Page 226 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 217

between different knots; and he proposed one, to be computed from a knot
diagram in terms of his crossing-type symbols.

A

65+64+63+262
1 A5 +A4+A3+ 2A2

Fig. 8. Tait's two different knots with the same complexions-symbol
Although his Complexions-Symbol had too many serious defects for it
to be of much use as a knot invariant, it posed a challenge to other workers
coming into the field: namely, to find good invariants. As such, it was perhaps
Listing's greatest contribution to Knot Theory. The major quest in twentieth
century knot research has been for ever stronger invariants.
Before describing this further work on invariants, however, we must men-
tion other work that was done in the final thirty years of the nineteenth cen-
tury. In particular, we shall describe the monumental achievements of P. G.
Tait and his collaborators; and they demand a section of their own. Passing
mention will first be made of several other items related to knot research.
H. Weith, in his inaugural dissertation which elaborated on Listing's work,
elegantly showed that there is an infinite number of different knot forms [93].
About that time, there was a curious connection made between knot
theory and psychic research. The mathematician Felix Klein appears to have
observed that no ordinary knot can exist knotted in a space of four dimensions;
one can always use the extra dimension in order to untie it, without, of course,
cutting the string [56], [83].
This proposition would be experimentally testable if only we had access
to a fourth dimension. Several very reputable and distinguished scientists,
including J. C. F. Zollner from Leipzig, began conducting knot experiments
which involved psychic mediums [98]. If the mediums were able to untie closed
knots, without cutting the string, it would be a reasonable conclusion that they
had some kind of access to a fourth dimension. In the account of Zollner's
investigations, a record is made that this kind of experiment was successfully
carried out in December 1877, by Mr. Slade, an American medium.