polynomial information

Combinatorial definitions of polynomial invariants

In the literature, several different conventions are used in the skein theoretic definitions of the polynomial invariants. We adopt the following for Jones, HOMFLY(PT), and Kauffman polynomials: denote the unknot by O and

positive crossing       negative crossing       smoothing
L+       L-       L0
  Variable(s)   Defining skein relation
Jones polynomial V(L) t V(O) = 1,   t-1 V(L+) - t V(L-) = (t1/2-t-1/2) V(L0)
HOMFLY(PT) polynomial P(L) v, z P(O) = 1,   v-1 P(L+) - v P(L-) = z P(L0)
Kauffman polynomial F(L) a, z F(L)=a-w(L) Λ(D) where w(L) is the writhe and
Λ(O) = 1,   Λ(positive crossing) + Λ(negative crossing) = z (Λ(smoothing) + Λ(smoothing)),
Λ(kink) = a Λ(kink removed), Λ(kink") = a-1 Λ(kink removed)

The polynomials are related as follows:   V(t)  =  P(v←t, z←t1/2-t-1/2)
 =  F(a←-t3/4, z←t1/4+t-1/4)

Our conventions agree with those in [1].

The following table compares other conventions with ours. In most cases other conventions are either identical with ours, or those for the mirror image. "Same" and "Mirror" designates the former and the latter, respectively.

Jones'
Bull. AMS paper [2]		 F-Y-H-L-M-O's
Bull. AMS paper [3]		 Kauffman's
Trans. AMS paper [4]		 Bar Natan's
KnotTheory [5]		 Wikipedia [6]
Jones
polynomial		 Mirror (t-1 is used in place of t) Same Same
HOMFLY(PT)
polynomial		 ℓP(L+)+ℓ−1P(L-)=−mP(L0)
xP(L+)+y−1P(L-)=−zP(L0) 		Mirror (a-1 is used in place of v) Mirror (α-1 is used in place of v)
Kauffman
polynomial		 Same Same Same

References

[1] Kawauchi, A., A survey of knot theory, Birkhauser Verlag, Basel, 1996. xxii+420 pp.

[2] Jones, V., A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103--111.

[3] Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett, K.; Ocneanu, A., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239-246.

[4] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417-471.

[5] Bar-Natan, D., The Mathematica Package KnotTheory`, http://katlas.math.toronto.edu/wiki/The_Mathematica_Package_KnotTheory%60

[6] Wikipedia, The Free Encyclopedia, retrived August 19, 2009.