Page 243 - J. C. Turner "History and Science of Knots"

234 History and Science of Knots

with violent means. The influence of combinatorial topology on knot theory
declined markedly during this period.

10. The Fifties and Sixties
Ralph Hartzler Fox (1913-1973) was a mathematician who fostered an impres-
sive mathematical environment around his person. Since Alexander's time,
Princeton University had been the great name in knot theory's geography;
and Fox's extensive publications on the subject made it even greater.
After the interruption in efforts caused by World War II, research came
to focus mainly on the knot group, its subgroups and the principal ideals of
its group ring. The way to represent a knot group by means of generators
and defining relations led Fox to discover a free differential calculus. The
ideas behind this calculus caused the Alexander polynomial to emerge as a
determinant value of a matrix in which the entries are `partial derivatives'
(in Fox's calculus) of the knot group's relators with respect to its generators.
The calculus came to be christened Fox's, and it led to the discovery of links
between hitherto unrelated other fields in mathematics. In knot theory itself,
it showed that the knot polynomial is determined by the group of the knot,
and provided a link between the combinatorial and geometric definitions of the
Alexander polynomial. On the practical side, the calculus supplied one more
method for calculating Alexander polynomials. Specifically, it became one of
the most important tools for studying knot groups defined by generators and
relations.
As a person Ralph Fox has left quite an impression. After his death
former students dedicated a 350-page book of their research papers to his
memory [100]. From his school came people like Joan Birman, whom we shall
meet later, and Lee Neuwirth working in knot groups; and Elvira Strasser
Rappaport who studied `knot-like' groups, addressing the question of which
groups are knot groups.
Many of the developments in topology during the 1950-1980 period came
to affect ideas about knots. Typifying the general development of knot theory
and its techniques is that the concept of `knot', so far treated as a polygonal
non-intersecting curve in 3-space (i.e. R3) upon which certain moves were per-
mitted became modernized to `knot' being an equivalence class of embeddings
of the unit circle Sl in S3. Topological studies had made it clear that R3
should be replaced by S3, in view of compactification properties of the lat-
ter. At the end of the fifties this led mathematicians such as Andre Haefliger
and Christopher Zeeman to elaborate upon a theme, traceable back to Emil
Artin's work, which considered mappings S" -> Sm, for which m - n = 2 [41] ,
[97]. These mappings `tied the n-dimensional unit sphere into a knot' in the
m-dimensional unit sphere. The objectives of this higher-dimensional, gener-