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

Jones Polynomial

The Jones polynomial, V(t), emerged from a study of finite dimensional von Neumann algebras. It is an invariant of oriented 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 +t1/4 .

If K* denotes the mirror image of a knot K, then VK*(t) = VK(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.


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

[2] Kauffman, L. H., "State models and the Jones polynomial," Topology, 26 (1987), 395-407.

[3] Lickorish, W. B. R., An introduction to knot theory. New York: Springer (1997).

[4] Murasugi, K., Knot theory and its applications. Boston, Massachusetts: Birkhauser Boston (1996).