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List of prime knots

From Wikipedia, the free encyclopedia

In the mathematical theory of knots, prime knots are those knots that are indecomposable under the operation of knot sum. They are listed here for quick comparison of their properties and varied naming schemes.

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Transcription

We found a new pattern in the primes! One that we didn't know was there. Primes don't like to repeat their last digits. It's really, really strange. A prime could end with a one, three, a seven, or a nine. I mean the exceptions are two itself, which obviously end in a two and five itself. But apart from that, they end with a one, a three, a seven, or a nine, right. And because we hope that primes are random, or they feel like they kind of appear randomly, that each one of those should be equally likely and something strange has happened, so some mathematicians in Stanford have looked at this and they've looked at consecutive primes and they've noticed things like if a prime ends in a nine, it's actually more likely to, for the next prime to end in a one, rather than a nine or a three or a seven. So you would expect all of those to equally likely to turn up next but when they did it and they looked at the first hundred million primes and when they did that, if you had a prime ending in a nine the chance the next prime ended in a nine was eighteen percent, which is not a quarter, and the chance that ended in a one was greater than a quarter, it was thirty-two percent. It shouldn't be! That shouldn't be how it is, but that's what they found out. Let's find out why this is. So if the primes are random, then these prime endings here: one, three, seven, and nine, they should each appear about a quarter of the time, and if we had the consecutive primes and we look at their prime endings we would have sixteen options: You have a one and a one, so that would be a prime ending in a one followed by a prime ending in a one. Or it could be a one followed by a three, a one followed by seven, a one followed by nine These are the options. Ok, there are sixteen options when you're looking at two consecutive primes so if it was random they should all appear one-sixteenth of the time equally likely of the time. That's not they've found at all. In particular, these diagonal entries here - so the one, one; three, three; seven, seven; nine, nine - are least likely to turn up. So, the primes aren't repeating themselves and there's a few explanations for why this could be the case which I'm going to dismiss. I'm going to give you a few explanations which aren't the reason why this is happening. So one might be: "Okay if you have a prime ending with a nine, then well you have to go through all the other numbers before you get to another prime ending with a nine." Or, you have to go through the numbers that end with a one, and a three, and a seven before you get to another number ending with a nine, so. [Brady Haran] It's further away! [James Grime] It's further away, right. And unfortunately, that isn't enough to explain the bias that we found. if that's the case then we're looking for prime gaps less than ten. Right? That means you have the next one in less than ten so you're going for a prime ending in a one next, or the prime ending with a three next. and prime gaps less than ten are not that many and so that's not enough to explain how biased this is. The bias is bigger than that, so that's no good. Another explanation might be: I said that these prime endings one, three, seven and nine if this was a random thing they should be appearing a quarter of the time each. Maybe the explanation is that that's not a true thing maybe the bias comes from the primes themselves and that doesn't work out either there is a slight bias in the primes and it's something we know about. It's called Chebyshev's bias. It says that primes ending in three and seven are slightly more likely. That is something we know about, we know why that happens, and it's such a slight thing. It is not enough. So there is this known bias in prime number endings, so under that assumption, the pair three-three and seven-seven should turn up more often, right? And the complete opposite is true. In fact, three-three and seven-seven are the least likely that are turning up. which is the exact opposite of what it should be, and nine-one are actually the most common pair! which is not what it should be at all. Ok, maybe, oh maybe it's just a base 10 thing, you know. Base ten, you know, who cares about base ten, right? If it was a fundamental property of primes, it would happen in any