The Jones Polynomial

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.