Page 271 - J. C. Turner "History and Science of Knots"

262 History and Science of Knots

A few decades later, Carl Friedrich Gauss (1777-1855), one of the greatest
mathematicians and physicists of all time, made sketches of knots*and began to
think of their properties. It is clear from notes found in his papers after he died
that during his working life he gave much thought to the problem of capturing
the essence of knots in mathematical terms. However, to our knowledge he
published only one paper which referred to knots; this paper dealt with a
problem in the theory of electrodynamics, and it involved the linking number
of two wires winding together in space. In 1847 his student (and later colleague,
at Gottingen University) Johann Benedict Listing (1806-1902) published the
first book on Topology [11]. The book was devoted primarily to knot theory;
and we may surmise that Listing was influenced by Gauss when developing
his ideas on this virgin subject.
The story of how topological knot theories have developed since 1847 is
told in the previous chapter of this book. It is a tale which deals almost
exclusively with the hunt for topological invariants of knots; that is, with
the search for mathematically derived numbers or expressions, obtained from
studies of diagrams of knots, which enable one knot to be distinguished from
another with mathematical certainty. Knots are truly mysterious objects; it is
hard to say unambiguously what they are, and even harder to describe their
properties. Before seeking a topological invariant for them, one first has to
find a way of defining a knot, in mathematical language. Then one has to
state very carefully what one means by saying `That knot is the same as this
knot.' Indeed, it turns out that there are many different, useful ways to define
`sameness', each being with reference to a different set of properties of a knot.
In short, one has to define equivalence relations for knots. Then, and only
then, can one attempt to classify knots, and to study their properties, using
mathematical models and/or experimental methods.
The kind of knot theory we have spoken about so far is `classical' or `topo-
logical' knot theory. Since the publication [7] of the Jones polynomial invariant
in 1986, with its applications in genetics and quantum physics, development
of this theory has become a burgeoning industry in Mathematical Science. We
submit, however, that it is only one part of the Science of Knots.
There is a great deal to learn about knots which does not hinge upon
purely topological notions. Indeed, we assert that the current topological knot
theory says nothing at all of interest or use to "the craftsman who fashions a
braid, a net or some knots" (see the above Vandermonde quotation).
The craft of braiding is perhaps as old as Humankind. It is highly likely
that Eve discovered how to braid her own tresses in the Garden of Eden. More-
over, it is still, today, a highly useful and decorative craft practised extensively
in virtually every society. Is it not strange, then, that until recently no useful

*A page from his notebooks, bearing the date 1794, is shown in the Preface to this book.