10 37
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![]() (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 37's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X4251 X10,4,11,3 X12,8,13,7 X8,12,9,11 X18,15,19,16 X16,5,17,6 X6,17,7,18 X20,13,1,14 X14,19,15,20 X2,10,3,9 |
Gauss code | 1, -10, 2, -1, 6, -7, 3, -4, 10, -2, 4, -3, 8, -9, 5, -6, 7, -5, 9, -8 |
Dowker-Thistlethwaite code | 4 10 16 12 2 8 20 18 6 14 |
Conway Notation | [2332] |
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, 11}, {2, 4}, {1, 3}, {5, 2}, {4, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 12}, {11, 5}, {12, 6}, {10, 1}] |
[edit Notes on presentations of 10 37]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 37"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X10,4,11,3 X12,8,13,7 X8,12,9,11 X18,15,19,16 X16,5,17,6 X6,17,7,18 X20,13,1,14 X14,19,15,20 X2,10,3,9 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -10, 2, -1, 6, -7, 3, -4, 10, -2, 4, -3, 8, -9, 5, -6, 7, -5, 9, -8 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 16 12 2 8 20 18 6 14 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[2332] |
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,-1,-2,1,3,-2,3,4,-3,4,4\})} |
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, 11}, {2, 4}, {1, 3}, {5, 2}, {4, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 12}, {11, 5}, {12, 6}, {10, 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 4 t^2-13 t+19-13 t^{-1} +4 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 4 z^4+3 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 | { 53, 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^5+2 q^4-4 q^3+7 q^2-8 q+9-8 q^{-1} +7 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} } |
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^2 a^4-a^4+z^4 a^2+z^2 a^2+a^2+2 z^4+3 z^2+1+z^4 a^{-2} +z^2 a^{-2} + a^{-2} -z^2 a^{-4} - a^{-4} } |
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 a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7+2 z^7 a^{-3} +2 a^4 z^6-3 a^2 z^6-3 z^6 a^{-2} +2 z^6 a^{-4} -10 z^6+a^5 z^5-2 a^3 z^5-2 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+2 a^2 z^4+2 z^4 a^{-2} -5 z^4 a^{-4} +14 z^4-3 a^5 z^3-3 a^3 z^3+a z^3+z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +3 a^4 z^2+3 z^2 a^{-4} -6 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} +2 z a^{-3} +2 z a^{-5} -a^4-a^2- a^{-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^{16}-2 q^{10}+2 q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} +2 q^{-8} -2 q^{-10} - q^{-16} } |
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^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-2 q^{70}-3 q^{68}+8 q^{66}-13 q^{64}+15 q^{62}-15 q^{60}+8 q^{58}+q^{56}-16 q^{54}+31 q^{52}-41 q^{50}+37 q^{48}-25 q^{46}-2 q^{44}+30 q^{42}-52 q^{40}+62 q^{38}-47 q^{36}+19 q^{34}+17 q^{32}-46 q^{30}+48 q^{28}-29 q^{26}+3 q^{24}+27 q^{22}-38 q^{20}+31 q^{18}+q^{16}-34 q^{14}+62 q^{12}-69 q^{10}+46 q^8-6 q^6-39 q^4+75 q^2-85+75 q^{-2} -39 q^{-4} -6 q^{-6} +46 q^{-8} -69 q^{-10} +62 q^{-12} -34 q^{-14} + q^{-16} +31 q^{-18} -38 q^{-20} +27 q^{-22} +3 q^{-24} -29 q^{-26} +48 q^{-28} -46 q^{-30} +17 q^{-32} +19 q^{-34} -47 q^{-36} +62 q^{-38} -52 q^{-40} +30 q^{-42} -2 q^{-44} -25 q^{-46} +37 q^{-48} -41 q^{-50} +31 q^{-52} -16 q^{-54} + q^{-56} +8 q^{-58} -15 q^{-60} +15 q^{-62} -13 q^{-64} +8 q^{-66} -3 q^{-68} -2 q^{-70} +3 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} } |
A1 Invariants.
