HOMFLY-PT polynomial in nLab

The HOMFLY-PT Polynomial

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. PP is an isotopy invariant (thus, unchanged by Reidemeister moves).

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

3. Let L +L_+, L −L_-, and L 0L_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⋅P(L +)+l −1⋅P(L −)+m⋅P(L 0)=0l \cdot P(L_+) + l^{-1} \cdot P(L_-) + m \cdot P(L_0) = 0.
   2. a⋅P(L +)−a −1⋅P(L −)=z⋅P(L 0)a \cdot P(L_+) - a^{-1} \cdot P(L_-) = z \cdot P(L_0). (Sometimes ν\nu is used instead of aa)
   3. α −1⋅P(L +)−α⋅P(L −)=z⋅P(L 0)\alpha^{-1} \cdot P(L_+) - \alpha \cdot P(L_-) = z \cdot P(L_0).
   4. Using three variables: x⋅P(L +)+y⋅P(L −)+z⋅P(L 0)=0x \cdot P(L_+) + y \cdot P(L_-) + z \cdot P(L_0) = 0.

   [[!include SVG skein positive crossing]] [[!include SVG skein negative crossing]] [[!include SVG skein no crossing]] L + L − L 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=α=a −1y = \alpha = a^{-1}
2. x=−α −1=−ax = - \alpha^{-1} = -a
3. a=−ila = - i l, l=ial = i a
4. z=imz = i m, m=−izm = - 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 −1a = q^{-1} and z=q 1/2−q −1/2z = q^{1/2} - q^{-1/2}
  2. α=q\alpha = q and z=q 1/2−q −1/2z = q^{1/2} - q^{-1/2}
  3. l=iq −1l = i q^{-1} and m=i(q −1/2−q 1/2)m = i (q^{-1/2} - q^{1/2})

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

  1. a=1a = 1
  2. α=1\alpha = 1
  3. l=il = i, m=−izm = -i z

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

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

• 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