Page 224 - J. C. Turner - History and Science of Knots

214 History and Science of Knots

him to introduce the concept of handedness for an oriented crossing (Fig. 6).
To each type he assigned a certain symbol distribution consisting of 8s and
As. This is illustrated below. The orientation becomes insignificant after the
regions have been assigned a type.
From some simple experiments with two- and three-stranded braids and
their closures, he came to consider the possibility of listing and classifying all
knot projections having fewer than seven crossing points.
Listing was the first to persist in representing knots as knotted circles, and
obtaining diagrams by projecting these onto a plane. By attaching symbols to
the crossings in a diagram, to indicate their types, and considering the resulting
symbol distribution in each of the diagram's regions, he was able to propose an
`invariant' for a knot. In general an invariant is a mathematical expression (it
may be just a number) which carries information about a system, and whose
values do not change when the system is transformed in some defined way.
An invariant in knot theory is generally an expression derived from a knot
diagram which depends solely on the knot or link under consideration, in any
of its forms, and not on any particular picture of them. The easiest invariant
to visualize, but one which is not very useful for distinguishing between knots,
is the number of components in an n-link L. By definition it equals n, and
remains so whatever continuous deformations L is subjected to.
Invariants are useful aids in the classification of knots for the following
reason. Suppose we compute the value of a particular invariant from two knot
diagrams, and obtain two different values. Then we can conclude that the
two knot forms from which the diagrams were obtained are different knots.
However, the converse is not true; diagrams having the same invariant value
may or may not come from the same knot. A perfect knot invariant, which
always takes the same value for any particular knot, and a different value for
any other knot, has yet to be discovered.
Listing concocted his invariant as follows. He called a region of a knot
diagram monotypical if all the angles on the region's boundary had been as-
signed the same type-symbol (that is, were all S or all A); in which case, the
region was itself given the same type-symbol. If a mixture of 6s and As had
occurred, he called the region amphitypical. In this way, Listing typified each
region, including the unbounded one (which he called the amplexum).
He defined a diagram to be in reduced form if it had a minimal number of
crossings, over all possible diagrams obtainable from the knot. He knew that if
one or more regions were amphitypical in a diagram, then the diagram would
not necessarily be in reduced form; but he gave no methods for reducing the
numbers of crossings in such diagrams.
He proposed an invariant for knots which have a monotypical diagram in
reduced form; and he gave it the name Complexions-Symbol. Later, we shall
give an example which shows that it is not a true invariant.