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 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 }+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.

[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).