Round handle

While writing this book, the authors found the water faucets (round handles) in the committee meeting room rest room (in a famous San Francisco hotel) opened backward from expected—... 

We later learned that water valves with lever handles (frequently seen in hospitals or in sinks intended for disabled persons) are designed for the lever to be pushed away from the user to close the valve. Thus, the cold water valve designed for a lever will turn to the left to close. If a round handle is fitted instead of the lever. . . splash ... 

Definition 2.1.4 The direct product of a circle with a two-dimensional disk on whose boundary two connected nonintersecting arcs li and I2 are distinguished will be called a round handle (Fig. 25). A round handle (full torus) is a thickened cylinder with feet (bases) li x and I2X S. ... 

Since, for the moment, the separatrix diagram of the critical circle ia supposed to be orientable, the first case (the Mobius strip) is impossible here. The tubular neighbourhood of the surface P. is therefore homeomorphic to a round handle. It is glued to Ca-g precisely in the fashion suggested by the definition of the round-handle -gluing operation (see above). The axes of both feet of the round handle are glued to two smooth circles 71 and 72 drawn on Ba-g by points A and B (Fig. 27) when the point x slides upon S. ... 

Note that a round handle may be glued to Pa- hi only two ways to one or two distinct tori. A thick Mobius strip may be glued only to one torus because... 

Lemma 2.1.5. Suppose on a critical level Ba there exists exactly one critical saddle circle S. 1) Let sd S ) be orientable. Consider a round handle corresponding to the critical circle and glued with both feet to Then each of the feet lies... 

Now we proceed to the proof of Lemma 2.1.6. We begin with the orientable case. Let a round handle corresponding to a saddle circle be glued to distinct tori Ti and T2 along rings whose axes are noncontractible (by virtue of Lemma 2.1.5) circles 71 and 72, respectively. 

Cutting the tori along these circles, we obtain two rings out of each torus. Gluing the round handle and considering the boundary of the obtained three-dimensional manifold, we get one torus as the upper component of the boundary. 

Gluing a round handle modifies one torus correspondingly into the following... 

Lemma 2.1.8. It may always be assumed fin the study of surgeiy on Liouville tori) that on each critical level Ba there exists exactly one critical saddle circle. In other words, it may always be assumed that round handles or thick Mobius strips are glued successively and not simultaneously. 

Make the wedges from pieces of hardwood that are about /sj in. thicker than the kerf (1). Taper the ends of the wedges slightly and round the corners for easy entrance into the kerf. Rounded handle ends with chamfered edges are easy on the hands, I paint my wedges a bright color to make them easy to spot if they drop into the sawdust. The paint helps preserve the wood, too. 

---