Page 248 - J. C. Turner "History and Science of Knots"

Topological Knot Theory 239

Figure 18 demonstrates the above process, producing the twist spectrum
for the 41 knot (Listing's). The final twists, with their orientations and their
corresponding polynomial terms, are shown on the right of the diagram.
An interesting connection between the Alexander polynomial A(t) of a
knot, and the twist spectrum T(u), is that the so-called determinant of the
knot, given by la(-I)/, is equal to the torsion coeficient value T(1). Also like
the Alexander polynomial, the twist spectrum of a composition of two knots is
equal to the product of the twist spectra of the two knots. Further, this time
like the Jones' polynomial, TK*(u) = TK(uP1) if I{* and I{ are mirror images.
So, for example, the trefoil and its mirror image are distinguished, since their
twist spectra coefficients-vectors are (I,(), 1,l) and (1,1,Q, 1).
Very soon after the twist spectrum was discovered, Vaughan Jones' great
knot polynomial discovery was announced [43]. As we shall see below, this
triggered an explosion of discoveries of polynomial invariants, and markedly
changed the face of topological knot theory, both pure and applied. Before
going on to describe these developments, it is necessary for us to review the
history and achievements of braid theory.

12. Researches in Braid Theory

The beginning of the 1920s witnessed an impasse in the theory of knots.
With the omnipotence of the knot group fatally punctured, and presentations
of knot groups stuck in generally unsolvable word problems, it was not strange
that knot theorists should seek new ways for achieving progress. The problems
of those days attracted some of the most prominent algebraists and topologists.
Minds like Seifert's pursued further research via Riemannian manifolds, while
the actions of others appeared more desperate. Reasoning that knots consist
of bits of knotted patterns, they broke them into smaller pieces which fulfilled
certain conditions and called these objects braids. Braids were not a new idea
when they entered the scene in the 1920s. Listing and Tait had already studied
procedures which generated simple knots after plaiting samples with two and
three strands. On the other hand the breaking up was something entirely new.
Emil Artin (1898-1962), with the help of Otto Schreier, formalized the ideas
and provided tools to carry on the halted quest. His landmark paper on them
[9], Theorie der Zopfe, appeared in 1925.
Basically, he provided an entirely new algebraic environment for knot
studies by introducing the so-called (algebraic) braid group, denoted by B,.
This is the set of all braids on n strings satisfying certain conditions, together
with a binary operation which consists of the simple process of concatenation
of two braids, joining the lower ends of one to the upper ends of the other.
Examples of 3- and Cstring braids appear in Fig. 19.
The geometric picture of a braid in R3 is easy to envisage. Consider n