Page 240 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 231

the remaining matrix, a polynomial in t is obtained. This is known as the
Alexander polynomial of the knot. We may denote it by AK(t), or simply AK.
Alexander was able to show that AK(t) is an invariant for the knot K (see
[6] for full details). In fact, Alexander presented a sequence of polynomials,
{An(t)}, with n = 1, 2, 3, ..., all invariants of the knot K. The first one (case
n = 1) is the one known as the Alexander polynomial.
Why associate a polynomial with a knot diagram? The schemes and
partitions which Tait, Kirkman and Little had worked with were unwieldy.
Listing's complexions-symbols were not quite what was needed to yield an
unambiguous invariant. But why Alexander's polynomial worked the way it
did was not clear at the outset. Alexander himself suspected that it was some
kind of shorthand for homology groups. A rather reasonable hunch, as later
work placed it on a sound homological base.
Alexander's polynomial proved to be a fairly powerful invariant of isotopy
in knots. Differently deformed versions of the same knot yield the same poly-
nomial AK. The following comments illustrate a few attractive aspects of the
polynomial's behaviour.
Given two prime knots with respective Alexander polynomials, the Alexan-
der polynomial for their knot-composition is given by the multiplication of the
two original polynomials. Another aspect almost amounts to a pun: for an
alternating knot, AK has coefficients of alternating sign [70]. These, and other
more refined pleasant properties, made it knot theory's main tool for almost
half a century. The Alexander polynomial's weak points are that it always
takes the same value for a knot and its mirror image; and that its power to
distinguish between knots terminates for certain example pairs and classes of
knots with more than 9 crossings. In 1934, classes of non-trivial knots with
trivial Alexander polynomial were discovered [78].

K_ Ko K+

Fig. 15. Set of Alexander crossings
Alexander also discovered a relationship between the polynomials of three
oriented knots whose diagrams are identical except within a neighborhood of
one fixed crossing where they are as shown in Fig. 15, [6, p. 301]. For further