1 00:00:20,249 --> 00:00:23,939 Let's change topic for a while 2 00:00:23,939 --> 00:00:30,939 These sailor knots are not knotted! The knots can escape from the string ends 3 00:00:31,369 --> 00:00:35,180 To capture the knots we have to glue the string ends together 4 00:00:38,980 --> 00:00:41,360 Now these are knots 5 00:00:41,460 --> 00:00:48,650 Closing the ends of the objects on the right we get more complex knots, where two components can be linked together 6 00:00:49,250 --> 00:00:53,240 Without originality, we call these objects links 7 00:00:55,540 --> 00:01:01,110 If we can untie them, we obtain a very simple link, called unlink 8 00:01:02,910 --> 00:01:08,050 From now on, we will say knots referring to both knots and links 9 00:01:10,620 --> 00:01:14,719 Look at this knot. If we take its mirror image, it looks different 10 00:01:17,319 --> 00:01:27,840 but we can deform the knot on the left into the one on the right without cutting the string 11 00:01:33,030 --> 00:01:40,000 Then, we will consider them the same knot even if they look different at first 12 00:01:48,000 --> 00:01:53,560 These two knots look different, too. But they are tied in the same manner 13 00:01:53,860 --> 00:01:57,740 They have different shapes but represent the same knot 14 00:01:58,240 --> 00:02:04,080 One can be deformed into the other without ever cutting the strings 15 00:02:04,880 --> 00:02:09,839 This knot is called trefoil 16 00:02:11,039 --> 00:02:14,819 As before, we can ask: is the trefoil the same as its mirror image? 17 00:02:16,290 --> 00:02:20,170 This is a very technical and difficult problem 18 00:02:20,170 --> 00:02:27,230 The two knots are not the same. They are called left-hand and right-hand trefoil 19 00:02:27,940 --> 00:02:35,030 How can we distinguish different knots or recognize different drawings of the same knot? 20 00:02:37,539 --> 00:02:46,520 A way to approach such a problem is to relate the realm of knots, closed strings, to that of braids 21 00:02:55,580 --> 00:03:02,080 It is easy to see that when we have a braid, we can tie the strand ends together 22 00:03:04,880 --> 00:03:11,540 and we can cross the bridge to the realm of knots 23 00:03:16,240 --> 00:03:22,490 And vice versa? Can we cross the bridge in the other direction? 24 00:03:22,490 --> 00:03:27,770 A theorem of Alexander ensures that it is possible and gives an algorithm to do it 25 00:03:28,700 --> 00:03:35,700 We describe it, even if more efficient ones are known 26 00:03:37,530 --> 00:03:44,530 We choose an axis. We will make a reel around it 27 00:03:44,740 --> 00:03:48,310 We choose a starting point and walk along the knot turning clockwise around the axis 28 00:03:54,010 --> 00:03:59,520 At some time the knot can turn and we will be walking anti-clockwise 29 00:04:00,060 --> 00:04:03,960 We color all the anti-clockwise pieces red 30 00:04:10,700 --> 00:04:20,790 Now we move each red piece in turn to the other side of the axis and color them yellow again 31 00:04:35,029 --> 00:04:41,020 In the end we have turned our knot into a reel around the axis 32 00:04:43,080 --> 00:04:51,509 Walking along the knot we will always be going in the same sense around the axis 33 00:05:00,070 --> 00:05:07,589 We take a half plane with the axis as border and cut the knot along it 34 00:05:08,049 --> 00:05:13,500 We open the strands keeping the endpoints fixed on the half planes 35 00:05:14,000 --> 00:05:19,099 The strings can never touch each other 36 00:05:19,099 --> 00:05:24,099 And here is our braid! 37 00:05:26,840 --> 00:05:31,899 When we close it, we get a knot equivalent to the original one 38 00:05:32,299 --> 00:05:36,079 that is, we can deform one into the other without cutting the string 39 00:05:37,079 --> 00:05:44,030 Why make life so difficult? The knot on the left seems simpler! 40 00:05:44,789 --> 00:05:49,750 But in this way we can exploit the group structure that we know on braids 41 00:05:50,549 --> 00:05:56,089 Alexander's theorem ensures that we can obtain any knot as the closure of a braid 42 00:05:56,589 --> 00:06:01,510 but two braids can be very different and still give the same knot 43 00:06:01,680 --> 00:06:07,210 For example, they don't even need to have the same number of strands! 44 00:06:08,510 --> 00:06:16,530 So the question now is: given two arbitrary braids, do they give the same knot, once they are closed? 45 00:06:18,010 --> 00:06:22,250 We introduce a new operation, called conjugation 46 00:06:22,550 --> 00:06:25,589 Choose a braid 47 00:06:25,789 --> 00:06:30,370 Take another one and its inverse and compose them in this manner: 48 00:06:30,570 --> 00:06:33,840 one on the left and the inverse on the right 49 00:06:34,540 --> 00:06:37,560 The new braid is called a conjugate of the first 50 00:06:40,560 --> 00:06:49,550 Note that the corresponding operation with numbers will not change the starting number: the product is commutative 51 00:06:54,050 --> 00:07:01,050 On the contrary, two braids can be different and still be conjugated 52 00:07:06,390 --> 00:07:13,350 Here is a simpler example: two generators of the braid group are surely different braids 53 00:07:14,050 --> 00:07:19,000 But look: they are conjugated 54 00:07:34,060 --> 00:07:38,010 In general understand whether two given braids are conjugate is an intriguing issue 55 00:07:44,010 --> 00:07:51,010 Let's go back to our problem: when do two braids close to the same knot? 56 00:07:53,040 --> 00:07:56,690 If we conjugate a braid with any other, 57 00:07:57,090 --> 00:08:01,000 when we close the