10 122
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 122's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1627 X7,15,8,14 X15,2,16,3 X5,12,6,13 X9,19,10,18 X3,11,4,10 X17,5,18,4 X19,9,20,8 X11,16,12,17 X13,1,14,20 |
| Gauss code | -1, 3, -6, 7, -4, 1, -2, 8, -5, 6, -9, 4, -10, 2, -3, 9, -7, 5, -8, 10 |
| Dowker-Thistlethwaite code | 6 10 12 14 18 16 20 2 4 8 |
| Conway Notation | [9*.20] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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 [{3, 11}, {5, 12}, {4, 6}, {2, 5}, {7, 3}, {6, 10}, {11, 8}, {9, 7}, {8, 1}, {10, 2}, {12, 9}, {1, 4}] |
[edit Notes on presentations of 10 122]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 122"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1627 X7,15,8,14 X15,2,16,3 X5,12,6,13 X9,19,10,18 X3,11,4,10 X17,5,18,4 X19,9,20,8 X11,16,12,17 X13,1,14,20 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 3, -6, 7, -4, 1, -2, 8, -5, 6, -9, 4, -10, 2, -3, 9, -7, 5, -8, 10 |
In[6]:=
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DTCode[K]
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Out[6]=
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6 10 12 14 18 16 20 2 4 8 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[9*.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}(4,\{1,1,2,-3,2,-1,-3,2,-3,2,-3\})} |
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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{ 4, 11, 4 } |
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, 11}, {5, 12}, {4, 6}, {2, 5}, {7, 3}, {6, 10}, {11, 8}, {9, 7}, {8, 1}, {10, 2}, {12, 9}, {1, 4}] |
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 -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 t^{-2} -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 -2 z^6-z^4+2 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 \left\{3,t^2+1\right\}} |
| Determinant and Signature | { 105, 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^6-5 q^5+9 q^4-13 q^3+17 q^2-17 q+17-13 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4} } |
| 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} -z^6+a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2+3 z^2 a^{-2} -2 z^2+4 a^{-2} -2 a^{-4} -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 4 z^9 a^{-1} +4 z^9 a^{-3} +18 z^8 a^{-2} +8 z^8 a^{-4} +10 z^8+11 a z^7+9 z^7 a^{-1} +3 z^7 a^{-3} +5 z^7 a^{-5} +8 a^2 z^6-42 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -13 z^6+4 a^3 z^5-14 a z^5-32 z^5 a^{-1} -25 z^5 a^{-3} -11 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+24 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +3 z^4-2 a^3 z^3+6 a z^3+18 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +2 a^2 z^2+2 z^2+2 z a^{-3} +2 z a^{-5} -4 a^{-2} -2 a^{-4} -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^{12}-2 q^{10}+q^8+q^6-4 q^4+3 q^2-2+2 q^{-2} +3 q^{-4} +5 q^{-8} -3 q^{-10} -3 q^{-16} + q^{-18} } |
| 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^{66}-3 q^{64}+6 q^{62}-10 q^{60}+10 q^{58}-8 q^{56}+q^{54}+15 q^{52}-31 q^{50}+51 q^{48}-63 q^{46}+56 q^{44}-32 q^{42}-14 q^{40}+73 q^{38}-133 q^{36}+183 q^{34}-198 q^{32}+149 q^{30}-34 q^{28}-132 q^{26}+304 q^{24}-402 q^{22}+375 q^{20}-211 q^{18}-57 q^{16}+321 q^{14}-469 q^{12}+426 q^{10}-200 q^8-114 q^6+360 q^4-422 q^2+253+68 q^{-2} -380 q^{-4} +538 q^{-6} -459 q^{-8} +166 q^{-10} +226 q^{-12} -552 q^{-14} +696 q^{-16} -597 q^{-18} +301 q^{-20} +97 q^{-22} -438 q^{-24} +626 q^{-26} -590 q^{-28} +366 q^{-30} -28 q^{-32} -290 q^{-34} +464 q^{-36} -423 q^{-38} +194 q^{-40} +131 q^{-42} -392 q^{-44} +458 q^{-46} -299 q^{-48} -24 q^{-50} +353 q^{-52} -547 q^{-54} +517 q^{-56} -287 q^{-58} -43 q^{-60} +323 q^{-62} -459 q^{-64} +419 q^{-66} -246 q^{-68} +30 q^{-70} +130 q^{-72} -204 q^{-74} +186 q^{-76} -115 q^{-78} +45 q^{-80} +12 q^{-82} -35 q^{-84} +34 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94} } |
Further Quantum Invariants
Further quantum knot invariants for 10_122.
