The Jones polynomial can be obtained from the Kauffman Bracket as follows:
Let w(D) denote the writhe of an oriented link diagram D of a link L (that is, the sum of the signs of the crossings of D). Let <D> be the Kauffman bracket of D (a laurent polynomial in the variable A). Then, with the substitution A2 = t½, the Jones polynomial of the link is:

VL(t) = (-A)-3w(D)<D>

Some particular evaluations of the Jones polynomial of interest at roots of unity are:

  • VL(1) = (-2)#L-1
  • VL(e2(pi)i/3) = (-1)#L-1
  • VL(e4(pi)i/3) = 1
  • VL(i) = (-sqrt(2))#L-1(-1)Arf(L) (or 0 where Arf(L) is undefined)
  • VL(-1) = DeltaL(-1)

Here, #L denotes the linking number of a link, Arf(L) denotes the Arf invariant, and DeltaL denotes the Conway polynomial.

It is not known whether the Jones polynomial distinguishes the unknot (that is, it is unknown whether there is a nontrivial link (knot) with Jones polynomial equal to 1). Finding such a knot, or proving that none such exist, is an important open question in knot theory.

Sources: Kawauchi, Akio. A Survey of Knot Theory. Birkhauser Verlag: 1996.
Lickorish, W.B. Raymond. An Introduction to Knot Theory. Springer: 1997.

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