Page 232 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 223

much aware of the difficulties which symmetries in mirroring brought along.
He introduced a Scottish verb, to flype, to denote an operation which can be
carried out on certain portions of knots or their diagrams. In fact, one of his
foremost contributions was to introduce nomenclature of this kind into knot
studies [58].
In order to develop the subject rigorously, however, he needed to discover
some form of knot invariant, which would help him to distinguish and identify
knot types. He gave no formal proofs that any of his methods actually came
to define or to implement one.
The main underlying problem which confronted Tait and his co-workers
was deciding when two knotted structures were isotopic, i.e. telling whether
either of them could be deformed, by a continuous transformation, into the
other. Two knots or links are said to be the same, or isotopy equivalent, if they
can be made to look exactly alike by pushing and pulling, but not cutting,
the string(s) in which they are realized. This problem of isotopy became
established as the central problem in knot theory, and it became known as the
Knot Problem. It was not to be dealt with satisfactorily until the advent of
algebraic topology.

Fig. 10. Knots from Mary Haseman's dissertation
Some thirty years after Tait's endeavours, Mary Gertrude Haseman tack-
led amphicheiral knots of 12-fold knottiness (Fig. 10). The results of her brave
expedition into the then uncharted regions of 12-crossing knot-projections
are presented in the charming dissertation [43], which gives a census of am-
phicheirals in that class of knottiness.

6. The Beginning of the 20th Century

By 1900 there were almost-complete tables available listing knots of up
to 11-fold crossings. They represented the fruits of the arduous labor by Tait