new braid 58 00:08:02,710 --> 00:08:08,010 we can shift the lateral pieces so that they cancel out 59 00:08:08,510 --> 00:08:12,550 since one is the inverse of the other 60 00:08:13,050 --> 00:08:17,560 In this way we get the same knot as closing the original braid 61 00:08:20,080 --> 00:08:24,590 We can modify our braid in another way: 62 00:08:24,790 --> 00:08:29,230 add a strand on the top and link the two top strands together 63 00:08:33,050 --> 00:08:41,020 The new braid, when closed, will give the same knot as the old one: 64 00:08:41,520 --> 00:08:48,020 we just need to undo the loop 65 00:08:49,520 --> 00:08:53,520 Of course we can do vice versa, too: 66 00:08:53,520 --> 00:09:00,590 cancel the last strand if it links just once with the second-to-last 67 00:09:00,790 --> 00:09:07,790 These operations are called stabilizations 68 00:09:09,440 --> 00:09:11,079 A Russian mathematician, Markov, noticed that: 69 00:09:11,279 --> 00:09:20,220 Two braids give the same knot if and only if they are related by a sequence of moves of the two kinds we have just seen 70 00:09:20,810 --> 00:09:28,850 This is now known as Markov theorem, even if the first proof is probably due to one of his students 71 00:09:30,690 --> 00:09:37,639 We didn't show the difficult part of this theorem, namely, that the two kinds of moves are enough 72 00:09:37,810 --> 00:09:44,870 We just make an example. We already know that these two braids give the same knot 73 00:09:45,020 --> 00:09:48,109 Now we can prove this, without passing through the realm of knots! 74 00:09:48,340 --> 00:10:00,370 We have to find a sequence of conjugations and stabilizations that transforms one braid into the other 75 00:10:12,500 --> 00:10:17,010 Markov theorem exactly says when two braids give the same knot, 76 00:10:17,010 --> 00:10:23,520 but in this form it is of no concrete use: finding a sequence of relations can be very difficult 77 00:10:25,300 --> 00:10:32,380 And, as in chapter 2, if we can't find such a sequence, it doesn't mean that none exists! 78 00:10:34,640 --> 00:10:40,310 Approaching knots through braids seems not to simplify things 79 00:10:40,310 --> 00:10:45,210 But one of the major results on knots was achieved just thanks to the braids! 80 00:10:45,210 --> 00:10:52,050 In 1984 Jones, studying braids, proved an outstanding result that revolutionized the theory of knots! 81 00:10:52,050 --> 00:10:59,070 It was so important that he won the Fields medal for it, the most important award for mathematicians 82 00:11:00,280 --> 00:11:06,230 Jones found a way to associate a formula, a mathematical expression, to each braid 83 00:11:06,520 --> 00:11:13,520 The powerful fact is that this permits us to distinguish the knots obtained closing the braids: 84 00:11:13,520 --> 00:11:19,080 if two braids have different formulas, then they give different knots 85 00:11:20,030 --> 00:11:29,000 On the other hand, if two braids differ by Markov moves, then they are associated to the same formula 86 00:11:30,050 --> 00:11:38,030 This means that the formula, called Jones polynomial, only depends on the knot and not on the braid used to get it! 87 00:11:40,040 --> 00:11:45,530 As an example, the trefoil and the figure eight knot have different Jones polynomials, so they are surely different knots 88 00:11:48,070 --> 00:11:54,090 Later another algorithm to calculate the Jones polynomial was discovered, not involving braids anymore 89 00:11:54,510 --> 00:11:58,750 Choose a direction to walk along the knot 90 00:11:58,950 --> 00:12:02,550 There are places where we see a crossing 91 00:12:02,550 --> 00:12:09,360 The crossings can be of two types depending on the strand that passes behind the other 92 00:12:09,560 --> 00:12:15,250 Resolving a crossing means to break the arcs and connect them in the other way, respecting their orientation 93 00:12:15,950 --> 00:12:20,980 Introduce a relation between these pieces 94 00:12:20,980 --> 00:12:24,470 The symbol V indicates the Jones polynomial 95 00:12:24,470 --> 00:12:27,680 Now, associate the polynomial 1 to the unknot 96 00:12:28,080 --> 00:12:33,090 Using just these two relations, we can calculate the polynomial on every knot 97 00:12:33,090 --> 00:12:38,190 Choose a crossing and apply the first relation to it 98 00:12:38,190 --> 00:12:40,790 Simplify... 99 00:12:44,090 --> 00:12:50,080 and apply the first relation again to the knot on the right to write a new equation 100 00:12:50,410 --> 00:12:57,400 Choose a new crossing and go on like this, writing equations and simplifying 101 00:13:03,510 --> 00:13:11,560 Using always the same relations, we can calculate the Jones' polynomial of the simplest knots 102 00:13:11,960 --> 00:13:18,550 Then, going back step by step, we can reconstruct the expression for the complex knots 103 00:13:18,600 --> 00:13:21,670 in our case the right trefoil 104 00:13:22,570 --> 00:13:30,500 We didn't check that this machinery is coherent, that is, making different choices always gives the same expression for a fixed knot 105 00:13:31,300 --> 00:13:38,430 This is the difficult part, and the interesting one: the Jones' polynomial is an invariant of knots: 106 00:13:38,630 --> 00:13:43,140 calculated on two equivalent knots, it is the same 107 00:13:53,800 --> 00:14:01,679 If we calculate the Jones polynomial on the left trefoil, we obtain an expression that is symmetric to the other, in some sense 108 00:14:01,679 --> 00:14:04,240 But not equal 109 00:14:04,240 --> 00:14:08,530 We have proved that the two trefoils are not equivalent!