A polynomial invariant of oriented links

Abstract

THE THEORY of classical knots and links of simple closed curves in the 3-dimensional sphere has, for very many years, occupied a pre-eminent position in the theory of low dimensional manifolds. It has been a motivation, an inspiration and a basis for copious examples. Knots have, in theory, been classified by Haken [lo] but the classification is by means of an algorithm that is too complex to use in practice. Thus one is led to seek simple invariants for knots which will distinguish large classes of specific examples. A knot (or link) invariant is a function from the isotopy classes of knots to some algebraic structure. Perhaps the most famous invariant of a knot K is the Alexander polynomial, AK(t), a Laurent polynomial in the variable t. This was introduced by .Alexander [l] who explained how to calculate the polynomial by taking the determinant of a matrix associated with a presentation (or picture) of the knot given by a suitably chosen projection of its spatial position to a plane. The Alexander polynomial is still remarkably efficacious in distinguishing specific knots and, being readily calculable by computer, is employed by modern compilers of prime knot tables as the fundamental invariant to distinguish between examples (see Thistlethwaite [20]). Of course other invariants, notably signatures and the sophisticated Casson-Gordon ‘invariants’ are now available as well. Nevertheless, AK(t) is still a most useful invariant. Much has been written on this polynomial during the last sixty years; and a modern definition might be as follows. If X is S3 - K, where K is now an oriented link, let X, be the covering of X corresponding to the kernel of the homomorphism xi(X) + H,(X) + i2 that sends meridians (with prefered orientation) to 1. Thus X, is acted upon by the infinitecyclicgroup, which is to be considered as a multiplicative group with generator denoted by t, acting as the deck transformations of the covering space. Then H,(X,) is a finitely generated module over the ring Z[t, t- ‘1, its order ideal is principal and AK(f) is a generator of that ideal. This defines AK(t) uniquely up to multiplication by an element of the form + tin, elements of this form being the units of the ring.