One of the goals of knot theory
is to find invariant
of knots. An invariant takes the same value on equivalent knot
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
Here are some of the properties of the Jones polynomial
taken from Jones' paper. For each knot K its Jones
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
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.)