CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 283 



from k upon R, must bring it back to I upon L through L. But this is clearly 

 impossible from want of a second connecting thread ; which proves the proposi- 

 tion. 



9. Knots Solid, Subsolid, and Unsolid. — A solid knot is a polyedron, 

 art. 7. 



Subsolid knots admit of no linear section but through the two crossings of 

 a flap ; through the projections of these a closed curve can be drawn meeting 

 the knot in no third point. 



The projections of an unsolid knot admit one or more linear sections either 

 through two crossings not on a flap, or through two edges coming from two 

 crossings not on the same flap. 



If a projected unsolid V admits a linear section through two edges coming 

 from two crossings on the same flap, V is made up of two portions, K and L, 

 connected like two links of an ordinary chain, so that K (or L) can be set free 

 in its completeness by breaking only one thread of the other. No knot V 

 constructed in these pages is such a compound of K and L. Such a compound 

 is easily drawn ; in such a V either K or L can be slipped along the thread of 

 the other, without twisting a tape, so as to occupy, if the other be unifilar, any 

 position upon it. All our unsolicls are composite ; but no severing of a single 

 thread will ever set free on one of them a portion which is a complete knot. 



Of solid knots we are not treating. If the apparent dignity of knots so 

 maintains itself as to make a treatise on these w-acra desirable, it will be no 

 difficult thing to show in a future memoir how to enumerate and construct 

 them to any value of n without omission or repetition. The beginner can 

 amuse himself with the regular 8-edron, which is trifilar, or with the unifilar of 

 eight crossings made by drawing within a square a square askew, and filling up 

 with eight triangles. 



10. I consider a knot as given by its projection upon and within any one, 

 2-gonal or ???-gonal, of its meshes drawn large, and as having the symmetry of 

 that projection. Nor do I trouble myself with inquiring how far that symmetry 

 is affected by the law of under and over at the crossings, because, in reading 

 the circle or circles of a projected knot, we can take any crossing q as our first, 

 and can on beginning to read take either of the threads at q as passing under 

 and over the other. A knot in space can be read only as given. 



In my description of the symmetry of our reticulations, I shall assume that 

 the reader understands the terms employed. They, with others not wanted 

 here, are necessary and sufficient ; they are the only such terms that ever have 

 been proposed ; and, for more than twenty years since they were introduced, 

 no more suitable terms have offered in their stead. I am quite ready to use 

 better ones when they are invented. The symmetry, however, of the figures 

 handled in this paper is of itself so evident that the reader will easily satisfy 



