2.2  Polynomial invariants 

The real recognition of KLs became possible after the introduction of polynomial invariants. The first of them, Alexander polynomial, was used by Alexander and Briggs (1927) to prove that knots with at most nine crossings claimed to be distinct in knot tables were actually distinct. K. Reidemeister completed the rigorous classification of knots with up to nine crossings in his book Knotentheorie published in 1932. For more than 40 years, Alexander polynomial remained only polynomial invariant able to distinguish KLs. The tangle approach was introduced by J. Conway in 1967, together with a new polynomial invariant, the Conway polynomial, based on a skein relation. J.W. Alexander knew about the skein relation, but J. Conway first
proved in 1967 that it can be used for an axiomatic definition of the polynomial (Conway, 1970). A modification of the skein relation resulted in the HOMFLYPT polynomial (1985). Probably the most famous polynomials are those of Jones and Kauffman. These made it possible to establish connections between knot theory and other branches of mathematics (the algebra of operators, braid theory), and especially physics (statistical models and quantum groups). Together with all these important achievements, there is one disappointing fact: every polynomial invariant sometimes fails, meaning that two (or more) different KLs may have equal polynomials. Even worse: some KLs that are really knotted are impossible to distinguish from the unknot by certain polynomial invariants. For example, there is an infinite number of non-alternating knots with Alexander polynomial equal to one, and an infinite number of non-alternating links with a trivial Jones polynomial. An infinite series of non-trivial non-alternating 2-component links: 

9*5 1 2:-1.-1.2.-1.-1:-5 -1 -2,
9*5 4 1 2:-1.-1.2.-1.-1:-5 -1 -4 -1 -2,
9*5 1 4 1 4 1 2:-1.-1.2.-1.-1:-5 -1 -4 -1 -4 -1 -2, etc.,

and 4-component links

16370*.-2:-5 1 2.-1:.-1.-1:.-1.5 1 2:-2,
16370*.-2:-5 1 4 1 2.-1:.-1.-1:.-1.5 1 4 1 2:-2,
16370*.-2:-5 1 4 1 4 1 2.-1:.-1.-1:.-1.5 1 4 1 4 1 2:-2,etc.,

where the links with trivial Jones polynomial are denoted by red color was recently discovered by S. Eliahou, L. Kauffman and M. Thistlethwaite (2003). The other links of these two series have trivial Jones polynomial up to some factor.