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Flag for Inappropriate Content. Download Now. Prasolov - Intuitive Topology For Later. Related titles. Carousel Previous Carousel Next. Adams C. Elements of Differential Geometry Millman-Parker. Hodge, Jonathan K. Probability Graduate Texts in - Albert N. Jump to Page. Search inside document. Mathematical World Intuitive Topology V. Prasolov Translated from the Russian by A. Deformations Chapter 2. Knots and Links Chapter 3. Homeomorphisms Chapter 6. Vector Fields on the Plane Chapter 7.
The initial aquaintance with this field is hindered by the fact that rigorous definitions of even the simplest notions of topology are rather abstract or very technical. For this reason the first really meaningful and in fact readily understandable topological theorems appear only after tedious preliminaries have been overcome. This preliminary work is mostly devoted to the detailed and accurate proofs of intuitively obvious statements: admittedly not a very exciting activity.
This book is an introductory course in topology of rather untraditional structure. We begin by defining the main notions in a tangible and perceptible way, on an everyday level, and as we go along we progressively make them more precise and rigorous, reaching the level of fairly sophisticated proofs.
This allows us to tackle meaningful problems from the very outset with some success. We do consider some invariants, but only simple and effective ones. The numerous illustrations are essential. In many parts of the book they are more important than the text, which is then little more than a commentary to the pictures. In the study of mathematics, problem solving plays a crucial role.
Reading ready-made proofs of theorems is a poor substitute for trying to prove them on your own, Many statements that the reader can profitably think about himself appear in the form of problems. These problems are an inherent part of our exposition, and therefore their solutions are presented at the end of each section. Among the books and articles that had the greatest influence on this book, I would like to name the book by Rolphsen  and the article by Viro .
As is usually done in mathematical books and papers, the symbol 01 marks the end of the proof of a proposition or a theorem. It should be mentioned that the present text is based on a series of lectures given by the author in the academic year to students of High School no.
Fleischer for useful discussions of the manuscript. We shall assume that the objects consid- ered are made from a very elastic material: their shape may be changed at will, you can bend, distort, stretch, and compress them as much as you like, but of course you may not tear them or glue parts of them together.
The deformations that you will be asked to find will seem impossible at first glance. But actually they are not difficult to visualize, as you can verify by reading their description in the solution section. However, we emphatically suggest that you try to find the solution on your own before looking at our answers. Problem 1.
Show that the elastic body represented in Figure 1. In other words, were the human body elastic enough, after making linked rings with your index fingers and thumbs, you could move your hands apart without separating the joined fingertips.
Ficure 1. A circle is drawn on a pretzel with two holes Figure 1. Show that it is possible to deform the pretzel so that the circle will be in the position represented in Figure 1.
Show that a punctured tube from a bicycle tire can be turned inside out. More precisely, this would be possible if the rubber from which the tube is made were elastic enough. In real life it is impossible to turn a punctured tube inside out. Show that the fancy pretzel represented in Figure 1. We sometimes indicate by arrows on the pictures the direction of motion or of deformation.
See Figure 1. Figure 1. First we perform the deformations shown in Figure 1. Once this is done, the previous defor- mations performed in reverse order result in the tube being turned inside out as required. First we perform the deformation shown in Figure 1.
The solid thus obtained provided it is elastic can clearly be deformed into the one shown in Figure 1. It now remains to apply the solution of Problem Ll. You can imagine a knot as a thin elastic string whose extremities have been glued together. One of the simplest knots is shown in Figure 2. It is known as the trefoil, or more precisely as the right trefoil. There is also a left trefoil, represented in Figure 2.
It can be proved that the left and right trefoils are different knots, ie. By a deformation of a knot we understand its deformation in space as a thin elastic string. Knots are usually represented by their plane projections often called knot diagrams , but it should be kept in mind that the projections of the same knot on different planes can look quite dissimilar. Draw the projections on the xy-plane and the xz-plane of the trefoil shown in Figure 2.
Ficure 2. The next knot after the trefoil in order of complexity is the figure eight knot, whose shape does indeed recall that of the digit 8 Figure 2. Figure 2. And even a fairly simple knot such as the trefoil or the eight, after it has been slightly tangled up by someone, may not be too easy to recognize. For example, it is not immediately obvious that all the knots shown on the top row of Figure 2. Moreover, several representations of the trefoil closely resemble some of the pictures of the figure eight knot.
We have placed such matching knots one under the other. As we have already mentioned, these two knots cannot be deformed into each other. The figure eight knot, however, behaves differently under mirror symmetry: it becomes a knot that can be transformed into the original one.
Indeed, it is easily seen that the first two knots from the left in the second row are mirror symmetric. If, instead of one string, we take several and glue together the extremities of each, we obtain a link. Two examples are shown in Figure 2. The first is known as the Hopf link, the second is the Whitehead link. CU a b Figure 2. Hence there exists a deformation that switches the components of the Hopf link.
It seems at first glance that no such deformation can exist for the Whitehead link. Indeed, if you cut string 1 near the top of the picture in one place, slip the other part of this string through the cut just once, and glue back the cut ends, you will then have no trouble unlinking the two strings from each other.
It seems just as obvious that this cannot be done with string 2. But in fact it can. Problem 2. The link shown in Figure 2. It is known as the Borromean rings because it appears on the coat of arms of the Borromeo clan, a famous and prosperous family from the ancient Italian nobility.!
A more interesting property of this link is that the three circles in it are pairwise unlinked: if any one of the rings is removed, the two remaining rings can be unlinked. An even stranger property is that if you link two of the rings in the simplest way so that they form a Hopf link , then the third ring can be slipped off these two Figure 2. Since the Borromean rings are pairwise unlinked, two of them thought of as being rigid hoops can be pulled far apart from each other; then the third which we can imagine as an elastic string or rope will wind itself around the two hoops.
Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. This book is an introduction to elementary topology presented in an intuitive way, emphasizing the visual aspect. Examples of nontrivial and often unexpected topological phenomena acquaint the reader with the picturesque world of knots, links, vector fields, and two-dimensional surfaces. The book begins with definitions presented in a tangible and perceptible way, on an everyday level, and progressively makes them more precise and rigorous, eventually reaching the level of fairly sophisticated proofs. This allows meaningful problems to be tackled from the outset.
V. V. Prasolov - Intuitive Topology 1995
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