base. And it does. That's what they found. So they checked it in other bases and they found the bias is still there, so it appears to be a fundamental property of the primes. For example, if we do it in base three. It will only end with a one or two unless it's three itself, right, we can ignore that one. You got two endings they should turn about fifty percent of the time each, which is about right, again Chebyshev's bias says just a little, slight bias toward the two, but its tiny. It's pretty much fifty-fifty, it's pretty much a coin toss. In fact that's what inspired this investigation, and the guys who did this investigation were thinking about coin tosses and said "Well primes in base three are like a coin toss. Let's see if that's the same thing because that's a random event." And then they found this completely different thing, this skew that primes don't like to repeat themselves. And that's not something that would happen in coin tosses. So if we did it in base three, we would have four prime endings, wouldn't we? We would have one-one, one-two, two-one, and two-two. And again they found the same thing. These ones with the repetition are the least likely to occur. They looked at the first million primes. And if you look at first million primes, then they should all be equally likely to turn up a quarter of the time: Two-hundred and fifty thousand. But no, they didn't get that, so these ones, with the repetition, were less than two hundred and fifty thousand. The ones without a repetition were more than two hundred and fifty thousand. So the mathematicians who have been investigating this have tried to come up with an explanation for this. And their explanation relies on a conjecture that goes back a hundred years. It's a conjecture by Hardy and Littlewood and they had a conjecture about the density of primes: how many primes you can find in patterns. So you can consider all kinds of patterns Like twin primes, that's a pattern, or cousin primes, which have gaps or four, or sexy primes, who have gaps of 6. So they had this conjecture about how many of these you should find, and the conjecture has not been proven There's a lot of evidence that supports that it's true, so if you look at the numerical evidence, it appears to be true but it hasn't been proven. So the mathematicians looking at this pattern used a modified version of that conjecture and they came up with a formula that they think might explain this idea. So that formula was the proportion of this pattern - let's say you've got prime endings a and b, so if we're doing it in base ten, this could be one-one or three-seven, or nine-one. So we're looking at the proportion of these endings and they come up with a formula. The formula was: one over the number of allowed endings, so this is like one sixteenth from what I've been doing in base ten right? So when it's equally likely, these are the allowed endings there. Sixteen of them. So the proportion is one over the allowed endings multiplied by a thing. Right, and what is this thing? That thing depends on if that pattern repeats, if you've got a equals b in that formula. If a equals b, that will affect what that thing is. I'll show you what it looks like in base three. It's kind of ok in base three. We're looking at proportions of these endings. If they're the same like this: a, a. So that would, in base three, the one-one endings and the two-two endings. The proportion is. If you are doing it when they're not equal, so if it was the one-two endings or the two-one, the proportion of a b, and I'm saying a is not the same as b Plus. So this formula they've got is still a conjectured formula because it is based on this Hardy-Littlewood formula which is still a conjectured formula. But it fits the evidence. Once we start going off to infinity, this bias becomes less and less important. This is a bias that is hanging around so in the great infinity of numbers, this bias is evening out, but even up to a trillion there is still a noticeable bias there. Audio books are a great way to pass time if, for example, you spend a lot of time commuting, and with over two hundred and fifty thousand titles in its collection, audible.com is the place to find something you'll enjoy. For your first book, why not go for The Humans by Matt Haig? It's a cracking story; it's funny, charming, and without giving too much away I think it will really appeal to Numberphile fans. It's been one of my favorite books in a while. Why not give them a try? You can actually sign up for a free 30 day trial at audible.com/numberphile Use that address and they'll know you came from here, and thanks to Audible for supporting this video

