10 10
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X19,7,20,6 X7,19,8,18 X9,17,10,16 X15,11,16,10 X17,9,18,8 X11,2,12,3 |
| Gauss code | -1, 10, -2, 1, -4, 5, -6, 9, -7, 8, -10, 2, -3, 4, -8, 7, -9, 6, -5, 3 |
| Dowker-Thistlethwaite code | 4 12 14 18 16 2 20 10 8 6 |
| Conway Notation | [51112] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 |
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 [{12, 5}, {1, 10}, {6, 11}, {10, 12}, {11, 4}, {5, 2}, {3, 1}, {4, 7}, {8, 6}, {7, 9}, {2, 8}, {9, 3}] |
[edit Notes on presentations of 10 10]
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 10"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X19,7,20,6 X7,19,8,18 X9,17,10,16 X15,11,16,10 X17,9,18,8 X11,2,12,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 10, -2, 1, -4, 5, -6, 9, -7, 8, -10, 2, -3, 4, -8, 7, -9, 6, -5, 3 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 14 18 16 2 20 10 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[51112] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(5,\{-1,-1,2,-1,2,2,3,-2,3,4,-3,4\})} |
In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 5, 12, 5 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{12, 5}, {1, 10}, {6, 11}, {10, 12}, {11, 4}, {5, 2}, {3, 1}, {4, 7}, {8, 6}, {7, 9}, {2, 8}, {9, 3}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^2-11 t+17-11 t^{-1} +3 t^{-2} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^4+z^2+1} |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 45, 0 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+2 q^6-3 q^5+5 q^4-6 q^3+7 q^2-7 q+6-4 q^{-1} +3 q^{-2} - q^{-3} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-2} +z^4 a^{-4} +z^4-a^2 z^2+2 z^2 a^{-4} -z^2 a^{-6} +z^2- a^{-2} +2 a^{-4} - a^{-6} +1} |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +5 z^8 a^{-4} +2 z^8 a^{-6} +4 z^7 a^{-1} +2 z^7 a^{-3} -z^7 a^{-5} +z^7 a^{-7} -7 z^6 a^{-2} -21 z^6 a^{-4} -10 z^6 a^{-6} +4 z^6+4 a z^5-7 z^5 a^{-1} -16 z^5 a^{-3} -10 z^5 a^{-5} -5 z^5 a^{-7} +3 a^2 z^4+5 z^4 a^{-2} +26 z^4 a^{-4} +15 z^4 a^{-6} -3 z^4+a^3 z^3-3 a z^3+3 z^3 a^{-1} +17 z^3 a^{-3} +17 z^3 a^{-5} +7 z^3 a^{-7} -2 a^2 z^2-4 z^2 a^{-2} -12 z^2 a^{-4} -8 z^2 a^{-6} -2 z^2-z a^{-1} -4 z a^{-3} -6 z a^{-5} -3 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1} |
| The A2 invariant | |
