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.
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.