Contents

Table of prime knots

Six or fewer crossings

Name Picture Alexander- Briggs-Rolfsen Dowker-

Thistlethwaite

Dowker notation Conway notation
Unknot
Blue Unknot.png
01 0a1
Trefoil knot
Blue Trefoil Knot.png
31 3a1 4 6 2 [3]
Figure-eight knot
Blue Figure-Eight Knot.png
41 4a1 4 6 8 2 [22]
Cinquefoil knot
Blue Cinquefoil Knot.png
51 5a2 6 8 10 2 4 [5]
Three-twist knot
Blue Three-Twist Knot.png
52 5a1 4 8 10 2 6 [32]
Stevedore knot
Blue Stevedore Knot.png
61 6a3 4 8 12 10 2 6 [42]
62 knot
Blue 6 2 Knot.png
62 6a2 4 8 10 12 2 6 [312]
63 knot
Blue 6 3 Knot.png
63 6a1 4 8 10 2 12 6 [2112]

Seven crossings

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
Blue 7 1 Knot.png
71 7a7 8 10 12 14 2 4 6 [7]
Blue 7 2 Knot.png
72 7a4 4 10 14 12 2 8 6 [52]
7-3 knot.svg
73 7a5 6 10 12 14 2 4 8 [43]
Celtic-knot-linear-7crossings.svg
74 7a6 6 10 12 14 4 2 8 [313]
7-5 knot.svg
75 7a3 4 10 12 14 2 8 6 [322]
7-6 knot.svg
76 7a2 4 8 12 2 14 6 10 [2212]
7-7 knot.svg
77 7a1 4 8 10 12 2 14 6 [21112]

Eight crossings

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
Blue 8 1 Knot.png
81 8a­11 4 10 16 14 12 2 8 6 [62]
Knot-8-2.png
82 8a8 4 10 12 14 16 2 6 8 [512]
83 8a­18 6 12 10 16 14 4 2 8 [44]
8-4 Knot.svg
84 8a­17 6 10 12 16 14 4 2 8 [413]
85 8a­13 6 8 12 2 14 16 4 10 [3,3,2]
8-6 knot.svg
86 8a­10 4 10 14 16 12 2 8 6 [332]
87 8a6 4 10 12 14 2 16 6 8 [4112]
88 8a4 4 8 12 2 16 14 6 10 [2312]
89 8a­16 6 10 12 14 16 4 2 8 [3113]
810 8a3 4 8 12 2 14 16 6 10 [3,21,2]
811 8a9 4 10 12 14 16 2 8 6 [3212]
8crossings-rose-limacon-knot.svg
812 8a5 4 8 14 10 2 16 6 12 [2222]
813 8a7 4 10 12 14 2 16 8 6 [31112]
814 8a1 4 8 10 14 2 16 6 12 [22112]
8crossings-two-trefoils.svg
815 8a2 4 8 12 2 14 6 16 10 [21,21,2]
8-16 knot.svg
816 8a­15 6 8 14 12 4 16 2 10 [.2.20]
8 17 Knot.svg
817 8a­14 6 8 12 14 4 16 2 10 [.2.2]
8crossing-symmetrical.svg
818 8a­12 6 8 10 12 14 16 2 4 [8*]
8crossing-symmetrical-nonalternating.svg
819 8n3 4 8 -12 2 -14 -16 -6 -10 [3,3,2-]
Knot 8 20.svg
820 8n1 4 8 -12 2 -14 -6 -16 -10 [3,21,2-]
Lissajous 8 21 Knot.png
821 8n2 4 8 -12 2 14 -6 16 10 [21,21,2-]

