THE MATHEMATICAL THEORY OF KNOTS In the sense of analysis situs (q.v.), a knot is any simple, closed curve in three dimensional space, a curve which never passes more than once through the same point of space and which may be thought of as starting at a point P and ultimately re turning to P. Two knots are said to be of the same class if there exists a continuous deformation of space carrying one knot into the position initially occupied by the other. If a knot is of the same class as a circle it is said to be unknotted, for it is con venient to treat an unknotted curve as a special case of a knot, just as it is convenient to treat a straight line as a special case of a curve. No general method has, as yet, been found for telling when two arbitrarily given knots are of the same class, or even for telling whether or not a curve is unknotted.
The structure of a perfectly general mathematical curve can be so complicated that, in most discussions on knots, attention is confined to curves of a reasonable degree of regularity. These last may be thought of, without serious error, as physical threads, arbitrarily twisted and tangled, and closed by having their two ends sealed together. From this physical point of view, the problem of classifying knots reduces to the problem of telling under what conditions it is possible, by a process of bending, stretching and shrinking, to deform one of two arbitrarily given threads into a thread of the same shape as the other.
The knot problem seems to have been originally proposed and studied by J. B. Listing in one of the earliest works ever pub lished on the subject of analysis situs. It was again attacked, in dependently, by P. G. Tait at a time when attempts were being made to interpret material atoms as vortex lines in the ether and to account for the differences in the various chemical ele ments by ascribing to the atoms of each a characteristic type of knottedness.
When we fix our eyes on a knotted thread we notice a certain number of apparent crossing points where, from our point of ob servation, one branch of the thread is seen to pass in front of another. This number of apparent crossing points may be varied either by deforming the knot or by shifting our point of ob servation, but if we start with any particular knot there is a certain minimal number k of crossing points which may be at tained by suitably deforming the knot, but below which it is impossible to pass. The number k is a knot invariant, called the
number of irreducible crossings of the knot. The simplest pos sible knot is one of zero crossings: that is to say, one which is unknotted. Next in order of simplicity come the two trefoil knots (fig. 2-Y) with three crossings each. These knots are not of the same class. The problem of effectively determining the num ber of irreducible crossings of an arbitrarily given knot is still unsolved.
A classification of the more elementary knots according to the number of their irreducible crossings was begun by Tait and later extended by Kirkman, Little and others, so that tables now exist which exhibit all possible kinds of knots of II or less crossings. These tables are not altogether satisfactory, how ever, for they were arrived at by essentially empirical methods. It has never actually been proved, for instance, that no two knots listed in the tables as of distinct classes can be transformed into one another.
The first effectively calculable knot invariants were discovered by J. W. Alexander and, later, independently by K. Reidemeister. With the aid of these invariants, Alexander and Briggs have shown that all knots of eight or less crossings listed as distinct by Tait actually do belong to different classes. However, the new invariants appear to be insufficient to solve the knot problem completely, for they fail to distinguish between certain knots of nine crossings which give every indication of belonging to different, classes. It is even doubtful whether they are sufficient to de termine whether or not an arbitrary thread is unknotted. A neat classification of braids has been made by Artin.