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^9-3 q^7+4 q^5-5 q^3+4 q+4 q^{-5} -4 q^{-7} +4 q^{-9} -4 q^{-11} + q^{-13} } |
| 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^{26}-3 q^{24}+q^{22}+7 q^{20}-14 q^{18}+7 q^{16}+23 q^{14}-40 q^{12}+q^{10}+52 q^8-43 q^6-24 q^4+54 q^2-9-35 q^{-2} +20 q^{-4} +27 q^{-6} -25 q^{-8} -20 q^{-10} +45 q^{-12} -3 q^{-14} -51 q^{-16} +40 q^{-18} +24 q^{-20} -56 q^{-22} +12 q^{-24} +36 q^{-26} -28 q^{-28} -10 q^{-30} +18 q^{-32} - q^{-34} -4 q^{-36} + q^{-38} } |
| 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^{51}-3 q^{49}+q^{47}+4 q^{45}-2 q^{43}-5 q^{41}+6 q^{39}+13 q^{37}-32 q^{35}-24 q^{33}+72 q^{31}+68 q^{29}-125 q^{27}-165 q^{25}+162 q^{23}+308 q^{21}-132 q^{19}-453 q^{17}+19 q^{15}+544 q^{13}+147 q^{11}-543 q^9-308 q^7+419 q^5+433 q^3-247 q-463 q^{-1} +53 q^{-3} +437 q^{-5} +119 q^{-7} -367 q^{-9} -253 q^{-11} +290 q^{-13} +360 q^{-15} -208 q^{-17} -441 q^{-19} +111 q^{-21} +511 q^{-23} +5 q^{-25} -542 q^{-27} -156 q^{-29} +525 q^{-31} +300 q^{-33} -420 q^{-35} -431 q^{-37} +258 q^{-39} +476 q^{-41} -67 q^{-43} -426 q^{-45} -97 q^{-47} +300 q^{-49} +184 q^{-51} -153 q^{-53} -177 q^{-55} +35 q^{-57} +118 q^{-59} +26 q^{-61} -57 q^{-63} -25 q^{-65} +12 q^{-67} +13 q^{-69} - q^{-71} -4 q^{-73} + q^{-75} } |
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^{12}-2 q^{10}+q^8+q^6-4 q^4+3 q^2-2+2 q^{-2} +3 q^{-4} +5 q^{-8} -3 q^{-10} -3 q^{-16} + q^{-18} } |
| 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^{36}-6 q^{34}+18 q^{32}-38 q^{30}+71 q^{28}-128 q^{26}+212 q^{24}-320 q^{22}+460 q^{20}-650 q^{18}+896 q^{16}-1186 q^{14}+1517 q^{12}-1834 q^{10}+2064 q^8-2134 q^6+1924 q^4-1414 q^2+586+492 q^{-2} -1670 q^{-4} +2834 q^{-6} -3788 q^{-8} +4442 q^{-10} -4690 q^{-12} +4518 q^{-14} -3944 q^{-16} +3034 q^{-18} -1890 q^{-20} +652 q^{-22} +538 q^{-24} -1540 q^{-26} +2230 q^{-28} -2566 q^{-30} +2544 q^{-32} -2260 q^{-34} +1804 q^{-36} -1300 q^{-38} +842 q^{-40} -480 q^{-42} +244 q^{-44} -104 q^{-46} +34 q^{-48} -8 q^{-50} + q^{-52} } |