Nine crossings

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
9-2 star polygon interlaced.svg
91 9a­41 10 12 14 16 18 2 4 6 8 [9]
92 knot 9a­27 4 12 18 16 14 2 10 8 6 [72]
93 9a­38 8 12 14 16 18 2 4 6 10 [63]
94 9a­35 6 12 14 18 16 2 4 10 8 [54]
95 9a­36 6 12 14 18 16 4 2 10 8 [513]
96 9a­23 4 12 14 16 18 2 10 6 8 [522]
97 9a­26 4 12 16 18 14 2 10 8 6 [342]
98 9a8 4 8 14 2 18 16 6 12 10 [2412]
99 9a­33 6 12 14 16 18 2 4 10 8 [423]
910 9a­39 8 12 14 16 18 2 6 4 10 [333]
911 9a­20 4 10 14 16 12 2 18 6 8 [4122]
912 9a­22 4 10 16 14 2 18 8 6 12 [4212]
913 9a­34 6 12 14 16 18 4 2 10 8 [3213]
914 9a­17 4 10 12 16 14 2 18 8 6 [41112]
915 9a­10 4 8 14 10 2 18 16 6 12 [2322]
916 9a­25 4 12 16 18 14 2 8 10 6 [3,3,2+]
917 9a­14 4 10 12 14 16 2 6 18 8 [21312]
918 9a­24 4 12 14 16 18 2 10 8 6 [3222]
919 9a3 4 8 10 14 2 18 16 6 12 [23112]
920 9a­19 4 10 14 16 2 18 8 6 12 [31212]
921 9a­21 4 10 14 16 12 2 18 8 6 [31122]
922 9a2 4 8 10 14 2 16 18 6 12 [211,3,2]
9crossing-knot symmetrical grid.svg
923 9a­16 4 10 12 16 2 8 18 6 14 [22122]
924 9a7 4 8 14 2 16 18 6 12 10 [3,21,2+]
925 9a4 4 8 12 2 16 6 18 10 14 [22,21,2]
926 9a­15 4 10 12 14 16 2 18 8 6 [311112]
927 9a­12 4 10 12 14 2 18 16 6 8 [212112]
928 9a5 4 8 12 2 16 14 6 18 10 [21,21,2+]
929 9a­31 6 10 14 18 4 16 8 2 12 [.2.20.2]
930 9a1 4 8 10 14 2 16 6 18 12 [211,21,2]
931 9a­13 4 10 12 14 2 18 16 8 6 [2111112]
932 9a6 4 8 12 14 2 16 18 10 6 [.21.20]
933 9a­11 4 8 14 12 2 16 18 10 6 [.21.2]
934 9a­28 6 8 10 16 14 18 4 2 12 [8*20]
9crossings-threesymmetric-other.svg
935 9a­40 8 12 16 14 18 4 2 6 10 [3,3,3]
936 9a9 4 8 14 10 2 16 18 6 12 [22,3,2]
937 9a­18 4 10 14 12 16 2 6 18 8 [3,21,21]
938 9a­30 6 10 14 18 4 16 2 8 12 [.2.2.2]
939 9a­32 6 10 14 18 16 2 8 4 12 [2:2:20]
Knot-9crossings-symmetrical.svg
940 9a­27 6 16 14 12 4 2 18 10 8 [9*]
9crossings-decorative-knot-threefold-incircle.svg
941 9a­29 6 10 14 12 16 2 18 4 8 [20:20:20]
942 9n4 4 8 10 −14 2 −16 −18 −6 −12 [22,3,2−]
943 9n3 4 8 10 14 2 −16 6 −18 −12 [211,3,2−]
944 9n1 4 8 10 −14 2 −16 −6 −18 −12 [22,21,2−]
945 9n2 4 8 10 −14 2 16 −6 18 12 [211,21,2−]
946 9n5 4 10 −14 −12 −16 2 −6 −18 −8 [3,3,21−]
9-crossing non-alternating 3-symmetrical.svg
947 9n7 6 8 10 16 14 −18 4 2 −12 [8*-20]
948 9n6 4 10 −14 −12 16 2 −6 18 8 [21,21,21−]
949 9n8 6 -10 −14 12 −16 −2 18 −4 −8 [−20:−20:−20]