Weight | Invariant |
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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^{11}+q^9-2 q^7+3 q^5-q^3+q+ q^{-1} - q^{-3} +3 q^{-5} -2 q^{-7} + q^{-9} - q^{-11} } |
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^{32}-q^{30}-q^{28}+3 q^{26}-3 q^{24}-3 q^{22}+9 q^{20}-4 q^{18}-10 q^{16}+13 q^{14}-14 q^{10}+9 q^8+5 q^6-8 q^4+2 q^2+7+2 q^{-2} -8 q^{-4} +5 q^{-6} +9 q^{-8} -14 q^{-10} +13 q^{-14} -10 q^{-16} -4 q^{-18} +9 q^{-20} -3 q^{-22} -3 q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-32} } |
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^{63}+q^{61}+q^{59}-2 q^{55}+q^{53}+2 q^{51}-3 q^{49}-4 q^{47}+6 q^{45}+9 q^{43}-8 q^{41}-18 q^{39}+8 q^{37}+31 q^{35}-2 q^{33}-41 q^{31}-15 q^{29}+50 q^{27}+27 q^{25}-48 q^{23}-44 q^{21}+39 q^{19}+52 q^{17}-26 q^{15}-53 q^{13}+14 q^{11}+47 q^9+2 q^7-36 q^5-13 q^3+26 q+26 q^{-1} -13 q^{-3} -36 q^{-5} +2 q^{-7} +47 q^{-9} +14 q^{-11} -53 q^{-13} -26 q^{-15} +52 q^{-17} +39 q^{-19} -44 q^{-21} -48 q^{-23} +27 q^{-25} +50 q^{-27} -15 q^{-29} -41 q^{-31} -2 q^{-33} +31 q^{-35} +8 q^{-37} -18 q^{-39} -8 q^{-41} +9 q^{-43} +6 q^{-45} -4 q^{-47} -3 q^{-49} +2 q^{-51} + q^{-53} -2 q^{-55} + q^{-59} + q^{-61} - q^{-63} } |
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^{104}-q^{102}-q^{100}-q^{96}+4 q^{94}-q^{92}+2 q^{88}-6 q^{86}+3 q^{84}-6 q^{82}+2 q^{80}+15 q^{78}-3 q^{76}+q^{74}-29 q^{72}-14 q^{70}+33 q^{68}+31 q^{66}+35 q^{64}-59 q^{62}-82 q^{60}-2 q^{58}+69 q^{56}+143 q^{54}-15 q^{52}-156 q^{50}-133 q^{48}+16 q^{46}+258 q^{44}+133 q^{42}-128 q^{40}-261 q^{38}-137 q^{36}+248 q^{34}+268 q^{32}+12 q^{30}-264 q^{28}-261 q^{26}+126 q^{24}+273 q^{22}+127 q^{20}-158 q^{18}-256 q^{16}-4 q^{14}+174 q^{12}+161 q^{10}-35 q^8-175 q^6-91 q^4+67 q^2+159+67 q^{-2} -91 q^{-4} -175 q^{-6} -35 q^{-8} +161 q^{-10} +174 q^{-12} -4 q^{-14} -256 q^{-16} -158 q^{-18} +127 q^{-20} +273 q^{-22} +126 q^{-24} -261 q^{-26} -264 q^{-28} +12 q^{-30} +268 q^{-32} +248 q^{-34} -137 q^{-36} -261 q^{-38} -128 q^{-40} +133 q^{-42} +258 q^{-44} +16 q^{-46} -133 q^{-48} -156 q^{-50} -15 q^{-52} +143 q^{-54} +69 q^{-56} -2 q^{-58} -82 q^{-60} -59 q^{-62} +35 q^{-64} +31 q^{-66} +33 q^{-68} -14 q^{-70} -29 q^{-72} + q^{-74} -3 q^{-76} +15 q^{-78} +2 q^{-80} -6 q^{-82} +3 q^{-84} -6 q^{-86} +2 q^{-88} - q^{-92} +4 q^{-94} - q^{-96} - q^{-100} - q^{-102} + q^{-104} } |
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^{155}+q^{153}+q^{151}+q^{147}-q^{145}-4 q^{143}-q^{141}+2 q^{139}+5 q^{135}+6 q^{133}-3 q^{131}-7 q^{129}-7 q^{127}-7 q^{125}+4 q^{123}+21 q^{121}+19 q^{119}-q^{117}-24 q^{115}-42 q^{113}-26 q^{111}+22 q^{109}+72 q^{107}+73 q^{105}+6 q^{103}-89 q^{101}-142 q^{99}-86 q^{97}+68 q^{95}+222 q^{93}+222 q^{91}+23 q^{89}-258 q^{87}-400 q^{85}-220 q^{83}+202 q^{81}+571 q^{79}+520 q^{77}-21 q^{75}-661 q^{73}-844 q^{71}-319 q^{69}+593 q^{67}+1155 q^{65}+750 q^{63}-381 q^{61}-1297 q^{59}-1189 q^{57}-17 q^{55}+1284 q^{53}+1545 q^{51}+447 q^{49}-1080 q^{47}-1710 q^{45}-867 q^{43}+751 q^{41}+1692 q^{39}+1151 q^{37}-378 q^{35}-1510 q^{33}-1271 q^{31}+39 q^{29}+1210 q^{27}+1247 q^{25}+223 q^{23}-881 q^{21}-1120 q^{19}-381 q^{17}+574 q^{15}+942 q^{13}+489 q^{11}-314 q^9-779 q^7-562 q^5+101 q^3+650 q+650 q^{-1} +101 q^{-3} -562 q^{-5} -779 q^{-7} -314 q^{-9} +489 q^{-11} +942 q^{-13} +574 q^{-15} -381 q^{-17} -1120 q^{-19} -881 q^{-21} +223 q^{-23} +1247 q^{-25} +1210 q^{-27} +39 q^{-29} -1271 q^{-31} -1510 q^{-33} -378 q^{-35} +1151 q^{-37} +1692 q^{-39} +751 q^{-41} -867 q^{-43} -1710 q^{-45} -1080 q^{-47} +447 q^{-49} +1545 q^{-51} +1284 q^{-53} -17 q^{-55} -1189 q^{-57} -1297 q^{-59} -381 q^{-61} +750 q^{-63} +1155 q^{-65} +593 q^{-67} -319 q^{-69} -844 q^{-71} -661 q^{-73} -21 q^{-75} +520 q^{-77} +571 q^{-79} +202 q^{-81} -220 q^{-83} -400 q^{-85} -258 q^{-87} +23 q^{-89} +222 q^{-91} +222 q^{-93} +68 q^{-95} -86 q^{-97} -142 q^{-99} -89 q^{-101} +6 q^{-103} +73 q^{-105} +72 q^{-107} +22 q^{-109} -26 q^{-111} -42 q^{-113} -24 q^{-115} - q^{-117} +19 q^{-119} +21 q^{-121} +4 q^{-123} -7 q^{-125} -7 q^{-127} -7 q^{-129} -3 q^{-131} +6 q^{-133} +5 q^{-135} +2 q^{-139} - q^{-141} -4 q^{-143} - q^{-145} + q^{-147} + q^{-151} + q^{-153} - q^{-155} } |
A2 Invariants.
Weight | Invariant |
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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^{16}-2 q^{10}+2 q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} +2 q^{-8} -2 q^{-10} - q^{-16} } |
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^{44}-2 q^{42}+6 q^{40}-12 q^{38}+21 q^{36}-34 q^{34}+50 q^{32}-74 q^{30}+102 q^{28}-134 q^{26}+166 q^{24}-194 q^{22}+209 q^{20}-206 q^{18}+172 q^{16}-114 q^{14}+27 q^{12}+70 q^{10}-180 q^8+282 q^6-359 q^4+420 q^2-426+420 q^{-2} -359 q^{-4} +282 q^{-6} -180 q^{-8} +70 q^{-10} +27 q^{-12} -114 q^{-14} +172 q^{-16} -206 q^{-18} +209 q^{-20} -194 q^{-22} +166 q^{-24} -134 q^{-26} +102 q^{-28} -74 q^{-30} +50 q^{-32} -34 q^{-34} +21 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} } |
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^{42}-q^{38}+2 q^{34}-4 q^{30}-q^{28}+6 q^{26}-8 q^{22}-2 q^{20}+7 q^{18}+2 q^{16}-11 q^{14}-q^{12}+6 q^{10}-2 q^8-3 q^6+5 q^4+6 q^2+2+6 q^{-2} +5 q^{-4} -3 q^{-6} -2 q^{-8} +6 q^{-10} - q^{-12} -11 q^{-14} +2 q^{-16} +7 q^{-18} -2 q^{-20} -8 q^{-22} +6 q^{-26} - q^{-28} -4 q^{-30} +2 q^{-34} - q^{-38} + q^{-42} } |
A3 Invariants.
