nLab 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: $x \cdot P(L_+) + y \cdot P(L_-) + z \cdot P(L_0) = 0$.
   $$\begin{array}{ccc} \begin{svg}[[!include SVG skein positive crossing]]\end{svg} & \begin{svg}[[!include SVG skein negative crossing]]\end{svg} & \begin{svg}[[!include SVG skein no crossing]]\end{svg} \\ L_+ & L_- & L_0 \end{array}$$

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. $l = i$, $m = i (q^{-1/2} - q^{1/2})$

Related entries

• Shum's theorem

References

See the wikipedia page for the origin of the name.

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

• N. D. Gilbert and T. Porter, Knots and Surfaces, Oxford U.P., 1994.

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
• Hugh Morton, Peter Samuelson, The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra, arxiv/1410.0859

category: knot theory