Funny Problems in Intuitive Topology

FUNNY PR OBLEMS IN INTUITIVE TOPOLOGY R. T A V AKOLI Abstract. The goal of this article is to in tro duce some b eautiful known rid- dles in intuitiv e top ology; hoping to make at least some fun for the reader. 1. Funny problems in intuitive topology The materials of this section are adapted from reference: [ 1 ]. Problem 1. ([ 1 ]) Show that the elastic b o dy represented in figure 1 (a) can b e deformed so as to b ecome the one shown in 1 (b). In other words, were the h uman b o dy elastic enough, after making linked rings with your index fingers and thum bs, y ou could mov e y our hands apart without separating the joined fingertips. Figure 1. Plot related to statement of problem 1 [ 1 ]. Problem 2. ([ 1 ]) A pretzel has tw o holes that ”hold” a doughnut (see figure 2 (a)). Show that the pretzel can be deformed in such a w a y that one of its ”handles” will unlink itself from the doughnut (figure 2 (b)). Problem 3. ([ 1 ]) A circle is dra wn on a pretzel with t w o holes ( 3 (a)). Sho w that it is p ossible to deform the pretzel so that the circle will b e in the p osition represen ted in 3 (b). Problem 4. ([ 1 ]) Show that a punctured tub e from a bicycle tire can be turned inside out. More precisely , this would b e p ossible if the rubb er from which the tub e is made were elastic enough. In real life it is imp ossible to turn a punctured tub e inside out. Problem 5. ([ 1 ]) Show that the fancy pretzel represen ted in 4 (a) can be deformed in to the ordinary pretzel with tw o holes ( 4 (b)). Department of Material Science and Engineering, Sharif Universit y of T echnology , T ehran, Iran, P .O. Box 11365-9466, tav@mehr.sharif.edu , roh tav@gmail.com . 1 2 R. T A V AKOLI Figure 2. Plot related to statement of problem 2 [ 1 ]. Figure 3. Plot related to statement of problem 3 [ 1 ]. Figure 4. Plot related to statement of problem 5 [ 1 ]. Do you can resolve these problems by yourself (please try and then go to the next page)? FUNNY PROBLEMS IN INTUITIVE TOPOLOGY 3 Graphical solutions to ab ov e problems are introduced in figures 5 through 10 . Figure 5. graphical solution for problem 1 [ 1 ]. Figure 6. graphical solution for problem 2 [ 1 ]. Regarding to the problem 4, first w e p erform the deformations shown in figure 8 . Then we can change the p osition of the obtained figure so that its ”inside” (sho wn in white) b ecomes its ”outside” (the shaded side of the surface) and vice- v ersa, simply by mo ving it as a rigid b o dy in space until the ho op 1 o ccupies the p osition of the ho op 2. Once this is done, the previous deformations p erformed in rev erse order result in the tub e b eing turned inside out as required. Note that this 4 R. T A V AKOLI Figure 7. graphical solution for problem 3 [ 1 ]. Figure 8. graphical solution for problem 4 [ 1 ]. pro cedure interc hanges the ”parallel” and the ”meridian” of the tube (see figure 9 ) [ 1 ]. Figure 9. graphical solution for problem 4 (contin ue) [ 1 ]. Regarding to the problem 5, first we p erform the deformation shown in figure 10 . The solid thus obtained (provided it is elastic) can clearly b e def.ermed into the one sho wn in figure 1 (a). It now remains to apply the solution of Problem 1 (cf. figure 5 ) T opologists call such ob jects isotopic (in space). More clearly , when we consider ob jects in space ”up to deformations”, i.e., w e did not distinguish ob jects that can b e deformed into each other by reshaping them, w e called such ob jects as isotopic. In the same w a y , the top ology aid us to identify isotopic knots. F or example all knots represented on the top ro w of figure ?? can be deformed in to each other. Similarly , all knots represented on the b ottom row of figure ?? can b e deformed in to each other. F or solution readers are refereed to c hapter 2 of [ 1 ]. References [1] VV Prasolov. Intuitive top olo gy . Orien t Blacksw an. 1 , 2 , 3 , 4 , 5 FUNNY PROBLEMS IN INTUITIVE TOPOLOGY 5 Figure 10. graphical solution for problem 5 [ 1 ]. Figure 11. All knots represen ted on the top ro w are isotopic, and the same for the b ottom row [ 1 ]. Dep ar tment of Ma terial Science and Engineering, Sharif University of Technology, Tehran, Iran, P.O. Box 11365-9466 E-mail address : tav@mehr.sharif.edu , rohtav@gmail.com