| 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^{32}-2 q^{30}-q^{28}+5 q^{26}-2 q^{24}-8 q^{22}+6 q^{20}+14 q^{18}-8 q^{16}-20 q^{14}+10 q^{12}+26 q^{10}-20 q^8-20 q^6+18 q^4+10 q^2-13-11 q^{-2} +17 q^{-4} -2 q^{-6} -6 q^{-8} +13 q^{-10} +5 q^{-12} -10 q^{-14} +12 q^{-16} +21 q^{-18} -17 q^{-20} -14 q^{-22} +11 q^{-24} +10 q^{-26} -21 q^{-28} -18 q^{-30} +19 q^{-32} +9 q^{-34} -12 q^{-36} -7 q^{-38} +7 q^{-40} +9 q^{-42} -3 q^{-44} -3 q^{-46} + q^{-48} } |
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^{28}-3 q^{26}+9 q^{22}-10 q^{20}-6 q^{18}+24 q^{16}-15 q^{14}-20 q^{12}+40 q^{10}-15 q^8-29 q^6+40 q^4-13 q^2-24+18 q^{-2} +4 q^{-4} -4 q^{-6} -4 q^{-8} +22 q^{-10} +17 q^{-12} -28 q^{-14} +11 q^{-16} +23 q^{-18} -43 q^{-20} +3 q^{-22} +26 q^{-24} -34 q^{-26} +10 q^{-28} +16 q^{-30} -17 q^{-32} +7 q^{-34} +3 q^{-36} -4 q^{-38} + q^{-40} } |
| 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^{15}-2 q^{13}+2 q^{11}-2 q^9+2 q^7-4 q^5+3 q^3-3 q+ q^{-1} + q^{-3} +2 q^{-5} +4 q^{-7} + q^{-9} +5 q^{-11} -3 q^{-13} + q^{-15} -4 q^{-17} + q^{-19} -3 q^{-21} + q^{-23} } |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | 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^{34}-2 q^{32}-2 q^{30}+6 q^{28}+q^{26}-9 q^{24}-q^{22}+14 q^{20}+q^{18}-22 q^{16}+4 q^{14}+30 q^{12}-13 q^{10}-29 q^8+26 q^6+17 q^4-36 q^2-12+27 q^{-2} -8 q^{-4} -30 q^{-6} +25 q^{-8} +31 q^{-10} -24 q^{-12} +10 q^{-14} +53 q^{-16} -11 q^{-18} -32 q^{-20} +28 q^{-22} +11 q^{-24} -42 q^{-26} -19 q^{-28} +23 q^{-30} -2 q^{-32} -25 q^{-34} +9 q^{-36} +19 q^{-38} -8 q^{-40} -5 q^{-42} +9 q^{-44} - q^{-46} -3 q^{-48} + q^{-50} } |
| 1,0,0,0 | 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^{18}-2 q^{16}+2 q^{14}-q^{12}-q^{10}+2 q^8-4 q^6+3 q^4-3 q^2+ q^{-4} +2 q^{-6} +3 q^{-8} +5 q^{-10} + q^{-12} +5 q^{-14} -3 q^{-16} + q^{-18} -3 q^{-20} -3 q^{-22} + q^{-24} -3 q^{-26} + q^{-28} } |
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^{28}-3 q^{26}+6 q^{24}-11 q^{22}+18 q^{20}-26 q^{18}+36 q^{16}-45 q^{14}+50 q^{12}-52 q^{10}+43 q^8-29 q^6+8 q^4+17 q^2-42+68 q^{-2} -86 q^{-4} +100 q^{-6} -100 q^{-8} +94 q^{-10} -75 q^{-12} +54 q^{-14} -25 q^{-16} - q^{-18} +25 q^{-20} -41 q^{-22} +50 q^{-24} -54 q^{-26} +50 q^{-28} -44 q^{-30} +31 q^{-32} -21 q^{-34} +11 q^{-36} -4 q^{-38} + q^{-40} } |
