282 REV. T. P. KIRKMAN ON THE ENUMERATION, DESCRIPTION, AND 



rested attention of the student, has unanimity been even so far secured. In 

 what Listing and Tait have contributed to this theory, there is an affirmation 

 of identity, or of equivalence, denoted by the symbol = , between two knots of 

 seven crossings which, by the above definition of sameness, are as unlike as 

 they can be. This, of course, is due to a mere difference in definitions. The 

 right definition in the view of Listing and Tait I find not easy to seize, and 1 

 cannot work on reticulations between equivalence and identity, nor pause to 

 consider the deformations of a knot A into an equivalent knot B that can be 

 effected by twisting the tape of A. I content myself by exhausting the forms 

 that differ according to my definition; and I leave to a more competent hand 

 the reductions to be made by twisting. 



6. The reader will judge for himself whether the number of different unifilar 

 knots of seven crossings is twelve, as I am compelled to believe, or at 

 most eight, as Tait prefers to say. Whatever be the decision of the reader, 

 I am highly delighted, while attempting to write on a theme so dry and tire- 

 some, that we have, at the outset, such a pretty little quarrel as it stands 

 wherewith to allure his attention. 



7. Every polyedron which is an w-acron having only tessarace summits is a 

 solid knot of n crossings, on which is neither linkage nor flap. Such a knot 

 can be projected on any of its triangular or m-gonal faces, so as to show all its 

 n crossings, and no more, within that face or at its summits. It has no linear 

 section, i.e., no plane can cut it in space, nor can any closed curve be drawn 

 upon lines of its projection, without meeting it in more than two points, on edge 

 or at crossing. 



In a knot which is no polyedron, we call a section that meets it in only two 

 projected summits or crossings, linear, as passing through two points only of 

 the figure ; although no crossing of a mesh in space can be cut through its 

 opposite angles without making four ends. 



8. No projected knot, solid or unsolid, can have a linear section through 

 one edge kl only, and through one crossing q only. For, if it can, let the two 

 threads at q on the left hand of the section be pq and rq, pq passing over rq at 

 q ; then ji><7 at the crossings passed under a thread, and rq at r passed over one. 

 Cut at q, making four ends ; reunite into one thread rp the two ends on the 

 left ; this shortened thread passes over a thread at r and under one at p, as 

 before, and its further course is unaltered in either direction. The same is 

 true of the shortened thread made by reuniting the two threads on the right of 

 the section. 



The figure is now the projection of a knot of n— 1 crossings having two 

 portions, L on the left and R on the right, which are connected by a single 

 thread kl, the law of under and over being observed in both L and R 

 The course of the thread kl, pursued along its circle small or great through R 