Weight | Invariant |
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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^{34}-q^{32}+q^{30}+2 q^{28}-4 q^{26}+3 q^{22}-10 q^{20}+9 q^{16}-11 q^{14}+2 q^{12}+11 q^{10}-6 q^8-q^6+7 q^4+q^2-2+ q^{-2} +7 q^{-4} - q^{-6} -6 q^{-8} +11 q^{-10} +2 q^{-12} -11 q^{-14} +9 q^{-16} -10 q^{-20} +3 q^{-22} -4 q^{-26} +2 q^{-28} + q^{-30} - q^{-32} + q^{-34} } |
1,0,0 |
B2 Invariants.
Weight | Invariant |
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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^{34}+q^{32}-3 q^{30}+4 q^{28}-6 q^{26}+8 q^{24}-11 q^{22}+12 q^{20}-12 q^{18}+11 q^{16}-7 q^{14}+4 q^{12}+3 q^{10}-8 q^8+15 q^6-19 q^4+23 q^2-24+23 q^{-2} -19 q^{-4} +15 q^{-6} -8 q^{-8} +3 q^{-10} +4 q^{-12} -7 q^{-14} +11 q^{-16} -12 q^{-18} +12 q^{-20} -11 q^{-22} +8 q^{-24} -6 q^{-26} +4 q^{-28} -3 q^{-30} + q^{-32} - q^{-34} } |
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^{56}-q^{52}-q^{50}+2 q^{48}+3 q^{46}-q^{44}-5 q^{42}-3 q^{40}+4 q^{38}+7 q^{36}-3 q^{34}-12 q^{32}-5 q^{30}+10 q^{28}+11 q^{26}-6 q^{24}-13 q^{22}-q^{20}+13 q^{18}+6 q^{16}-8 q^{14}-7 q^{12}+6 q^{10}+8 q^8-2 q^6-7 q^4+2 q^2+9+2 q^{-2} -7 q^{-4} -2 q^{-6} +8 q^{-8} +6 q^{-10} -7 q^{-12} -8 q^{-14} +6 q^{-16} +13 q^{-18} - q^{-20} -13 q^{-22} -6 q^{-24} +11 q^{-26} +10 q^{-28} -5 q^{-30} -12 q^{-32} -3 q^{-34} +7 q^{-36} +4 q^{-38} -3 q^{-40} -5 q^{-42} - q^{-44} +3 q^{-46} +2 q^{-48} - q^{-50} - q^{-52} + q^{-56} } |
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^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-2 q^{70}-3 q^{68}+8 q^{66}-13 q^{64}+15 q^{62}-15 q^{60}+8 q^{58}+q^{56}-16 q^{54}+31 q^{52}-41 q^{50}+37 q^{48}-25 q^{46}-2 q^{44}+30 q^{42}-52 q^{40}+62 q^{38}-47 q^{36}+19 q^{34}+17 q^{32}-46 q^{30}+48 q^{28}-29 q^{26}+3 q^{24}+27 q^{22}-38 q^{20}+31 q^{18}+q^{16}-34 q^{14}+62 q^{12}-69 q^{10}+46 q^8-6 q^6-39 q^4+75 q^2-85+75 q^{-2} -39 q^{-4} -6 q^{-6} +46 q^{-8} -69 q^{-10} +62 q^{-12} -34 q^{-14} + q^{-16} +31 q^{-18} -38 q^{-20} +27 q^{-22} +3 q^{-24} -29 q^{-26} +48 q^{-28} -46 q^{-30} +17 q^{-32} +19 q^{-34} -47 q^{-36} +62 q^{-38} -52 q^{-40} +30 q^{-42} -2 q^{-44} -25 q^{-46} +37 q^{-48} -41 q^{-50} +31 q^{-52} -16 q^{-54} + q^{-56} +8 q^{-58} -15 q^{-60} +15 q^{-62} -13 q^{-64} +8 q^{-66} -3 q^{-68} -2 q^{-70} +3 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} } |
.
