# The HOMFLY-PT Polynomial

## Idea

The HOMFLY-PT polynomial is a knot and link invariant. Confusingly, there are several variants depending on exactly which relationships are used to define it. All are related by simple substitutions.

## Definition

To compute the HOMFLY-PT polynomial, one starts from an oriented link diagram? and uses the following rules:

1. $P$ is an isotopy invariant (thus, unchanged by Reidemeister moves).

2. $P(\text{unknot}) = 1$

3. Let $L_+$, $L_-$, and $L_0$ be links which are the same except for one part where they differ according to the diagrams below. Then, depending on the choice of variables:

1. $l \cdot P(L_+) + l^{-1} \cdot P(L_-) + m \cdot P(L_0) = 0$.
2. $a \cdot P(L_+) - a^{-1} \cdot P(L_-) = z \cdot P(L_0)$. (Sometimes $\nu$ is used instead of $a$)
3. $\alpha^{-1} \cdot P(L_+) - \alpha \cdot P(L_-) = z \cdot P(L_0)$.
4. Using three variables: $x \cdot P(L_+) + y \cdot P(L_-) + z \cdot P(L_0) = 0$.
$\begin{array}{ccc} \begin{svg}\end{svg} & \begin{svg}\end{svg} & \begin{svg}\end{svg} \\ L_+ & L_- & L_0 \end{array}$

From the rules, one can read off the relationships between the different formulations:

1. $y = \alpha = a^{-1}$
2. $x = - \alpha^{-1} = -a$
3. $a = - i l$, $l = i a$
4. $z = i m$, $m = - i z$.

## Properties

The HOMFLY polynomial generalises both the Jones polynomial and the Alexander polynomial (equivalently, the Conway polynomial?).

• To get the Jones polynomial, make one of the following substitutions:

1. $a = q^{-1}$ and $z = q^{1/2} - q^{-1/2}$
2. $\alpha = q$ and $z = q^{1/2} - q^{-1/2}$
3. $l = i q^{-1}$ and $m = i (q^{-1/2} - q^{1/2})$
• To get the Conway polynomial?, make one of the following substitutions:

1. $a = 1$
2. $\alpha = 1$
3. $l = i$, $m = -i z$
• To get the Alexander polynomial, make one of the following substitutions:

1. $a = 1$, $z = q^{1/2} - q^{-1/2}$
2. $\alpha = 1$, $z = q^{1/2} - q^{-1/2}$
3. $l = i$, $m = i (q^{-1/2} - q^{1/2})$

## References

Some fairly elementary discussion of the HOMFLY polynomial is given introductory texts such as

The original work was published as

• P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, and A. Ocneanu. (1985). A New Polynomial Invariant of Knots and Links Bulletin of the American Mathematical Society 12 (2): 239–246.

More recent work includes:

• A.Mironov, A.Morozov, An.Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, arxiv/1112.5754

category: knot theory

Revised on December 1, 2012 17:12:38 by Tim Porter (95.147.237.68)