| The G2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+8 q^{38}-10 q^{36}+11 q^{34}-7 q^{32}+q^{30}+5 q^{28}-10 q^{26}+14 q^{24}-14 q^{22}+11 q^{20}-6 q^{18}-2 q^{16}+9 q^{14}-10 q^{12}+13 q^{10}-11 q^8+8 q^6-3 q^4-3 q^2+9-10 q^{-2} +9 q^{-4} -3 q^{-6} - q^{-8} +6 q^{-10} -7 q^{-12} +3 q^{-14} +5 q^{-16} -13 q^{-18} +15 q^{-20} -14 q^{-22} + q^{-24} +14 q^{-26} -26 q^{-28} +28 q^{-30} -23 q^{-32} +10 q^{-34} +6 q^{-36} -22 q^{-38} +29 q^{-40} -26 q^{-42} +18 q^{-44} -2 q^{-46} -10 q^{-48} +19 q^{-50} -13 q^{-52} +12 q^{-54} -10 q^{-58} +14 q^{-60} -9 q^{-62} +13 q^{-66} -22 q^{-68} +24 q^{-70} -15 q^{-72} - q^{-74} +15 q^{-76} -27 q^{-78} +27 q^{-80} -21 q^{-82} +6 q^{-84} +5 q^{-86} -15 q^{-88} +17 q^{-90} -14 q^{-92} +8 q^{-94} - q^{-96} -3 q^{-98} +3 q^{-100} -4 q^{-102} +3 q^{-104} - q^{-106} + q^{-108} } |
Further Quantum Invariants
Further quantum knot invariants for 10_10.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+2 q^5-q^3+2 q- q^{-1} + q^{-5} - q^{-7} +2 q^{-9} - q^{-11} + q^{-13} - q^{-15} } |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-2 q^{18}-q^{16}+4 q^{14}-3 q^{12}+4 q^8-5 q^6+q^4+5 q^2-4+5 q^{-4} - q^{-6} -3 q^{-8} +2 q^{-10} +3 q^{-12} -3 q^{-14} -2 q^{-16} +5 q^{-18} -2 q^{-20} -5 q^{-22} +5 q^{-24} + q^{-26} -6 q^{-28} +4 q^{-30} +4 q^{-32} -5 q^{-34} +3 q^{-38} -2 q^{-40} - q^{-42} + q^{-44} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{39}+2 q^{37}+q^{35}-2 q^{33}-3 q^{31}+q^{29}+6 q^{27}-2 q^{25}-4 q^{23}-2 q^{21}+4 q^{19}+2 q^{17}-5 q^{13}-4 q^{11}+6 q^9+9 q^7-2 q^5-11 q^3+12 q^{-1} +6 q^{-3} -8 q^{-5} -8 q^{-7} +11 q^{-11} +6 q^{-13} -10 q^{-15} -10 q^{-17} +7 q^{-19} +13 q^{-21} -6 q^{-23} -14 q^{-25} +3 q^{-27} +14 q^{-29} -12 q^{-33} -2 q^{-35} +11 q^{-37} +7 q^{-39} -9 q^{-41} -11 q^{-43} +4 q^{-45} +13 q^{-47} + q^{-49} -14 q^{-51} -9 q^{-53} +12 q^{-55} +13 q^{-57} -5 q^{-59} -15 q^{-61} + q^{-63} +14 q^{-65} +4 q^{-67} -10 q^{-69} -7 q^{-71} +5 q^{-73} +6 q^{-75} -2 q^{-77} -4 q^{-79} +2 q^{-83} + q^{-85} - q^{-87} } |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{64}-2 q^{62}-q^{60}+2 q^{58}+q^{56}+5 q^{54}-7 q^{52}-5 q^{50}+q^{48}+4 q^{46}+18 q^{44}-11 q^{42}-15 q^{40}-8 q^{38}+7 q^{36}+37 q^{34}-4 q^{32}-28 q^{30}-33 q^{28}+2 q^{26}+63 q^{24}+23 q^{22}-32 q^{20}-65 q^{18}-21 q^{16}+77 q^{14}+61 q^{12}-12 q^{10}-85 q^8-59 q^6+57 q^4+79 q^2+31-57 q^{-2} -78 q^{-4} +2 q^{-6} +49 q^{-8} +55 q^{-10} +2 q^{-12} -49 q^{-14} -37 q^{-16} -9 q^{-18} +38 q^{-20} +47 q^{-22} + q^{-24} -41 q^{-26} -42 q^{-28} +11 q^{-30} +55 q^{-32} +27 q^{-34} -38 q^{-36} -51 q^{-38} +56 q^{-42} +35 q^{-44} -40 q^{-46} -53 q^{-48} -6 