Ten crossings

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
101 knot 10a­75 4 12 20 18 16 14 2 10 8 6 [82]
102 10a­59 4 12 14 16 18 20 2 6 8 10 [712]
103 10a­­117 6 14 12 20 18 16 4 2 10 8 [64]
104 10a­­113 6 12 14 20 18 16 4 2 10 8 [613]
105 10a­56 4 12 14 16 18 2 20 6 8 10 [6112]
106 10a­70 4 12 16 18 20 14 2 10 6 8 [532]
107 10a­65 4 12 14 18 16 20 2 10 8 6 [5212]
108 10a­­114 6 14 12 16 18 20 4 2 8 10 [514]
109 10a­­110 6 12 14 16 18 20 4 2 8 10 [5113]
1010 10a­64 4 12 14 18 16 2 20 10 8 6 [51112]
1011 10a­­116 6 14 12 18 20 16 4 2 10 8 [433]
1012 10a­43 4 10 14 16 2 20 18 6 8 12 [4312]
1013 10a­54 4 10 18 16 12 2 20 8 6 14 [4222]
1014 10a­33 4 10 12 16 18 2 20 6 8 14 [42112]
1015 10a­68 4 12 16 18 14 2 10 20 6 8 [4132]
1016 10a­­115 6 14 12 16 18 20 4 2 10 8 [4123]
1017 10a­­107 6 12 14 16 18 2 4 20 8 10 [4114]
1018 10a­63 4 12 14 18 16 2 10 20 8 6 [41122]
1019 10a­­108 6 12 14 16 18 2 4 20 10 8 [41113]
1020 10a­74 4 12 18 20 16 14 2 10 8 6 [352]
1021 10a­60 4 12 14 16 18 20 2 6 10 8 [3412]
1022 10a­­112 6 12 14 18 20 16 4 2 10 8 [3313]
1023 10a­57 4 12 14 16 18 2 20 6 10 8 [33112]
1024 10a­71 4 12 16 18 20 14 2 10 8 6 [3232]
Knot-10-25-sm.png
1025 10a­61 4 12 14 16 18 20 2 10 8 6 [32212]
1026 10a­­111 6 12 14 16 18 20 4 2 10 8 [32113]
1027 10a­58 4 12 14 16 18 2 20 10 8 6 [321112]
1028 10a­44 4 10 14 16 2 20 18 8 6 12 [31312]
1029 10a­53 4 10 16 18 12 2 20 8 6 14 [31222]
1030 10a­34 4 10 12 16 18 2 20 8 6 14 [312112]
1031 10a­69 4 12 16 18 14 2 10 20 8 6 [31132]
1032 10a­55 4 12 14 16 18 2 10 20 8 6 [311122]
1033 10a­­109 6 12 14 16 18 4 2 20 10 8 [311113]
1034 10a­19 4 8 14 2 20 18 16 6 12 10 [2512]
1035 10a­23 4 8 16 10 2 20 18 6 14 12 [2422]
1036 10a5 4 8 10 16 2 20 18 6 14 12 [24112]
1037 10a­49 4 10 16 12 2 8 20 18 6 14 [2332]
1038 10a­29 4 10 12 16 2 8 20 18 6 14 [23122]
1039 10a­26 4 10 12 14 18 2 6 20 8 16 [22312]
1040 10a­30 4 10 12 16 2 20 6 18 8 14 [222112]
1041 10a­35 4 10 12 16 20 2 8 18 6 14 [221212]
1042 10a­31 4 10 12 16 2 20 8 18 6 14 [2211112]
1043 10a­52 4 10 16 14 2 20 8 18 6 12 [212212]
1044 10a­32 4 10 12 16 14 2 20 18 8 6 [2121112]
1045 10a­25 4 10 12 14 16 2 20 18 8 6 [21111112]
1046 10a­81 6 8 14 2 16 18 20 4 10 12 [5,3,2]
1047 10a­15 4 8 14 2 16 18 20 6 10 12 [5,21,2]
