Perhaps one of the hardest problems in knot theory is determining whether two knots are isotopic. One method of solving this problem is the polynomial invariant.
The ideal polynomial invariant would yield a unique polynomial for each isotopic class that is consistant through each projection of the class. Most of the polynomial invariants described on this site are based on skein relations. The Alexander Polynomial is the exception; it was originally designed to use matrices. However, another mathematician, known as John H. Conway, found a way to calculate the Alexander Polynomial through a skein relation.
The earliest polynomial invariant was the Alexander Polynomial, named after its creator, James Waddell Alexander II. The Alexander Polynomial was based on the concept that two knots can be distinguished as different through linear color tests. Unfortunately, it did not distinguish between a knot and its mirror image, and thus assumed that all knots were achiral. The Alexander Polynomial, published in 1928, remained the only knot polynomial invariant for over five decades.
In 1984, Vaughan F. Jones created a new polynomial invariant, named the Jones Polynomial. The amazing part, however, was that he "discovered" the polynomial when he noticed that some equations in knot theory looked similar to those found in operator algebras, which are associated with quantum mechanics. The Jones Polynomial was also the first polynomial invariant to use skein relations.
In 1987, L. H. Kauffman discovered his own polynomial derived from the Bracket Polynomial. The Bracket Polynomial recognizes the regularly isotopic projections of an isotopic class as the members of the same isotopic class, but the Bracket Polynomial is not an isotopy invariant because it is not invariant under the first Reidemeister Move. Kauffman, however, found an expression to multiply by the Bracket Polynomial to obtain an isotopically invariant polynomial, and another expression to multiply by the Kauffman Polynomial to obtain the Jones Polynomial.
The publication of the Jones Polynomial excited the mathematical community to the point that new polynomial invariants were being created at a stupendous rate. One of the objectives of the time was to find a polynomial that generalized the Alexander and Jones polynomial. The Oriented Polynomial, or the HOMFLY Polynomial was a successful solution, published by several groups of mathematicians simultaneously. The paper was published under the names of Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter. The HOMFLY Polynomial uses a skein relation, like the Jones Polynomial, but the new polynomial uses two variables, unlike the Alexander and Jones Polynomials.
The polynomial invariant, probably the most important aspect of knot classification, still has many questions surrounding it. What kind of information is being expressed in the polynomials discovered so far? Is there a polynomial invariant simple enough to calculate, no matter how many crossings are involved? Is there a polynomial invariant that is a generalization of the HOMFLY Polynomial? Does an invariant that is ideal for both knots and links exist? Only the right application of the right knowledge will reveal the answers.
The currently published polynomial invariants include:
The Alexander Polynomial
The Bracket and Kauffman Polynomials
The Oriented/HOMFLY Polynomial
The Jones Polynomial