10 105
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 105's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X4251 X12,4,13,3 X20,8,1,7 X16,5,17,6 X6,15,7,16 X10,17,11,18 X18,9,19,10 X8,14,9,13 X14,20,15,19 X2,12,3,11 |
| Gauss code | 1, -10, 2, -1, 4, -5, 3, -8, 7, -6, 10, -2, 8, -9, 5, -4, 6, -7, 9, -3 |
| Dowker-Thistlethwaite code | 4 12 16 20 18 2 8 6 10 14 |
| Conway Notation | [21:20:20] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 |
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 [{3, 10}, {2, 6}, {1, 3}, {12, 8}, {9, 7}, {8, 5}, {6, 11}, {10, 12}, {4, 9}, {5, 2}, {11, 4}, {7, 1}] |
[edit Notes on presentations of 10 105]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 105"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X12,4,13,3 X20,8,1,7 X16,5,17,6 X6,15,7,16 X10,17,11,18 X18,9,19,10 X8,14,9,13 X14,20,15,19 X2,12,3,11 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -10, 2, -1, 4, -5, 3, -8, 7, -6, 10, -2, 8, -9, 5, -4, 6, -7, 9, -3 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 16 20 18 2 8 6 10 14 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[21:20:20] |
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,-2,1,3,2,2,-4,-3,2,-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[{3, 10}, {2, 6}, {1, 3}, {12, 8}, {9, 7}, {8, 5}, {6, 11}, {10, 12}, {4, 9}, {5, 2}, {11, 4}, {7, 1}] |
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 t^3-8 t^2+22 t-29+22 t^{-1} -8 t^{-2} + t^{-3} } |
| 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 z^6-2 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 | { 91, 2 } |
| 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^7-4 q^6+8 q^5-12 q^4+15 q^3-15 q^2+14 q-11+7 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^6 a^{-2} +2 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -3 z^2+a^2+ a^{-2} -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 2 z^9 a^{-1} +2 z^9 a^{-3} +11 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8+3 a z^7+6 z^7 a^{-1} +13 z^7 a^{-3} +10 z^7 a^{-5} +a^2 z^6-19 z^6 a^{-2} -3 z^6 a^{-4} +8 z^6 a^{-6} -7 z^6-8 a z^5-24 z^5 a^{-1} -33 z^5 a^{-3} -13 z^5 a^{-5} +4 z^5 a^{-7} -3 a^2 z^4+2 z^4 a^{-2} -9 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -z^4+7 a z^3+18 z^3 a^{-1} +19 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} +3 a^2 z^2+4 z^2 a^{-2} +5 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1} |
| The A2 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^{10}-q^6+3 q^4-2 q^2+2 q^{-2} -3 q^{-4} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} - q^{-20} + q^{-22} } |
| 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^{46}-2 q^{44}+6 q^{42}-10 q^{40}+13 q^{38}-13 q^{36}+4 q^{34}+18 q^{32}-47 q^{30}+82 q^{28}-97 q^{26}+75 q^{24}-8 q^{22}-95 q^{20}+200 q^{18}-257 q^{16}+230 q^{14}-110 q^{12}-81 q^{10}+262 q^8-360 q^6+332 q^4-170 q^2-47+233 q^{-2} -311 q^{-4} +242 q^{-6} -67 q^{-8} -131 q^{-10} +261 q^{-12} -259 q^{-14} +124 q^{-16} +92 q^{-18} -293 q^{-20} +397 q^{-22} -359 q^{-24} +181 q^{-26} +73 q^{-28} -324 q^{-30} +472 q^{-32} -465 q^{-34} +308 q^{-36} -48 q^{-38} -210 q^{-40} +372 q^{-42} -381 q^{-44} +242 q^{-46} -23 q^{-48} -174 q^{-50} +266 q^{-52} -210 q^{-54} +44 q^{-56} +152 q^{-58} -277 q^{-60} +283 q^{-62} -164 q^{-64} -29 q^{-66} +203 q^{-68} -305 q^{-70} +301 q^{-72} -199 q^{-74} +53 q^{-76} +85 q^{-78} -173 q^{-80} +192 q^{-82} -159 q^{-84} +96 q^{-86} -26 q^{-88} -29 q^{-90} +57 q^{-92} -66 q^{-94} +54 q^{-96} -33 q^{-98} +16 q^{-100} + q^{-102} -8 q^{-104} +10 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} } |
Further Quantum Invariants
Further quantum knot invariants for 10_105.