1048 10a­79 6 8 14 2 16 18 4 20 10 12 [41,3,2]
1049 10a­13 4 8 14 2 16 18 6 20 10 12 [41,21,2]
1050 10a­82 6 8 14 2 16 18 20 4 12 10 [32,3,2]
1051 10a­16 4 8 14 2 16 18 20 6 12 10 [32,21,2]
1052 10a­80 6 8 14 2 16 18 4 20 12 10 [311,3,2]
1053 10a­14 4 8 14 2 16 18 6 20 12 10 [311,21,2]
1054 10a­48 4 10 16 12 2 8 18 20 6 14 [23,3,2]
1055 10a9 4 8 12 2 16 6 20 18 10 14 [23,21,2]
1056 10a­28 4 10 12 16 2 8 18 20 6 14 [221,3,2]
1057 10a6 4 8 12 2 14 18 6 20 10 16 [221,21,2]
1058 10a­20 4 8 14 10 2 18 6 20 12 16 [22,22,2]
10-59 knot theory square.svg
1059 10a2 4 8 10 14 2 18 6 20 12 16 [22,211,2]
10-60 knot theory square.svg
1060 10a1 4 8 10 14 2 16 18 6 20 12 [211,211,2]
1061 10a­­123 8 10 16 14 2 18 20 6 4 12 [4,3,3]
1062 10a­41 4 10 14 16 2 18 20 6 8 12 [4,3,21]
1063 10a­51 4 10 16 14 2 18 8 6 20 12 [4,21,21]
1064 10a­­122 8 10 14 16 2 18 20 6 4 12 [31,3,3]
1065 10a­42 4 10 14 16 2 18 20 8 6 12 [31,3,21]
1066 10a­40 4 10 14 16 2 18 8 6 20 12 [31,21,21]
1067 10a­37 4 10 14 12 18 2 6 20 8 16 [22,3,21]
1068 10a­67 4 12 16 14 18 2 20 6 10 8 [211,3,3]
1069 10a­38 4 10 14 12 18 2 16 6 20 8 [211,21,21]
1070 10a­22 4 8 16 10 2 18 20 6 14 12 [22,3,2+]
1071 10a­10 4 8 12 2 18 14 6 20 10 16 [22,21,2+]
1072 10a4 4 8 10 16 2 18 20 6 14 12 [211,3,2+]
1073 10a3 4 8 10 14 2 18 16 6 20 12 [211,21,2+]
1074 10a­62 4 12 14 16 20 18 2 8 6 10 [3,3,21+]
Vodicka knot modified.svg
1075 10a­27 4 10 12 14 18 2 16 6 20 8 [21,21,21+]
1076 10a­73 4 12 18 20 14 16 2 10 8 6 [3,3,2++]
1077 10a­18 4 8 14 2 18 20 16 6 12 10 [3,21,2++]
1078 10a­17 4 8 14 2 18 16 6 12 20 10 [21,21,2++]
1079 10a­78 6 8 12 2 16 4 18 20 10 14 [(3,2)(3,2)]
1080 10a8 4 8 12 2 16 6 18 20 10 14 [(3,2)(21,2)]
1081 10a7 4 8 12 2 16 6 18 10 20 14 [(21,2)(21,2)]
1082 10a­83 6 8 14 16 4 18 20 2 10 12 [.4.2]
1083 10a­84 6 8 16 14 4 18 20 2 12 10 [.31.20]
1084 10a­50 4 10 16 14 2 8 18 20 12 6 [.22.2]
1085 10a­86 6 8 16 14 4 18 20 2 10 12 [.4.20]
1086 10a­87 6 8 14 16 4 18 20 2 12 10 [.31.2]
1087 10a­39 4 10 14 16 2 8 18 20 12 6 [.22.20]
1088 10a­11 4 8 12 14 2 16 20 18 10 6 [.21.21]
1089 10a­21 4 8 14 12 2 16 20 18 10 6 [.21.210]
1090 10a­92 6 10 14 2 16 20 18 8 4 12 [.3.2.2]
1091 10a­­106 6 10 20 14 16 18 4 8 2 12 [.3.2.20]
1092 10a­46 4 10 14 18 2 16 8 20 12 6 [.21.2.20]
1093 10a­­101 6 10 16 20 14 4 18 2 12 8 [.3.20.2]
1094 10a­91 6 10 14 2 16 18 20 8 4 12 [.30.2.2]
1095 10a­47 4 10 14 18 2 16 20 8 12 6 [.210.2.2]
1096 10a­24 4 8 18 12 2 16 20 6 10 14 [.2.21.2]
1097 10a­12 4 8 12 18 2 16 20 6 10 14 [.2.210.2]
1098 10a­96 6 10 14 18 2 16 20 4 8 12 [.2.2.2.20]
1099 10a­­103 6 10 18 14 2 16 20 8 4 12 [.2.2.20.20]
10100 10a­­104 6 10 18 14 16 4 20 8 2 12 [3:2:2]
10101 10a­45 4 10 14 18 2 16 6 20 8 12 [21:2:2]
10102 10a­97 6 10 14 18 16 4 20 2 8 12 [3:2:20]
10103 10a­­105 6 10 18 16 14 4 20 8 2 12 [30:2:2]
10104 10a­­118 6 16 12 14 18 4 20 2 8 10 [3:20:20]
10105 10a­72 4 12 16 20 18 2 8 6 10 14 [21:20:20]
10106 10a­95 6 10 14 16 18 4 20 2 8 12 [30:2:20]
10107 10a­66 4 12 16 14 18 2 8 20 10 6 [210:2:20]
10108 10a­­119 6 16 12 14 18 4 20 2 10 8 [30:20:20]
10109 10a­93 6 10 14 16 2 18 4 20 8 12 [2.2.2.2]
10110 10a­­100 6 10 16 20 14 2 18 4 8 12 [2.2.2.20]
10111 10a­98 6 10 16 14 2 18 8 20 4 12 [2.2.20.2]
10112 10a­76 6 8 10 14 16 18 20 2 4 12 [8*3]
10113 10a­36 4 10 14 12 2 16 18 20 8 6 [8*21]
10114 10a­77 6 8 10 14 16 20 18 2 4 12 [8*30]
10115 10a­94 6 10 14 16 4 18 2 20 12 8 [8*20.20]
Triquetra-heart-knot.svg
10116 10a­­120 6 16 18 14 2 4 20 8 10 12 [8*2:2]
10117 10a­99 6 10 16 14 18 4 20 2 12 8 [8*2:20]
10118 10a­88 6 8 18 14 16 4 20 2 10 12 [8*2:.2]
10119 10a­85 6 8 14 18 16 4 20 10 2 12 [8*2:.20]
Two-trefoils-on-loop doubly-interlinked 10crossings.svg
10120 10a­­102 6 10 18 12 4 16 20 8 2 14 [8*20::20]
10121 10a­90 6 10 12 20 18 16 8 2 4 14 [9*20]
10crossings-two-triquetras-joined.svg
10122 10a­89 6 10 12 14 18 16 20 2 4 8 [9*.20]
Floral fivefold knot green (geometry).svg
10123 10a­­121 8 10 12 14 16 18 20 2 4 6 [10*]
10124 10n­21 4 8 -14 2 -16 -18 -20 -6 -10 -12 [5,3,2-]
10125 10n­15 4 8 14 2 -16 -18 6 -20 -10 -12 [5,21,2-]
10126 10n­17 4 8 -14 2 -16 -18 -6 -20 -10 -12 [41,3,2-]
10127 10n­16 4 8 -14 2 16 18 -6 20 10 12 [41,21,2-]
10128 10n­22 4 8 -14 2 -16 -18 -20 -6 -12 -10 [32,3,2-]
10129 10n­18 4 8 14 2 -16 -18 6 -20 -12 -10 [32,21,-2]
10130 10n­20 4 8 -14 2 -16 -18 -6 -20 -12 -10 [311,3,2-]
10131 10n­19 4 8 -14 2 16 18 -6 20 12 10 [311,21,2-]
Knot-10-132-sm.png
10132 10n­13 4 8 -12 2 -16 -6 -20 -18 -10 -14 [23,3,2-]
10133 10n4 4 8 12 2 -14 -18 6 -20 -10 -16 [23,21,2-]
10134 10n6 4 8 -12 2 -14 -18 -6 -20 -10 -16 [221,3,2-]
