Page 244 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 235

alized knot theory included classification of knots with respect to isotopy. A
difficulty was that the construction of these knots could not be visualized by
simple-minded drawings of knot projections. Classification, and hence finding
invariants, had therefore to be coupled to construction methods, showing how
the invariants were realizable.
The more formal demands on smoothness of mappings brought in the
notions of tame and wild knots. A knot is tame if it is equivalent to some
polygonal knot; otherwise it is wild. The distinction was of vital importance;
the principal invariants of knot type, namely the elementary ideals and the
knot polynomials, were not necessarily defined for a wild knot. Knot theory
was largely confined to the study of polygonal knots, and it was natural to
ask what kinds of knot other than these were tame. An early theorem, and
partial answer to this question was: If a knot parametrized by arc length is
continuously differentiable, then it is tame.
There are infinite classes of wild knots, and their study forms a field of
its own within the topological theory of knots.

11. John Conway's Tangling

As we have seen above, the problem of distinguishing knot-types for given
numbers n of crossings, and tabulating them, was first tackled by the three
men Tait, Kirkman and Little, in the final fifteen years of the 19th century.
They succeeded in resolving the problem, by largely empirical methods, for
n = 3, ... ., 1and for most of the alternating knots on 11 crossings.
There the matter rested for some seventy years, until John Horton Conway
devised entirely new methods for studying knots, based on a construct called
a tangle. Essentially, a tangle is a portion of a knot-diagram from which a
number, usually four, free-end strands emanate; an example (Fig. 17) is given
below.

Fig. 17. An example of a Conway tangle