Page 239 - J. C. Turner "History and Science of Knots"
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230 History and Science of Knots
revealing its shortcomings, the group of a knot was still a powerful invariant.
And in the late 1960s the role of the peripheral system was finally clarified;
it was shown to be a complete invariant. This demonstration resulted from
Waldhausen's work on irreducible, sufficiently large, 3-manifolds, which in turn
was based on earlier ideas by Haken [42], [44].
The knot group, even though it was an unwieldy mathematical object,
formed the basis for much further research on knots. The new approach via
knot groups effectively brought the unknot U into the picture (it was a knot
that had been ignored or not taken seriously by early researchers). This knot,
a kind of `limit' in the class of knots, now became an important one in the
knot tables, because its group turned out to be a special one, namely the
infinite cyclic group with one generator, which is isomorphic to the group of
integers under addition. The proof of this was settled in 1956 by the great
mathematician Papakyriakopoulos, who also proved that the group of a knot
determines the homotopy type of its complement.
Equivalent knots which are projected into distinct diagrams can yield dif-
ferent presentations of their knot group. These presentations must, as theory
tells us, relate to isomorphic groups. However, there is no general algorithm
which will enable us to decide whether two given representations relate to two
isomorphic groups. It is known that no such general algorithm is possible.
Nevertheless, in working to resolve particular cases, the main efforts in knot
research came to concentrate on the problem of finding reduced presentations
of knot-groups; in the process, the problem of knot-equivalence was cast into
an (algebraic) word-problem mould. The main question became: When are
two presentations of knot groups equivalent? The complexity of this problem
(which is, as already noted, generally unsolvable) led to a quest for simpler
invariants, ones more tractable than the knot-group.
This research direction began with a discovery by J. W. Alexander; in
1928 he `launched' the knot polynomial which was later named after him. It
was a totally new idea. He described a method for associating with each knot
a polynomial, such that if one form of a knot can be topologically transformed
into another form, both will have the same associated polynomial; it quickly
proved to be an especially powerful tool in knot theory. For example, the
polynomial was able to distinguish 76 knots out of the first 84 in the knot-
tables; they were found to have unique Alexander polynomials.
Alexander first obtained his polynomial of a knot K by labelling the
regions in the plane bounded by an oriented knot-diagram of K having n
crossings. By noting the types of crossing around the knot, in relation to
the arc labels, he extracted a certain n x n matrix (now called the Alexander
matrix). All of the elements in an Alexander matrix are either 0, or -1, or
t, or 1 - t, where t is a dummy variable or parameter. By removing the
last row, and the last column, of the matrix, and taking the determinant of