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+4 q^3-4 q+3 q^{-1} - q^{-3} +3 q^{-7} -4 q^{-9} +4 q^{-11} -3 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^{22}-2 q^{20}-q^{18}+9 q^{16}-7 q^{14}-14 q^{12}+26 q^{10}+q^8-38 q^6+28 q^4+23 q^2-44+10 q^{-2} +33 q^{-4} -26 q^{-6} -12 q^{-8} +24 q^{-10} +5 q^{-12} -26 q^{-14} +2 q^{-16} +36 q^{-18} -27 q^{-20} -24 q^{-22} +46 q^{-24} -11 q^{-26} -31 q^{-28} +28 q^{-30} +2 q^{-32} -15 q^{-34} +8 q^{-36} + q^{-38} -3 q^{-40} + q^{-42} } |
| 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^{45}-2 q^{43}-q^{41}+4 q^{39}+6 q^{37}-10 q^{35}-20 q^{33}+15 q^{31}+48 q^{29}-4 q^{27}-88 q^{25}-38 q^{23}+126 q^{21}+108 q^{19}-128 q^{17}-202 q^{15}+86 q^{13}+289 q^{11}-4 q^9-330 q^7-106 q^5+325 q^3+209 q-278 q^{-1} -279 q^{-3} +199 q^{-5} +314 q^{-7} -110 q^{-9} -315 q^{-11} +27 q^{-13} +295 q^{-15} +53 q^{-17} -249 q^{-19} -135 q^{-21} +190 q^{-23} +212 q^{-25} -111 q^{-27} -283 q^{-29} +9 q^{-31} +328 q^{-33} +103 q^{-35} -326 q^{-37} -213 q^{-39} +284 q^{-41} +279 q^{-43} -197 q^{-45} -296 q^{-47} +100 q^{-49} +262 q^{-51} -24 q^{-53} -192 q^{-55} -15 q^{-57} +115 q^{-59} +27 q^{-61} -61 q^{-63} -20 q^{-65} +30 q^{-67} +6 q^{-69} -10 q^{-71} -3 q^{-73} +5 q^{-75} + q^{-77} -3 q^{-79} + q^{-81} } |
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^6+3 q^4-2 q^2+2 q^{-2} -3 q^{-4} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} - q^{-20} + q^{-22} } |
| 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^{28}-2 q^{24}-q^{22}+5 q^{20}+5 q^{18}-7 q^{16}-9 q^{14}+9 q^{12}+10 q^{10}-12 q^8-14 q^6+13 q^4+19 q^2-14-11 q^{-2} +19 q^{-4} +5 q^{-6} -13 q^{-8} -3 q^{-10} +9 q^{-12} -5 q^{-14} -4 q^{-16} +11 q^{-18} -4 q^{-20} -12 q^{-22} +12 q^{-24} +15 q^{-26} -20 q^{-28} -10 q^{-30} +20 q^{-32} +7 q^{-34} -17 q^{-36} -9 q^{-38} +16 q^{-40} +4 q^{-42} -10 q^{-44} - q^{-46} +5 q^{-48} +2 q^{-50} -3 q^{-52} - q^{-54} + q^{-56} } |
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^{20}-2 q^{18}+2 q^{16}+4 q^{14}-10 q^{12}+8 q^{10}+9 q^8-24 q^6+18 q^4+12 q^2-32+21 q^{-2} +13 q^{-4} -29 q^{-6} +9 q^{-8} +14 q^{-10} -12 q^{-12} -5 q^{-14} +6 q^{-16} +12 q^{-18} -13 q^{-20} -9 q^{-22} +32 q^{-24} -18 q^{-26} -18 q^{-28} +34 q^{-30} -15 q^{-32} -17 q^{-34} +23 q^{-36} -5 q^{-38} -10 q^{-40} +9 q^{-42} -3 q^{-46} + q^{-48} } |
| 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^9-q^7+3 q^5-3 q^3+2 q-2 q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} + q^{-9} - q^{-11} +2 q^{-13} - q^{-15} +3 q^{-17} -3 q^{-19} +3 q^{-21} -2 q^{-23} - q^{-27} + 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^{20}-2 q^{18}+6 q^{16}-10 q^{14}+18 q^{12}-26 q^{10}+33 q^8-38 q^6+40 q^4-36 q^2+26-13 q^{-2} -5 q^{-4} +25 q^{-6} -45 q^{-8} +62 q^{-10} -72 q^{-12} +77 q^{-14} -72 q^{-16} +62 q^{-18} -45 q^{-20} +27 q^{-22} -6 q^{-24} -12 q^{-26} +26 q^{-28} -36 q^{-30} +39 q^{-32} -39 q^{-34} +33 q^{-36} -25 q^{-38} +18 q^{-40} -11 q^{-42} +6 q^{-44} -3 q^{-46} + q^{-48} } |
| 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^{34}-2 q^{30}-2 q^{28}+4 q^{26}+7 q^{24}-2 q^{22}-14 q^{20}-6 q^{18}+19 q^{16}+20 q^{14}-14 q^{12}-33 q^{10}-2 q^8+40 q^6+22 q^4-32 q^2-36+14 q^{-2} +42 q^{-4} +5 q^{-6} -36 q^{-8} -17 q^{-10} +25 q^{-12} +21 q^{-14} -17 q^{-16} -22 q^{-18} +11 q^{-20} +25 q^{-22} -6 q^{-24} -28 q^{-26} +31 q^{-30} +9 q^{-32} -30 q^{-34} -20 q^{-36} +28 q^{-38} +32 q^{-40} -17 q^{-42} -41 q^{-44} +40 q^{-48} +18 q^{-50} -27 q^{-52} -31 q^{-54} +8 q^{-56} +28 q^{-58} +8 q^{-60} -15 q^{-62} -14 q^{-64} +3 q^{-66} +10 q^{-68} +3 q^{-70} -3 q^{-72} -3 q^{-74} + q^{-78} } |
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^{46}-2 q^{44}+6 q^{42}-10 q^{40}+13 q^{38}-13 q^{36}+4 q^{34}+18 q^{32}-47 q^{30}+82 q^{28}-97 q^{26}+75 q^{24}-8 q^{22}-95 q^{20}+200 q^{18}-257 q^{16}+230 q^{14}-110 q^{12}-81 q^{10}+262 q^8-360 q^6+332 q^4-170 q^2-47+233 q^{-2} -311 q^{-4} +242 q^{-6} -67 q^{-8} -131 q^{-10} +261 q^{-12} -259 q^{-14} +124 q^{-16} +92 q^{-18} -293 q^{-20} +397 q^{-22} -359 q^{-24} +181 q^{-26} +73 q^{-28} -324 q^{-30} +472 q^{-32} -465 q^{-34} +308 q^{-36} -48 q^{-38} -210 q^{-40} +372 q^{-42} -381 q^{-44} +242 q^{-46} -23 q^{-48} -174 q^{-50} +266 q^{-52} -210 q^{-54} +44 q^{-56} +152 q^{-58} -277 q^{-60} +283 q^{-62} -164 q^{-64} -29 q^{-66} +203 q^{-68} -305 q^{-70} +301 q^{-72} -199 q^{-74} +53 q^{-76} +85 q^{-78} -173 q^{-80} +192 q^{-82} -159 q^{-84} +96 q^{-86} -26 q^{-88} -29 q^{-90} +57 q^{-92} -66 q^{-94} +54 q^{-96} -33 q^{-98} +16 q^{-100} + q^{-102} -8 q^{-104} +10 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} } |
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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["10 105"];
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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 t^3-8 t^2+22 t-29+22 t^{-1} -8 t^{-2} + t^{-3} } |
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 z^6-2 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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{ 91, 2 } |
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^7-4 q^6+8 q^5-12 q^4+15 q^3-15 q^2+14 q-11+7 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^6 a^{-2} +2 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -3 z^2+a^2+ a^{-2} -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 2 z^9 a^{-1} +2 z^9 a^{-3} +11 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8+3 a z^7+6 z^7 a^{-1} +13 z^7 a^{-3} +10 z^7 a^{-5} +a^2 z^6-19 z^6 a^{-2} -3 z^6 a^{-4} +8 z^6 a^{-6} -7 z^6-8 a z^5-24 z^5 a^{-1} -33 z^5 a^{-3} -13 z^5 a^{-5} +4 z^5 a^{-7} -3 a^2 z^4+2 z^4 a^{-2} -9 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -z^4+7 a z^3+18 z^3 a^{-1} +19 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} +3 a^2 z^2+4 z^2 a^{-2} +5 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n163,}
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]:=