| 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}-3 q^{42}-3 q^{40}+3 q^{38}+10 q^{36}+3 q^{34}-14 q^{32}-16 q^{30}+7 q^{28}+30 q^{26}+12 q^{24}-31 q^{22}-35 q^{20}+14 q^{18}+52 q^{16}+10 q^{14}-50 q^{12}-33 q^{10}+34 q^8+45 q^6-18 q^4-49 q^2-1+44 q^{-2} +11 q^{-4} -36 q^{-6} -18 q^{-8} +31 q^{-10} +27 q^{-12} -19 q^{-14} -26 q^{-16} +22 q^{-18} +38 q^{-20} -8 q^{-22} -45 q^{-24} -6 q^{-26} +48 q^{-28} +27 q^{-30} -44 q^{-32} -52 q^{-34} +18 q^{-36} +55 q^{-38} +6 q^{-40} -48 q^{-42} -28 q^{-44} +29 q^{-46} +34 q^{-48} -9 q^{-50} -24 q^{-52} -4 q^{-54} +14 q^{-56} +7 q^{-58} -4 q^{-60} -4 q^{-62} + q^{-66} } |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | 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^{38}-3 q^{36}+3 q^{34}-4 q^{32}+10 q^{30}-14 q^{28}+14 q^{26}-19 q^{24}+30 q^{22}-33 q^{20}+31 q^{18}-39 q^{16}+45 q^{14}-37 q^{12}+29 q^{10}-26 q^8+13 q^6+4 q^4-18 q^2+26-50 q^{-2} +61 q^{-4} -66 q^{-6} +76 q^{-8} -76 q^{-10} +84 q^{-12} -62 q^{-14} +70 q^{-16} -49 q^{-18} +39 q^{-20} -21 q^{-22} +6 q^{-24} -26 q^{-28} +29 q^{-30} -40 q^{-32} +38 q^{-34} -44 q^{-36} +44 q^{-38} -37 q^{-40} +33 q^{-42} -25 q^{-44} +18 q^{-46} -11 q^{-48} +7 q^{-50} -4 q^{-52} + q^{-54} } |
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^{66}-3 q^{64}+6 q^{62}-10 q^{60}+10 q^{58}-8 q^{56}+q^{54}+15 q^{52}-31 q^{50}+51 q^{48}-63 q^{46}+56 q^{44}-32 q^{42}-14 q^{40}+73 q^{38}-133 q^{36}+183 q^{34}-198 q^{32}+149 q^{30}-34 q^{28}-132 q^{26}+304 q^{24}-402 q^{22}+375 q^{20}-211 q^{18}-57 q^{16}+321 q^{14}-469 q^{12}+426 q^{10}-200 q^8-114 q^6+360 q^4-422 q^2+253+68 q^{-2} -380 q^{-4} +538 q^{-6} -459 q^{-8} +166 q^{-10} +226 q^{-12} -552 q^{-14} +696 q^{-16} -597 q^{-18} +301 q^{-20} +97 q^{-22} -438 q^{-24} +626 q^{-26} -590 q^{-28} +366 q^{-30} -28 q^{-32} -290 q^{-34} +464 q^{-36} -423 q^{-38} +194 q^{-40} +131 q^{-42} -392 q^{-44} +458 q^{-46} -299 q^{-48} -24 q^{-50} +353 q^{-52} -547 q^{-54} +517 q^{-56} -287 q^{-58} -43 q^{-60} +323 q^{-62} -459 q^{-64} +419 q^{-66} -246 q^{-68} +30 q^{-70} +130 q^{-72} -204 q^{-74} +186 q^{-76} -115 q^{-78} +45 q^{-80} +12 q^{-82} -35 q^{-84} +34 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94} } |
.
