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 nonalternating
knots with Alexander polynomial equal to one, and an infinite number of
nonalternating links with a trivial Jones polynomial. An infinite series
of nontrivial nonalternating 2component 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 4component links:
16370^{*}.2:3.1:.1.1:.1.3:2, 

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.