10135 10n5 4 8 -12 2 14 18 -6 20 10 16 [221,21,2-]
10136 10n3 4 8 10 -14 2 -18 -6 -20 -12 -16 [22,22,2-]
10137 10n2 4 8 10 -14 2 -16 -18 -6 -20 -12 [22,211,2-]
10138 10n1 4 8 10 -14 2 16 18 -6 20 12 [211,211,2-]
10139 10n­27 4 10 -14 -16 2 -18 -20 -6 -8 -12 [4,3,3-]
10140 10n­29 4 10 -14 -16 2 18 20 -8 -6 12 [4,3,21-]
10141 10n­25 4 10 -14 -16 2 18 -8 -6 20 12 [4,21,21-]
10142 10n­30 4 10 -14 -16 2 -18 -20 -8 -6 -12 [31,3,3-]
10143 10n­26 4 10 -14 -16 2 -18 -8 -6 -20 -12 [31,3,21-]
10144 10n­28 4 10 14 16 2 -18 -20 8 6 -12 [31,21,21-]
10145 10n­14 4 8 -12 -18 2 -16 -20 -6 -10 -14 [22,3,3-]
10146 10n­23 4 8 -18 -12 2 -16 -20 -6 -10 -14 [22,21,21-]
10147 10n­24 4 10 -14 12 2 16 18 -20 8 -6 [211,3,21-]
10148 10n­12 4 8 -12 2 -16 -6 -18 -20 -10 -14 [(3,2)(3,2-)]
10149 10n­11 4 8 -12 2 16 -6 18 20 10 14 [(3,2)(21,2-)]
10150 10n9 4 8 -12 2 -16 -6 -18 -10 -20 -14 [(21,2)(3,2-)]
10151 10n8 4 8 -12 2 16 -6 18 10 20 14 [(21,2)(21,2-)]
10152 10n­36 6 8 12 2 -16 4 -18 -20 -10 -14 [(3,2)-(3,2)]
10153 10n­10 4 8 12 2 -16 6 -18 -20 -10 -14 [(3,2)-(21,2)]
10154 10n7 4 8 12 2 -16 6 -18 -10 -20 -14 [(21,2)-(21,2)]
10155 10n­39 6 10 14 16 18 4 -20 2 8 -12 [-3:2:2]
10156 10n­32 4 12 16 -14 18 2 -8 20 10 6 [-3:2:20]
10157 10n­42 6 -10 -18 14 -2 -16 20 8 -4 12 [-3:20:20]
10158 10n­41 6 -10 -16 14 -2 -18 8 20 -4 -12 [-30:2:2]
10159 10n­34 6 8 10 14 16 -18 -20 2 4 -12 [-30:2:20]
10160 10n­33 4 12 -16 -14 -18 2 -8 -20 -10 -6 [-30:20:20]
10-161 knot (Perko 1).svg
10161[a] 10n­31 4 12 -16 14 -18 2 8 -20 -10 -6 [3:-20:-20]
10162[b] 10n­40 6 10 14 18 16 4 -20 2 8 -12 [-30:-20:-20]
10163[c] 10n­35 6 8 10 14 16 -20 -18 2 4 -12 [8*-30]
10164[d] 10n­38 6 -10 -12 14 -18 -16 20 -2 -4 -8 [8*2:-20]
10165[e] 10n­37 6 8 14 18 16 4 -20 10 2 -12 [8*2:.-20]

Higher

 Kinoshita–Terasaka & Conway knots
Kinoshita–Terasaka & Conway knots

Notes

  1. ^ Originally listed as both 10161 and 10162 in the Rolfsen table. The error was discovered by Kenneth Perko (see Perko pair).
  2. ^ Listed as 10163 in the Rolfsen table.
  3. ^ Listed as 10164 in the Rolfsen table.
  4. ^ Listed as 10165 in the Rolfsen table.
  5. ^ Listed as 10166 in the Rolfsen table.

See also

External links

This page was last modified on 22 March 2017, at 18:58.
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