q^{-50} +56 q^{-52} +48 q^{-54} -30 q^{-56} -55 q^{-58} -27 q^{-60} +38 q^{-62} +58 q^{-64} - q^{-66} -31 q^{-68} -42 q^{-70} -4 q^{-72} +39 q^{-74} +22 q^{-76} +13 q^{-78} -27 q^{-80} -35 q^{-82} -4 q^{-84} +8 q^{-86} +42 q^{-88} +15 q^{-90} -22 q^{-92} -30 q^{-94} -29 q^{-96} +28 q^{-98} +38 q^{-100} +15 q^{-102} -15 q^{-104} -45 q^{-106} -6 q^{-108} +20 q^{-110} +27 q^{-112} +12 q^{-114} -25 q^{-116} -17 q^{-118} -4 q^{-120} +12 q^{-122} +16 q^{-124} -4 q^{-126} -6 q^{-128} -6 q^{-130} +6 q^{-134} + q^{-136} -2 q^{-140} - q^{-142} + q^{-144} } |
| 5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{95}+2 q^{93}+q^{91}-2 q^{89}-q^{87}-3 q^{85}+q^{83}+6 q^{81}+6 q^{79}-4 q^{77}-10 q^{75}-10 q^{73}+2 q^{71}+19 q^{69}+19 q^{67}-32 q^{63}-32 q^{61}+2 q^{59}+41 q^{57}+55 q^{55}+9 q^{53}-68 q^{51}-83 q^{49}-11 q^{47}+81 q^{45}+114 q^{43}+34 q^{41}-112 q^{39}-160 q^{37}-42 q^{35}+137 q^{33}+203 q^{31}+69 q^{29}-161 q^{27}-261 q^{25}-103 q^{23}+180 q^{21}+316 q^{19}+152 q^{17}-172 q^{15}-354 q^{13}-219 q^{11}+135 q^9+373 q^7+281 q^5-61 q^3-345 q-325 q^{-1} -34 q^{-3} +268 q^{-5} +333 q^{-7} +131 q^{-9} -158 q^{-11} -292 q^{-13} -193 q^{-15} +30 q^{-17} +210 q^{-19} +224 q^{-21} +77 q^{-23} -114 q^{-25} -207 q^{-27} -145 q^{-29} +19 q^{-31} +172 q^{-33} +180 q^{-35} +37 q^{-37} -137 q^{-39} -184 q^{-41} -65 q^{-43} +111 q^{-45} +183 q^{-47} +74 q^{-49} -114 q^{-51} -185 q^{-53} -69 q^{-55} +124 q^{-57} +205 q^{-59} +80 q^{-61} -137 q^{-63} -235 q^{-65} -107 q^{-67} +139 q^{-69} +264 q^{-71} +147 q^{-73} -116 q^{-75} -283 q^{-77} -199 q^{-79} +71 q^{-81} +282 q^{-83} +241 q^{-85} -6 q^{-87} -248 q^{-89} -273 q^{-91} -66 q^{-93} +190 q^{-95} +273 q^{-97} +126 q^{-99} -108 q^{-101} -237 q^{-103} -166 q^{-105} +21 q^{-107} +170 q^{-109} +173 q^{-111} +46 q^{-113} -80 q^{-115} -131 q^{-117} -90 q^{-119} -2 q^{-121} +69 q^{-123} +84 q^{-125} +53 q^{-127} +10 q^{-129} -41 q^{-131} -74 q^{-133} -66 q^{-135} -19 q^{-137} +43 q^{-139} +87 q^{-141} +77 q^{-143} +7 q^{-145} -71 q^{-147} -101 q^{-149} -58 q^{-151} +28 q^{-153} +92 q^{-155} +90 q^{-157} +21 q^{-159} -58 q^{-161} -90 q^{-163} -52 q^{-165} +16 q^{-167} +63 q^{-169} +62 q^{-171} +16 q^{-173} -34 q^{-175} -51 q^{-177} -27 q^{-179} +7 q^{-181} +29 q^{-183} +28 q^{-185} +6 q^{-187} -14 q^{-189} -17 q^{-191} -7 q^{-193} +2 q^{-195} +8 q^{-197} +8 q^{-199} -4 q^{-203} -3 q^{-205} - q^{-207} +2 q^{-211} + q^{-213} - q^{-215} } |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{10}+q^8+q^6-q^4+2 q^2- q^{-6} + q^{-8} + q^{-12} +2 q^{-14} - q^{-16} - q^{-22} } |
