\ / \ / | | \ / \ / | | \ / | | / \ / \ | | / \ / \ | | L+ L- Lo

Then these links will be mapped to polynomials of *t* which will be
related by a relation

*A(t) f(L+) + B(t) f(L-) = C(t) f(Lo)*

where *L+* is the link where there is a positive crossing in the
distinct
neighborhood, *L-* is the link with the negative crossing, and *Lo*
is the link
without the crossing.

One may use these relations to calculate *f(L)* for any link as
follows. When one applies a skein relation to a neighborhood that contains one
crossing, one obtains one link with one crossing less, and one link with the
same number of crossings but one crossing reversed. One may thus continue
until one gets links with fewer crossings than the original, and one link where
crossings are arranged so as to render the link trivial. This is always
possible, since if a link consists of the components *K1*, *K2*,...,
*Kn* such that for *i* less than *j*, all crossings of *Ki*
are above all crossings of *Kj*, and if in each component *Ki* one
may move along the loop in such a way that overcrossings always appear before
undercrossings, the link consists of *n* unlinked circles. One may show
inductively that the polynomial of such a link is

/A(t) + B(t)\ n-1 |-------------| \ C(t) /

if one uses the convention that the polynomial of the trivial knot is 1.

For the Alexander-Conway polynomial one may show that *A(t)*=1 and
*B(t)*=-1. In 1984 **V.F.R. Jones** generalized these relations and
thus obtained the *Jones Polynomial*, which is more efficient in
distinguishing inequivalent knots than the Alexander-Conway. In 1985 however a
number of mathematicians showed independently of each other, and almost
simultaneously, that the three terms *A(t)*, *B(t)* and *C(t)*
did not have to be related to each other through some parameter *t* in
order for the skein relations to define a knot invariant. One may thus define
a two-parameter invariant *P(s,t)*, which is known as the **homflypt**
polynomial from the initials of the mathematicians that discovered it.

In addition to being more efficient than both the Alexander-Conway and the
Jones polynomial in distinguishing inequivalent knots, the homflypt
polynomial was the first known knot invariant to distinguish mirror
symmetric knots. Since the terms *A* and *B* in the skein relations
are unrelated, one obtains the polynomial of the mirror symmetric knot by
exchanging *A* with *B* and vice versa. If these polynomials are
distinct, the knot is *chiral* (inequivalent to its mirror symmetric).

To continue this discussion and see how one may develop a computer program
to calculate the homflypt polynomial, click
**here**
.

*Charilaos Aneziris, charilaos_aneziris@standardandpoors.com*

**Copyright 1996**

Educational institutions are encouraged to reproduce and distribute these materials for educational use free of charge as long as credit and notification are provided. For any other purpose except educational, such as commercial etc, use of these materials is prohibited without prior written permission.