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A POLYNOMIAL INVARIANT OF ORIENTED LINKS W. B. R. LICKORISHand KENNETH C. MILLET-I
 

Summary: A POLYNOMIAL INVARIANT OF ORIENTED LINKS
W. B. R. LICKORISHand KENNETH C. MILLET-I
(Receiced 15 February 1985: recked 21 July 1985)
$0. ISTRODUCTION
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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics