The HOMFLYPT Polynomial

While in the 1920's Alexander was able to distinguish inequivalent knots by calculating the determinant of a linear homogeneous system, in the 1960's Conway showed that there is an easier way to obtain the Alexander polynomial. Instead of obtaining the system and calculating its determinant, one could proceed inductively by using the skein relations. Namely, let three regular projections of links are everywhere the same except in one neighborhood, and if in that neighborhood they look like
\   /           \   /           |    |
 \ /		 \ /            |    |
  \		  /             |    |     
 / \		 / \            |    |
/   \		/   \           |    |

  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,

Copyright 1996


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