|
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["10 105"];
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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 t^3-8 t^2+22 t-29+22 t^{-1} -8 t^{-2} + t^{-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^7-4 q^6+8 q^5-12 q^4+15 q^3-15 q^2+14 q-11+7 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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{K11n163,} |
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, 0) |
| 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=} 2 is the signature of 10 105. 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^{20}-4 q^{19}+4 q^{18}+8 q^{17}-27 q^{16}+21 q^{15}+34 q^{14}-86 q^{13}+41 q^{12}+91 q^{11}-156 q^{10}+38 q^9+154 q^8-190 q^7+10 q^6+185 q^5-171 q^4-26 q^3+171 q^2-112 q-49+117 q^{-1} -45 q^{-2} -44 q^{-3} +51 q^{-4} -6 q^{-5} -19 q^{-6} +11 q^{-7} + q^{-8} -3 q^{-9} + q^{-10} } |
| 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}-4 q^{38}+4 q^{37}+4 q^{36}-7 q^{35}-11 q^{34}+20 q^{33}+28 q^{32}-57 q^{31}-52 q^{30}+108 q^{29}+116 q^{28}-187 q^{27}-229 q^{26}+276 q^{25}+402 q^{24}-349 q^{23}-625 q^{22}+375 q^{21}+878 q^{20}-344 q^{19}-1122 q^{18}+262 q^{17}+1307 q^{16}-119 q^{15}-1441 q^{14}-30 q^{13}+1479 q^{12}+204 q^{11}-1463 q^{10}-355 q^9+1365 q^8+506 q^7-1221 q^6-623 q^5+1023 q^4+711 q^3-797 q^2-738 q+545+712 q^{-1} -310 q^{-2} -622 q^{-3} +114 q^{-4} +488 q^{-5} +16 q^{-6} -329 q^{-7} -89 q^{-8} +200 q^{-9} +90 q^{-10} -93 q^{-11} -71 q^{-12} +36 q^{-13} +40 q^{-14} -9 q^{-15} -19 q^{-16} +3 q^{-17} +5 q^{-18} + q^{-19} -3 q^{-20} + q^{-21} } |
| 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}-4 q^{63}+4 q^{62}+4 q^{61}-11 q^{60}+9 q^{59}-12 q^{58}+24 q^{57}+7 q^{56}-76 q^{55}+40 q^{54}+10 q^{53}+129 q^{52}-10 q^{51}-373 q^{50}+22 q^{49}+199 q^{48}+630 q^{47}+47 q^{46}-1278 q^{45}-515 q^{44}+512 q^{43}+2147 q^{42}+863 q^{41}-2840 q^{40}-2437 q^{39}+91 q^{38}+4733 q^{37}+3454 q^{36}-3927 q^{35}-5700 q^{34}-2233 q^{33}+6969 q^{32}+7567 q^{31}-3134 q^{30}-8620 q^{29}-6107 q^{28}+7270 q^{27}+11269 q^{26}-621 q^{25}-9553 q^{24}-9721 q^{23}+5650 q^{22}+12982 q^{21}+2218 q^{20}-8484 q^{19}-11785 q^{18}+3161 q^{17}+12649 q^{16}+4506 q^{15}-6213 q^{14}-12283 q^{13}+428 q^{12}+10833 q^{11}+6174 q^{10}-3204 q^9-11434 q^8-2333 q^7+7759 q^6+6975 q^5+220 q^4-9063 q^3-4430 q^2+3766 q+6190+3049 q^{-1} -5330 q^{-2} -4744 q^{-3} +101 q^{-4} +3740 q^{-5} +3936 q^{-6} -1629 q^{-7} -3115 q^{-8} -1653 q^{-9} +1024 q^{-10} +2760 q^{-11} +363 q^{-12} -1016 q^{-13} -1372 q^{-14} -357 q^{-15} +1071 q^{-16} +543 q^{-17} +51 q^{-18} -489 q^{-19} -401 q^{-20} +188 q^{-21} +168 q^{-22} +150 q^{-23} -62 q^{-24} -131 q^{-25} +10 q^{-26} +8 q^{-27} +42 q^{-28} +3 q^{-29} -22 q^{-30} +3 q^{-31} -3 q^{-32} +5 q^{-33} + q^{-34} -3 q^{-35} + q^{-36} } |
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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