6 1: Difference between revisions
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{{4D Invariants}} |
{{4D Invariants}} |
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{{Polynomial Invariants}} |
{{Polynomial Invariants}} |
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== "Similar" Knots (within the Atlas) == |
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Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]: |
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{[[9_46]], [[K11n67]], [[K11n97]], [[K11n139]], ...} |
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{{Vassiliev Invariants}} |
{{Vassiliev Invariants}} |
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Revision as of 16:11, 29 August 2005
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Visit 6 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 6 1's page at Knotilus! Visit 6 1's page at the original Knot Atlas! 6_1 is also known as "Stevedore's Knot" (see e.g. [1]), and as the pretzel knot P(5,-1,-1). |
Knot presentations
Planar diagram presentation | X1425 X7,10,8,11 X3948 X9,3,10,2 X5,12,6,1 X11,6,12,7 |
Gauss code | -1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5 |
Dowker-Thistlethwaite code | 4 8 12 10 2 6 |
Conway Notation | [42] |
Length is 7, width is 4. Braid index is 4. |
Three dimensional invariants
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[edit Notes for 6 1's three dimensional invariants]
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Four dimensional invariants
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Polynomial invariants
Alexander polynomial | |
Conway polynomial | |
2nd Alexander ideal (db, data sources) | |
Determinant and Signature | { 9, 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^2-q+2-2 q^{-1} + q^{-2} - 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 a^4-z^2 a^2-a^2-z^2+ a^{-2} } |
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^3 z^5+a z^5+a^4 z^4+2 a^2 z^4+z^4-3 a^3 z^3-2 a z^3+z^3 a^{-1} -3 a^4 z^2-4 a^2 z^2+z^2 a^{-2} +2 a^3 z+2 a z+a^4+a^2- a^{-2} } |
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^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-6} + q^{-8} } |
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}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-14} - q^{-18} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} } |
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^9-q^3+ q^{-1} + q^{-5} } |
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}-q^{22}-q^{16}+q^{12}+q^8+q^6+ q^{-2} - q^{-4} - q^{-6} + q^{-8} + q^{-14} } |
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}-q^{47}-q^{45}+q^{41}-q^{37}+q^{33}+q^{31}-q^{27}+q^{25}+q^{23}-q^{19}-q^{13}-q^{11}+q^7+q^3+ q^{-1} +2 q^{-3} + q^{-5} - q^{-7} - q^{-9} + q^{-11} + q^{-13} - q^{-15} - q^{-17} + q^{-27} } |
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^{84}-q^{80}-q^{78}-q^{76}+q^{74}+q^{72}+q^{70}-2 q^{66}+q^{62}+q^{60}+q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+2 q^{48}-q^{44}-2 q^{42}+q^{38}-q^{34}-2 q^{32}+2 q^{28}+q^{26}+q^{24}+q^{20}+2 q^{18}-q^{14}-q^{12}-1- q^{-2} +2 q^{-4} + q^{-6} + q^{-8} - q^{-10} -2 q^{-12} + q^{-14} +2 q^{-16} +3 q^{-18} -2 q^{-22} + q^{-28} - q^{-32} - q^{-36} + q^{-44} } |