| 1,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-4 q^{26}+8 q^{24}-12 q^{22}+18 q^{20}-28 q^{18}+34 q^{16}-36 q^{14}+39 q^{12}-42 q^{10}+42 q^8-34 q^6+32 q^4-32 q^2+30-24 q^{-2} +21 q^{-4} -12 q^{-6} +22 q^{-10} -44 q^{-12} +76 q^{-14} -102 q^{-16} +124 q^{-18} -137 q^{-20} +146 q^{-22} -140 q^{-24} +116 q^{-26} -91 q^{-28} +54 q^{-30} -20 q^{-32} -22 q^{-34} +56 q^{-36} -74 q^{-38} +92 q^{-40} -92 q^{-42} +85 q^{-44} -70 q^{-46} +52 q^{-48} -38 q^{-50} +21 q^{-52} -12 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60} } |
| 2,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}-q^{24}-2 q^{22}+q^{20}+2 q^{18}-q^{16}-3 q^{14}+4 q^{12}+4 q^{10}-5 q^8-4 q^6+5 q^4+2 q^2-3+4 q^{-4} +2 q^{-6} - q^{-8} +2 q^{-10} + q^{-12} - q^{-14} +3 q^{-16} + q^{-18} -4 q^{-20} -2 q^{-22} +2 q^{-24} -4 q^{-28} -2 q^{-30} +4 q^{-32} +3 q^{-34} -2 q^{-36} - q^{-38} +3 q^{-40} +2 q^{-42} -2 q^{-44} -3 q^{-46} + q^{-48} + q^{-50} - q^{-52} - q^{-54} + q^{-58} } |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-2 q^{20}-q^{18}+4 q^{16}-3 q^{14}-2 q^{12}+6 q^{10}-q^8-4 q^6+6 q^4+q^2-4+3 q^{-2} + q^{-4} -3 q^{-6} - q^{-8} +2 q^{-10} +2 q^{-12} -3 q^{-14} +2 q^{-16} +5 q^{-18} -3 q^{-20} + q^{-22} +5 q^{-24} -3 q^{-26} +2 q^{-30} -4 q^{-32} - q^{-34} + q^{-36} -3 q^{-38} + q^{-40} + q^{-42} - q^{-44} + q^{-46} } |
| 1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{13}+q^{11}+q^7-q^5+2 q^3+ q^{-1} - q^{-7} - q^{-9} + q^{-11} +2 q^{-15} + q^{-17} +2 q^{-19} - q^{-21} - q^{-25} - q^{-29} } |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{22}+2 q^{20}-3 q^{18}+4 q^{16}-5 q^{14}+6 q^{12}-6 q^{10}+7 q^8-6 q^6+6 q^4-3 q^2+3 q^{-2} -5 q^{-4} +9 q^{-6} -11 q^{-8} +14 q^{-10} -14 q^{-12} +13 q^{-14} -12 q^{-16} +9 q^{-18} -7 q^{-20} +3 q^{-22} + q^{-24} -3 q^{-26} +6 q^{-28} -6 q^{-30} +8 q^{-32} -7 q^{-34} +7 q^{-36} -5 q^{-38} +3 q^{-40} -3 q^{-42} + q^{-44} - q^{-46} } |
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-2 q^{32}-2 q^{30}+q^{28}+4 q^{26}+q^{24}-4 q^{22}-4 q^{20}+2 q^{18}+6 q^{16}+3 q^{14}-4 q^{12}-5 q^{10}+q^8+7 q^6+2 q^4-5 q^2-4+3 q^{-2} +5 q^{-4} - q^{-6} -6 q^{-8} - q^{-10} +5 q^{-12} +3 q^{-14} -3 q^{-16} -2 q^{-18} +3 q^{-20} +3 q^{-22} -2 q^{-24} -3 q^{-26} +3 q^{-28} +5 q^{-30} - q^{-32} -7 q^{-34} -2 q^{-36} +7 q^{-38} +6 q^{-40} -3 q^{-42} -8 q^{-44} +7 q^{-48} +4 q^{-50} -5 q^{-52} -6 q^{-54} + q^{-56} +5 q^{-58} -4 q^{-62} -2 q^{-64} +2 