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 37"];
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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 4 t^2-13 t+19-13 t^{-1} +4 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 4 z^4+3 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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{ 53, 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^5+2 q^4-4 q^3+7 q^2-8 q+9-8 q^{-1} +7 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} } |
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^2 a^4-a^4+z^4 a^2+z^2 a^2+a^2+2 z^4+3 z^2+1+z^4 a^{-2} +z^2 a^{-2} + a^{-2} -z^2 a^{-4} - a^{-4} } |
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 a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7+2 z^7 a^{-3} +2 a^4 z^6-3 a^2 z^6-3 z^6 a^{-2} +2 z^6 a^{-4} -10 z^6+a^5 z^5-2 a^3 z^5-2 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+2 a^2 z^4+2 z^4 a^{-2} -5 z^4 a^{-4} +14 z^4-3 a^5 z^3-3 a^3 z^3+a z^3+z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +3 a^4 z^2+3 z^2 a^{-4} -6 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} +2 z a^{-3} +2 z a^{-5} -a^4-a^2- a^{-2} - a^{-4} +1} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_28,}
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}} ): {}
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 37"];
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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 4 t^2-13 t+19-13 t^{-1} +4 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^5+2 q^4-4 q^3+7 q^2-8 q+9-8 q^{-1} +7 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} } } |
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_28,} |
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: | (3, 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 ). 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 37. 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
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^{15}-2 q^{14}+5 q^{12}-8 q^{11}+17 q^9-21 q^8-6 q^7+40 q^6-34 q^5-20 q^4+63 q^3-38 q^2-33 q+73-33 q^{-1} -38 q^{-2} +63 q^{-3} -20 q^{-4} -34 q^{-5} +40 q^{-6} -6 q^{-7} -21 q^{-8} +17 q^{-9} -8 q^{-11} +5 q^{-12} -2 q^{-14} + q^{-15} } |
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^{30}+2 q^{29}-q^{27}-3 q^{26}+5 q^{25}+q^{24}-6 q^{23}-4 q^{22}+15 q^{21}+4 q^{20}-23 q^{19}-14 q^{18}+41 q^{17}+27 q^{16}-56 q^{15}-53 q^{14}+67 q^{13}+92 q^{12}-79 q^{11}-128 q^{10}+71 q^9+175 q^8-66 q^7-206 q^6+44 q^5+242 q^4-33 q^3-251 q^2+6 q+265+6 q^{-1} -251 q^{-2} -33 q^{-3} +242 q^{-4} +44 q^{-5} -206 q^{-6} -66 q^{-7} +175 q^{-8} +71 q^{-9} -128 q^{-10} -79 q^{-11} +92 q^{-12} +67 q^{-13} -53 q^{-14} -56 q^{-15} +27 q^{-16} +41 q^{-17} -14 q^{-18} -23 q^{-19} +4 q^{-20} +15 q^{-21} -4 q^{-22} -6 q^{-23} + q^{-24} +5 q^{-25} -3 q^{-26} - q^{-27} +2 q^{-29} - q^{-30} } |
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^{50}-2 q^{49}+q^{47}-q^{46}+6 q^{45}-7 q^{44}+q^{43}+3 q^{42}-9 q^{41}+15 q^{40}-16 q^{39}+9 q^{38}+16 q^{37}-27 q^{36}+19 q^{35}-46 q^{34}+24 q^{33}+63 q^{32}-29 q^{31}+23 q^{30}-140 q^{29}+q^{28}+143 q^{27}+42 q^{26}+97 q^{25}-298 q^{24}-140 q^{23}+166 q^{22}+191 q^{21}+339 q^{20}-423 q^{19}-401 q^{18}+33 q^{17}+315 q^{16}+724 q^{15}-403 q^{14}-657 q^{13}-243 q^{12}+318 q^{11}+1111 q^{10}-256 q^9-803 q^8-528 q^7+220 q^6+1363 q^5-78 q^4-816 q^3-724 q^2+80 q+1447+80 q^{-1} -724 q^{-2} -816 q^{-3} -78 q^{-4} +1363 q^{-5} +220 q^{-6} -528 q^{-7} -803 q^{-8} -256 q^{-9} +1111 q^{-10} +318 q^{-11} -243 q^{-12} -657 q^{-13} -403 q^{-14} +724 q^{-15} +315 q^{-16} +33 q^{-17} -401 q^{-18} -423 q^{-19} +339 q^{-20} +191 q^{-21} +166 q^{-22} -140 q^{-23} -298 q^{-24} +97 q^{-25} +42 q^{-26} +143 q^{-27} + q^{-28} -140 q^{-29} +23 q^{-30} -29 q^{-31} +63 q^{-32} +24 q^{-33} -46 q^{-34} +19 q^{-35} -27 q^{-36} +16 q^{-37} +9 q^{-38} -16 q^{-39} +15 q^{-40} -9 q^{-41} +3 q^{-42} + q^{-43} -7 q^{-44} +6 q^{-45} - q^{-46} + q^{-47} -2 q^{-49} + q^{-50} } |
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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