Please see the polynomial description page for up-to-date descriptions of the conventions used in KnotInfo for the Jones, Homfly, and Kauffman polynomials.

The Jones polynomial, V(t), emerged from a study of finite dimensional von Neumann algebras. It is an invariant of oreinted knots and links.

Shortly after its formulation by Jones, Kauffman gave a combinatorial definition using the bracket polynomial. In fact, the Jones polynomial can be obtained from the Kauffman bracket polynomial by evaluating at t ^{-1/4}. It can also be obtained from the Kauffman polynomial by substituting a= -t^{ -3/4 } and z= t^{ -1/4 }+t^{1/4 }.

If K_{*} denotes the mirror image of a knot K, then V_{K*}(t) = V_{K}(t^{-1}). Thus the Jones polynomial can sometimes distinguish a knot from its mirror image and so is distinct from the Alexander polynomial. However, both are 1 variable specializations of the HOMFLY polynomial.

**References**

V. F. R. Jones, *A new knot polynomial and von Neumann algebras,* Bull. Amer. Math. Soc., **33** (1986), 219-225.

L. H. Kauffman, *State models and the Jones polynomial,* Topology. **26** (1987), 395-407.

W. B. R. Lickorish, *An introduction to knot theory*. New York : Springer, c1997.

K. Murasugi, *Knot theory and its applications*. Boston, Massachusetts : Birkhauser Boston, 1996.