nLab HOMFLY-PT polynomial

The HOMFLY-PT Polynomial

The HOMFLY-PT Polynomial


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.


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. lP(L +)+l 1P(L )+mP(L 0)=0l \cdot P(L_+) + l^{-1} \cdot P(L_-) + m \cdot P(L_0) = 0.
    2. aP(L +)a 1P(L )=zP(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 1P(L +)αP(L )=zP(L 0)\alpha^{-1} \cdot P(L_+) - \alpha \cdot P(L_-) = z \cdot P(L_0).
    4. Using three variables: xP(L +)+yP(L )+zP(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.


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


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

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

Last revised on October 7, 2017 at 06:52:15. See the history of this page for a list of all contributions to it.