q^{-66} +2 q^{-68} - q^{-70} - q^{-72} + q^{-76} } |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+8 q^{38}-10 q^{36}+11 q^{34}-7 q^{32}+q^{30}+5 q^{28}-10 q^{26}+14 q^{24}-14 q^{22}+11 q^{20}-6 q^{18}-2 q^{16}+9 q^{14}-10 q^{12}+13 q^{10}-11 q^8+8 q^6-3 q^4-3 q^2+9-10 q^{-2} +9 q^{-4} -3 q^{-6} - q^{-8} +6 q^{-10} -7 q^{-12} +3 q^{-14} +5 q^{-16} -13 q^{-18} +15 q^{-20} -14 q^{-22} + q^{-24} +14 q^{-26} -26 q^{-28} +28 q^{-30} -23 q^{-32} +10 q^{-34} +6 q^{-36} -22 q^{-38} +29 q^{-40} -26 q^{-42} +18 q^{-44} -2 q^{-46} -10 q^{-48} +19 q^{-50} -13 q^{-52} +12 q^{-54} -10 q^{-58} +14 q^{-60} -9 q^{-62} +13 q^{-66} -22 q^{-68} +24 q^{-70} -15 q^{-72} - q^{-74} +15 q^{-76} -27 q^{-78} +27 q^{-80} -21 q^{-82} +6 q^{-84} +5 q^{-86} -15 q^{-88} +17 q^{-90} -14 q^{-92} +8 q^{-94} - q^{-96} -3 q^{-98} +3 q^{-100} -4 q^{-102} +3 q^{-104} - q^{-106} + q^{-108} } |
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Computer Talk
The above data is available with the Mathematica package
KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 10"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^2-11 t+17-11 t^{-1} +3 t^{-2} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^4+z^2+1} |
In[6]:=
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Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
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Out[6]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 45, 0 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+2 q^6-3 q^5+5 q^4-6 q^3+7 q^2-7 q+6-4 q^{-1} +3 q^{-2} - q^{-3} } |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-2} +z^4 a^{-4} +z^4-a^2 z^2+2 z^2 a^{-4} -z^2 a^{-6} +z^2- a^{-2} +2 a^{-4} - a^{-6} +1} |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +5 z^8 a^{-4} +2 z^8 a^{-6} +4 z^7 a^{-1} +2 z^7 a^{-3} -z^7 a^{-5} +z^7 a^{-7} -7 z^6 a^{-2} -21 z^6 a^{-4} -10 z^6 a^{-6} +4 z^6+4 a z^5-7 z^5 a^{-1} -16 z^5 a^{-3} -10 z^5 a^{-5} -5 z^5 a^{-7} +3 a^2 z^4+5 z^4 a^{-2} +26 z^4 a^{-4} +15 z^4 a^{-6} -3 z^4+a^3 z^3-3 a z^3+3 z^3 a^{-1} +17 z^3 a^{-3} +17 z^3 a^{-5} +7 z^3 a^{-7} -2 a^2 z^2-4 z^2 a^{-2} -12 z^2 a^{-4} -8 z^2 a^{-6} -2 z^2-z a^{-1} -4 z a^{-3} -6 z a^{-5} -3 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_164,}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 10"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^2-11 t+17-11 t^{-1} +3 t^{-2} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+2 q^6-3 q^5+5 q^4-6 q^3+7 q^2-7 q+6-4 q^{-1} +3 q^{-2} - q^{-3} } } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{10_164,} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{} |
