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
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