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^{125}-q^{121}-q^{119}-q^{117}+q^{113}+2 q^{111}+q^{109}-q^{105}-2 q^{103}-q^{101}+q^{99}+2 q^{97}+q^{95}-q^{91}-2 q^{89}-q^{87}+2 q^{83}+2 q^{81}+q^{79}-q^{77}-3 q^{75}-2 q^{73}+2 q^{69}+2 q^{67}-2 q^{63}-2 q^{61}-q^{59}+2 q^{57}+4 q^{55}+2 q^{53}-q^{51}-q^{49}-q^{47}+q^{45}+2 q^{43}+q^{41}-2 q^{39}-3 q^{37}-2 q^{35}+q^{31}-q^{27}-q^{25}+q^{23}+2 q^{21}+q^{19}-q^{13}+q^{11}+q^9+2 q^5+2 q^3- q^{-1} - q^{-3} + q^{-7} +2 q^{-9} + q^{-11} - q^{-13} -4 q^{-15} -3 q^{-17} +3 q^{-21} +4 q^{-23} + q^{-25} -2 q^{-27} -4 q^{-29} - q^{-31} +2 q^{-33} +3 q^{-35} +2 q^{-37} - q^{-41} - q^{-43} + q^{-47} - q^{-55} - q^{-57} + q^{-65} } |
6 | 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^{174}-q^{170}-q^{168}-q^{166}+2 q^{160}+2 q^{158}+q^{156}-q^{152}-2 q^{150}-3 q^{148}+q^{144}+2 q^{142}+2 q^{140}+q^{138}-3 q^{134}-2 q^{132}-q^{130}+q^{126}+3 q^{124}+3 q^{122}-q^{118}-3 q^{116}-3 q^{114}-3 q^{112}+q^{110}+4 q^{108}+3 q^{106}+2 q^{104}-q^{102}-3 q^{100}-4 q^{98}-q^{96}+2 q^{94}+4 q^{92}+5 q^{90}+2 q^{88}-2 q^{86}-5 q^{84}-3 q^{82}-q^{80}+2 q^{78}+4 q^{76}+2 q^{74}-q^{72}-5 q^{70}-4 q^{68}-2 q^{66}+q^{64}+4 q^{62}+3 q^{60}-3 q^{56}-2 q^{54}+2 q^{50}+4 q^{48}+3 q^{46}-3 q^{42}-q^{40}+q^{38}+q^{36}+q^{34}-q^{30}-2 q^{28}+q^{24}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}+q^6+2 q^4+2 q^2+2- q^{-2} -2 q^{-4} - q^{-8} +2 q^{-10} +4 q^{-12} +5 q^{-14} + q^{-16} -3 q^{-18} -4 q^{-20} -5 q^{-22} - q^{-24} +4 q^{-26} +8 q^{-28} +4 q^{-30} - q^{-32} -5 q^{-34} -8 q^{-36} -4 q^{-38} + q^{-40} +6 q^{-42} +4 q^{-44} + q^{-46} - q^{-48} -4 q^{-50} -3 q^{-52} +3 q^{-56} +2 q^{-58} + q^{-60} + q^{-62} - q^{-64} - q^{-66} + q^{-70} + q^{-76} - q^{-78} - q^{-80} - q^{-82} + q^{-90} } |
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^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-6} + q^{-8} } |
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}+2 q^{32}-2 q^{30}+2 q^{28}-2 q^{26}-2 q^{22}-2 q^{20}-2 q^{16}+4 q^{14}+q^{12}+6 q^{10}+4 q^6-2 q^4-2-2 q^{-2} - q^{-4} -2 q^{-6} +2 q^{-8} +2 q^{-12} +2 q^{-16} + q^{-20} } |
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^{36}+q^{34}+q^{32}-q^{30}-q^{28}-q^{26}-q^{24}-q^{22}-q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+2 q^{10}+q^8+q^6+q^4- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} } |
3,0 | 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}+q^{64}+q^{62}-2 q^{58}-2 q^{56}-2 q^{54}+2 q^{42}+2 q^{40}+2 q^{38}+q^{32}+2 q^{30}+q^{28}-q^{24}-q^{20}-3 q^{18}-3 q^{16}-3 q^{14}-q^{12}-q^{10}+q^8+2 q^6+3 q^4+3 q^2+3+3 q^{-2} +2 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} -3 q^{-18} -3 q^{-20} - q^{-22} + q^{-26} + q^{-30} + q^{-32} + q^{-34} + q^{-36} } |
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^{28}+q^{24}-q^{20}+q^{12}+2 q^{10}-q^4-q^2-1- q^{-2} + q^{-4} + q^{-8} +2 q^{-10} + q^{-12} + q^{-16} } |
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^{19}+q^{17}+q^{15}-q^9-q^7-q^5+ q^{-3} + q^{-7} + q^{-9} + q^{-11} } |
A4 Invariants.
