# 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   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( ) + Λ( ) = z (Λ( ) + Λ( )), Λ( ) = a Λ( ), Λ( ) = a-1 Λ( )

 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 .

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  F-Y-H-L-M-O's Bull. AMS paper  Kauffman's Trans. AMS paper  Bar Natan's KnotTheory  Wikipedia  Jonespolynomial 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) Kauffmanpolynomial Same Same Same

### References

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

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

 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.

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

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

 Wikipedia, The Free Encyclopedia, retrived August 19, 2009.