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["10 122"];
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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 -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 t^{-2} -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 -2 z^6-z^4+2 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 \left\{3,t^2+1\right\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 105, 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^6-5 q^5+9 q^4-13 q^3+17 q^2-17 q+17-13 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4} } |
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} -z^6+a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2+3 z^2 a^{-2} -2 z^2+4 a^{-2} -2 a^{-4} -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 4 z^9 a^{-1} +4 z^9 a^{-3} +18 z^8 a^{-2} +8 z^8 a^{-4} +10 z^8+11 a z^7+9 z^7 a^{-1} +3 z^7 a^{-3} +5 z^7 a^{-5} +8 a^2 z^6-42 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -13 z^6+4 a^3 z^5-14 a z^5-32 z^5 a^{-1} -25 z^5 a^{-3} -11 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+24 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +3 z^4-2 a^3 z^3+6 a z^3+18 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +2 a^2 z^2+2 z^2+2 z a^{-3} +2 z a^{-5} -4 a^{-2} -2 a^{-4} -1} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n185,}
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["10 122"];
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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 -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 t^{-2} -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^6-5 q^5+9 q^4-13 q^3+17 q^2-17 q+17-13 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4} } } |
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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{K11n185,} |
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: | (2, 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 10 122. 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^{18}-5 q^{17}+3 q^{16}+20 q^{15}-33 q^{14}-15 q^{13}+84 q^{12}-57 q^{11}-83 q^{10}+164 q^9-41 q^8-174 q^7+212 q^6+7 q^5-239 q^4+207 q^3+59 q^2-246 q+152+85 q^{-1} -183 q^{-2} +74 q^{-3} +66 q^{-4} -88 q^{-5} +23 q^{-6} +25 q^{-7} -25 q^{-8} +7 q^{-9} +4 q^{-10} -4 q^{-11} + q^{-12} } |
| 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^{36}-5 q^{35}+3 q^{34}+14 q^{33}-42 q^{31}-29 q^{30}+97 q^{29}+92 q^{28}-125 q^{27}-241 q^{26}+121 q^{25}+429 q^{24}-9 q^{23}-638 q^{22}-208 q^{21}+788 q^{20}+534 q^{19}-856 q^{18}-897 q^{17}+799 q^{16}+1254 q^{15}-631 q^{14}-1578 q^{13}+413 q^{12}+1801 q^{11}-125 q^{10}-1978 q^9-139 q^8+2034 q^7+443 q^6-2048 q^5-682 q^4+1920 q^3+929 q^2-1730 q-1066+1404 q^{-1} +1145 q^{-2} -1050 q^{-3} -1080 q^{-4} +677 q^{-5} +910 q^{-6} -360 q^{-7} -683 q^{-8} +152 q^{-9} +438 q^{-10} -39 q^{-11} -243 q^{-12} +6 q^{-13} +111 q^{-14} + q^{-15} -50 q^{-16} +10 q^{-17} +15 q^{-18} -7 q^{-19} -5 q^{-20} +3 q^{-21} +4 q^{-22} -4 q^{-23} + q^{-24} } |
| 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^{60}-5 q^{59}+3 q^{58}+14 q^{57}-6 q^{56}-9 q^{55}-56 q^{54}+8 q^{53}+126 q^{52}+73 q^{51}+14 q^{50}-370 q^{49}-285 q^{48}+281 q^{47}+586 q^{46}+741 q^{45}-701 q^{44}-1468 q^{43}-673 q^{42}+880 q^{41}+2997 q^{40}+747 q^{39}-2369 q^{38}-3644 q^{37}-1547 q^{36}+4943 q^{35}+4858 q^{34}+171 q^{33}-6159 q^{32}-7414 q^{31}+2969 q^{30}+8585 q^{29}+6684 q^{28}-4473 q^{27}-13248 q^{26}-3251 q^{25}+8247 q^{24}+13620 q^{23}+1429 q^{22}-15524 q^{21}-10334 q^{20}+4000 q^{19}+17807 q^{18}+8401 q^{17}-14277 q^{16}-15584 q^{15}-1524 q^{14}+19134 q^{13}+14190 q^{12}-11277 q^{11}-18735 q^{10}-6782 q^9+18435 q^8+18492 q^7-7057 q^6-19850 q^5-11705 q^4+15253 q^3+20794 q^2-1299 q-17650-15351 q^{-1} +8955 q^{-2} +19206 q^{-3} +4533 q^{-4} -11417 q^{-5} -15187 q^{-6} +1703 q^{-7} +13052 q^{-8} +7028 q^{-9} -3926 q^{-10} -10460 q^{-11} -2413 q^{-12} +5657 q^{-13} +5148 q^{-14} +477 q^{-15} -4606 q^{-16} -2335 q^{-17} +1225 q^{-18} +1994 q^{-19} +1092 q^{-20} -1214 q^{-21} -882 q^{-22} +41 q^{-23} +342 q^{-24} +451 q^{-25} -215 q^{-26} -141 q^{-27} -5 q^{-28} -13 q^{-29} +97 q^{-30} -43 q^{-31} -2 q^{-32} +12 q^{-33} -17 q^{-34} +13 q^{-35} -9 q^{-36} +3 q^{-37} +4 q^{-38} -4 q^{-39} + q^{-40} } |
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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