Vassiliev invariants
| V2 and V3: | (1, 2) |
| V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
| The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 10 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1}
-dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)}
)
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{21}-2 q^{20}-q^{19}+6 q^{18}-5 q^{17}-6 q^{16}+15 q^{15}-5 q^{14}-16 q^{13}+22 q^{12}-q^{11}-26 q^{10}+25 q^9+6 q^8-33 q^7+24 q^6+12 q^5-34 q^4+19 q^3+14 q^2-28 q+14+10 q^{-1} -19 q^{-2} +10 q^{-3} +4 q^{-4} -10 q^{-5} +6 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{42}+2 q^{41}+q^{40}-2 q^{39}-5 q^{38}+4 q^{37}+9 q^{36}-3 q^{35}-17 q^{34}+q^{33}+23 q^{32}+7 q^{31}-30 q^{30}-15 q^{29}+33 q^{28}+25 q^{27}-31 q^{26}-36 q^{25}+28 q^{24}+40 q^{23}-19 q^{22}-45 q^{21}+13 q^{20}+42 q^{19}-3 q^{18}-41 q^{17}+32 q^{15}+9 q^{14}-27 q^{13}-11 q^{12}+15 q^{11}+17 q^{10}-8 q^9-17 q^8-2 q^7+17 q^6+8 q^5-12 q^4-13 q^3+9 q^2+8 q+2-7 q^{-1} -3 q^{-2} -3 q^{-3} +11 q^{-4} +4 q^{-5} -6 q^{-6} -13 q^{-7} +10 q^{-8} +9 q^{-9} -4 q^{-10} -11 q^{-11} +4 q^{-12} +7 q^{-13} -2 q^{-14} -3 q^{-15} - q^{-16} +3 q^{-17} - q^{-18} } |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{70}-2 q^{69}-q^{68}+2 q^{67}+q^{66}+6 q^{65}-8 q^{64}-7 q^{63}+2 q^{62}+3 q^{61}+26 q^{60}-12 q^{59}-23 q^{58}-11 q^{57}-5 q^{56}+63 q^{55}+3 q^{54}-30 q^{53}-37 q^{52}-44 q^{51}+93 q^{50}+33 q^{49}-7 q^{48}-47 q^{47}-101 q^{46}+92 q^{45}+41 q^{44}+30 q^{43}-20 q^{42}-135 q^{41}+80 q^{40}+10 q^{39}+38 q^{38}+20 q^{37}-126 q^{36}+97 q^{35}-33 q^{34}+31 q^{32}-96 q^{31}+156 q^{30}-53 q^{29}-65 q^{28}+3 q^{27}-71 q^{26}+234 q^{25}-45 q^{24}-127 q^{23}-44 q^{22}-58 q^{21}+309 q^{20}-24 q^{19}-183 q^{18}-95 q^{17}-45 q^{16}+374 q^{15}+4 q^{14}-227 q^{13}-148 q^{12}-44 q^{11}+416 q^{10}+50 q^9-236 q^8-195 q^7-72 q^6+404 q^5+101 q^4-183 q^3-201 q^2-119 q+324+122 q^{-1} -95 q^{-2} -153 q^{-3} -141 q^{-4} +208 q^{-5} +96 q^{-6} -22 q^{-7} -80 q^{-8} -125 q^{-9} +110 q^{-10} +52 q^{-11} +11 q^{-12} -25 q^{-13} -85 q^{-14} +49 q^{-15} +17 q^{-16} +16 q^{-17} - q^{-18} -44 q^{-19} +19 q^{-20} +2 q^{-21} +9 q^{-22} +3 q^{-23} -15 q^{-24} +5 q^{-25} - q^{-26} +3 q^{-27} + q^{-28} -3 q^{-29} + q^{-30} } |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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