
PROFESSOR TAIT ON KNOTS. 149° 
§3. If we affix letters to the various crossings, and, going continuously 
round the curve, write down the name of each crossing in the order in which 
we reach it, we have, as will be proved later, the means of drawing without 
ambiguity the projection of the knot. If, in addition, we are told whether we 
passed over or under on each occasion of reaching a crossing we can, again 
without any ambiguity, construct the knot in wire or cord. Passing over is, in 
what follows, indicated by a + subscribed to the letter dénoting the crossing— 
passing under bya —. Any specification which includes these two pieces 
of information is necessarily fully descriptive of the knot ; and when it is given 
in the particular form now to be explained we shall call it the Scheme. 
If in accordance with § 1 we make the crossings alternately over and 
under, it is obvious that the odd places and even places of the scheme will each 
contain all the crossings. As the choice of letters is at our disposal, we may 
therefore call the crossings in the odd places A, B, C, &c., in alphabetical 
_ order, starting from any crossing we please, and going round the knotted wire 
in any of the four possible ways, 7.¢., starting from any crossing by any of the 
four paths which lead from it, put the successive letters at the first, third, fifth, 
&c., crossings as we meet them. Then it is obvious that the essential character of 
the projected knot must depend only upon the way in which the letters are arranged 
in the even places of the scheme. Of course, the nature and reducibility (7.¢., capa- 
_ bility of being simplified by the removal of nugatory crossings) of the knot itself 
_ depend also upon the subscribed signs. [In general there will be four different 
_ schemes for any one knot, but in the simpler cases these are often identical two 
| and two, sometimes all four. ] 
§4. Here we may remark that it is obvious that when the crossings are 
| alternately + and — no reduction is possible, unless there be essentially nuga- 
tory crossings, as explained in §2. For the only way of getting rid of such 
alternations of + and — along the same cord is by wntwisting ; and this process, 
| except in the essentially nugatory cases, gets rid of a crossing at one place only 
| by introducing it at another. It will be seen later that this process may in certain 
' cases be employed to change the scheme of a knot, and thus to show that in 
| these cases there may be more than four different schemes representing the same 
_ knot; though, as we have already seen, a scheme is perfectly definite as to the 
knot it represents. Hence, in the first part of our work, we shall suppose that 
| the crossings are taken alternately + and —, so that no reduction is possible. 
But it will afterwards be shown that, even when all essentially nugatory cross- 
| ings are removed, it is not always necessary to have the regular alternation of 
/ + and — in order that the knot may not be farther reducible. It is easy to 
| see a reason for this, if we think of a knot made up of different knots on the 
| same string, whether separate from one another or linked together. For the 
irreducibility of each separate knot depends only upon the alternations of + and 
VOL. XXVIII. PART I. . Pat) 