Weight | Invariant |
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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^{38}+q^{36}+q^{34}+q^{32}+q^{30}-q^{28}-2 q^{26}-2 q^{24}-q^{22}-q^{20}+3 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^8-q^6-2 q^4-2 q^2-2-2 q^{-2} - q^{-4} + q^{-6} + q^{-10} +2 q^{-12} +2 q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} } |
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^{24}+q^{22}+q^{20}+q^{18}-q^{12}-q^{10}-q^8-q^6+ q^{-4} + q^{-8} + q^{-10} + q^{-12} + q^{-14} } |
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^{28}+q^{24}+q^{20}-q^{12}-2 q^8-q^4+q^2-1+ q^{-2} + q^{-4} + q^{-8} + q^{-12} + q^{-16} } |
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}+q^{38}-q^{34}-q^{32}+q^{20}+q^{18}+q^{16}+q^{12}-q^6-q^4- q^{-2} - q^{-4} + q^{-6} + q^{-14} + q^{-16} + q^{-18} + q^{-26} } |
D4 Invariants.
Weight | Invariant |
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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}+q^{34}+q^{30}+q^{16}+q^{12}-q^{10}-q^6-2 q^2- q^{-2} + q^{-6} + q^{-10} + q^{-12} +2 q^{-14} + q^{-16} + q^{-18} + q^{-22} } |
G2 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^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-14} - q^{-18} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} } |
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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["6 1"];
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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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In[5]:=
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Conway[K][z]
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Out[5]=
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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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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 9, 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^2-q+2-2 q^{-1} + q^{-2} - 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 a^4-z^2 a^2-a^2-z^2+ a^{-2} } |
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^3 z^5+a z^5+a^4 z^4+2 a^2 z^4+z^4-3 a^3 z^3-2 a z^3+z^3 a^{-1} -3 a^4 z^2-4 a^2 z^2+z^2 a^{-2} +2 a^3 z+2 a z+a^4+a^2- a^{-2} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {9_46, K11n67, K11n97, K11n139, ...}
Vassiliev invariants
V2 and V3: | (-2, 1) |
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 6 1. 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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Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session.
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{Include}(\textrm{ColouredJonesM.mhtml})}
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 29, 2005, 15:27:48)... | |
In[2]:= | Crossings[Knot[6, 1]] |
Out[2]= | 6 |
In[3]:= | PD[Knot[6, 1]] |
Out[3]= | PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[5, 12, 6, 1], X[11, 6, 12, 7]] |
In[4]:= | GaussCode[Knot[6, 1]] |
Out[4]= | GaussCode[-1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5] |
In[5]:= | BR[Knot[6, 1]] |
Out[5]= | BR[4, {-1, -1, -2, 1, 3, -2, 3}] |
In[6]:= | alex = Alexander[Knot[6, 1]][t] |
Out[6]= | 2 |
In[7]:= | Conway[Knot[6, 1]][z] |
Out[7]= | 2 1 - 2 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]} |
In[9]:= | {KnotDet[Knot[6, 1]], KnotSignature[Knot[6, 1]]} |
Out[9]= | {9, 0} |
In[10]:= | J=Jones[Knot[6, 1]][q] |
Out[10]= | -4 -3 -2 2 2 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[6, 1]} |
In[12]:= | A2Invariant[Knot[6, 1]][q] |
Out[12]= | -14 -12 -6 -4 2 6 8 q + q - q - q + q + q + q |
In[13]:= | Kauffman[Knot[6, 1]][a, z] |
Out[13]= | 2 3-2 2 4 3 z 2 2 4 2 z |
In[14]:= | {Vassiliev[2][Knot[6, 1]], Vassiliev[3][Knot[6, 1]]} |
Out[14]= | {-2, 1} |
In[15]:= | Kh[Knot[6, 1]][q, t] |
Out[15]= | 1 1 1 1 1 1 5 2 |
See/edit the Rolfsen_Splice_Template.