Torus knot

In fact the trajectories are always knotted if p, q>2 have no common factors. The resulting curves are called p. q torus knots. 

Fig. C11.2 This table shows all possible prime knots with up to 8 crossings on the projection. For chiral knots, only one of the mirror images is shown. There are many ways to identify certain separate classes of knots for instance, knots 3i, 5i, 7i are called torus knots, because they can be nicely placed on the surface of a doughnut (and there is obviously a torus knot with any odd number of crossings). But the classification of knots relevant for their physics is yet to be developed. The figure is courtesy of R. Scharein the knot images were produced by his software KnotPlot (see http //www.knotplot.com).

We can use this same approach to prove that other molecular knots and links are topologically chiral. For example, consider the molecular link illustrated in Figure 18. This catenane was synthesized by Nierengarten et al. [12]. For this molecule the set T(G) consists of many unlinks together with many copies of the (4,2)-torus link, illustrated as L in Figure 12. However we saw earlier that this unoriented link is topologically chiral. Therefore, the molecular (4,2)-torus link is topologically chiral as well. 

Scheme 3 Diagram for the circular musculature of sphincteric cylinder (canalis egestorius). R.P.L., right pyloric (canalis) loop L.P.L., left pyloric (canalis) loop P.M.K., pyloric muscle knot (torus) S, stomach D.B., duodenal bulb. (Ring of circular musculature surrounding commencement of duodenum not shown.) (From Ref. f)...

When plotted on the torus, the same trajectory gives. .. a trefoil knot Figure 8.6.5 shows a trefoil, alongside a top view of a torus with a trefoil wound around it. 

Do you see why this knot corresponds to p = 3, <7 = 27 Follow the knotted trajectory in Figure 8.6.5, and count the number of revolutions made by during the time that 0, makes one revolution, where 0, is latitude and 0 is longitude. Starting on the outer equator, the trajectory moves onto the top surface, dives into the hole, travels along the bottom surface, and then reappears on the outer equator, two-thirds of the way around the torus. Thus 62 makes two-thirds of a revolution while 0 makes one revolution hence p = 3, q = 2. 

There are also a number of other fairly obvious stereoisomeric possibilities (i.e., ones transparent to graph theory) that have as yet received no attention, e.g., multiple toroidal rings (or Klein bottles) with a catenane type of interlocking knots tied in the torus tube concentric multilayered toroidal tubes. 

Finally, we mention that nonplanar graphs having one unavoidable intersection of edges can be represented on the surface of a torus without such an intersection. The nonplanar graph as well as the skeleton graph of the Mobius compound synthesized by Walba et dl (which is homeomorphic to /Ca ) can be drawn on a torus surface without intersections. The surface of the torus is also important for other unusual chemical compounds The skeletons of catenanes, rotaxanes and knots cannot be embedded in the surface of a three-dimensional sphere, but in that of a torus. 

Compressive forces in children can result in cortical bone buckling. These fractures, which are commonly also referred to as buckle or torus injuries, are incomplete and the cortex is intact (Fig. 8.6a,b). Torus is derived from the Latin meaning a protuberance or knot and typically involves both cortical surfaces, while a buckle fracture may only involve a... 

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