8 2

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8 1.gif

8_1

8 3.gif

8_3

8 2.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 2 at Knotilus!

[edit 8 2 Quick Notes]
[edit 8 2 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9
Gauss code -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6
Dowker-Thistlethwaite code 4 10 12 14 16 2 6 8
Conway Notation [512]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 8, width is 3,

Braid index is 3

8 2 ML.gif 8 2 AP.gif
[{10, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {3, 5}, {4, 6}, {5, 7}, {6, 9}, {7, 1}]

[edit Notes on presentations of 8 2]

Knot 8_2.
A graph, knot 8_2.
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`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["8 2"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6
In[6]:= DTCode[K]
Out[6]= 4 10 12 14 16 2 6 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [512]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(3,\{-1,-1,-1,-1,-1,2,-1,2\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 3, 8, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
8 2 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {3, 5}, {4, 6}, {5, 7}, {6, 9}, {7, 1}]
In[14]:= Draw[ap]
8 2 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,5\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-11][1]
Hyperbolic Volume 4.93524
A-Polynomial See Data:8 2/A-polynomial

[edit Notes for 8 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math]
Topological 4 genus [math]\displaystyle{ 2 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -4

[edit Notes for 8 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 17, -4 }
Jones polynomial [math]\displaystyle{ 1- q^{-1} +2 q^{-2} -2 q^{-3} +3 q^{-4} -3 q^{-5} +2 q^{-6} -2 q^{-7} + q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^6+3 z^2 a^6+a^6-z^6 a^4-5 z^4 a^4-7 z^2 a^4-3 a^4+z^4 a^2+4 z^2 a^2+3 a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^2 a^{10}+2 z^3 a^9-z a^9+2 z^4 a^8-z^2 a^8+2 z^5 a^7-2 z^3 a^7-z a^7+2 z^6 a^6-5 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-2 z^5 a^5-z^3 a^5+z a^5+3 z^6 a^4-12 z^4 a^4+12 z^2 a^4-3 a^4+z^7 a^3-4 z^5 a^3+3 z^3 a^3+z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 a^2 }[/math]
The A2 invariant [math]\displaystyle{ q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^6+q^4+q^2+1 }[/math]
The G2 invariant [math]\displaystyle{ q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2 q^{120}-2 q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2 q^{92}-q^{84}+q^{82}-2 q^{78}+q^{76}-q^{74}+q^{70}-4 q^{68}+2 q^{66}-3 q^{64}-3 q^{58}+3 q^{56}-2 q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2 q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2 q^{30}-2 q^{28}+3 q^{26}-q^{24}+q^{22}+3 q^{20}-3 q^{18}+4 q^{16}+q^{12}+q^{10}-q^8+2 q^6+q^2 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_2.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{17}-q^{15}-q^{11}+q^7+q^3+ q^{-1} }[/math]
2 [math]\displaystyle{ q^{46}-q^{44}-q^{42}+q^{40}-q^{38}+q^{36}+q^{34}-q^{32}+q^{28}-q^{24}-q^{18}-q^{16}+q^{14}+2 q^8+q^6-q^4+q^2+1- q^{-2} + q^{-6} }[/math]
3 [math]\displaystyle{ q^{87}-q^{85}-q^{83}+q^{79}+q^{77}-q^{75}-q^{71}+q^{69}+q^{67}-2 q^{63}+2 q^{59}-q^{55}-q^{53}+q^{49}+2 q^{47}-q^{45}-q^{43}+2 q^{39}-q^{37}-2 q^{35}+q^{31}-q^{29}-q^{27}+q^{23}+q^{21}-q^{19}-q^{17}+q^{15}+2 q^{13}+q^{11}-q^9-q^7+2 q^5+2 q^3-2 q^{-1} +2 q^{-5} + q^{-7} - q^{-9} - q^{-11} + q^{-15} }[/math]
4 [math]\displaystyle{ q^{140}-q^{138}-q^{136}+3 q^{130}-q^{128}-q^{126}-2 q^{124}-q^{122}+5 q^{120}+q^{118}-q^{116}-4 q^{114}-2 q^{112}+5 q^{110}+4 q^{108}-q^{106}-6 q^{104}-4 q^{102}+4 q^{100}+5 q^{98}+2 q^{96}-4 q^{94}-6 q^{92}+3 q^{88}+4 q^{86}-2 q^{82}-q^{80}-2 q^{78}+q^{76}+4 q^{74}+2 q^{72}-q^{70}-3 q^{68}-q^{66}+3 q^{64}+2 q^{62}-2 q^{60}-3 q^{58}-q^{56}+3 q^{54}+2 q^{52}-3 q^{50}-3 q^{48}+4 q^{44}+3 q^{42}-3 q^{40}-4 q^{38}-2 q^{36}+2 q^{34}+3 q^{32}-q^{30}-q^{28}-2 q^{26}+3 q^{22}+2 q^{20}+2 q^{18}-q^{16}-2 q^{14}+4 q^8+2 q^6-q^4-2 q^2-3+2 q^{-2} +3 q^{-4} +2 q^{-6} -4 q^{-10} - q^{-12} + q^{-14} +2 q^{-16} +2 q^{-18} - q^{-20} - q^{-22} - q^{-24} + q^{-28} }[/math]
5 [math]\displaystyle{ q^{205}-q^{203}-q^{201}+2 q^{195}+q^{193}-q^{191}-3 q^{189}-q^{187}+q^{185}+4 q^{183}+3 q^{181}-q^{179}-5 q^{177}-5 q^{175}+3 q^{173}+6 q^{171}+4 q^{169}-2 q^{167}-9 q^{165}-7 q^{163}+5 q^{161}+11 q^{159}+5 q^{157}-5 q^{155}-11 q^{153}-9 q^{151}+5 q^{149}+14 q^{147}+10 q^{145}-2 q^{143}-12 q^{141}-10 q^{139}-q^{137}+9 q^{135}+12 q^{133}+5 q^{131}-5 q^{129}-9 q^{127}-8 q^{125}-2 q^{123}+4 q^{121}+5 q^{119}+4 q^{117}-5 q^{113}-5 q^{111}-3 q^{109}+2 q^{107}+6 q^{105}+6 q^{103}-q^{101}-5 q^{99}-3 q^{97}+q^{95}+5 q^{93}+4 q^{91}-3 q^{89}-5 q^{87}-q^{85}+3 q^{83}+6 q^{81}+3 q^{79}-4 q^{77}-8 q^{75}-3 q^{73}+4 q^{71}+8 q^{69}+5 q^{67}-3 q^{65}-10 q^{63}-7 q^{61}+q^{59}+8 q^{57}+7 q^{55}-8 q^{51}-9 q^{49}-3 q^{47}+7 q^{45}+9 q^{43}+4 q^{41}-3 q^{39}-8 q^{37}-5 q^{35}+5 q^{31}+6 q^{29}+2 q^{27}-q^{25}-3 q^{23}-3 q^{21}-q^{19}+2 q^{17}+4 q^{15}+3 q^{13}+3 q^{11}-3 q^7-4 q^5-2 q^3+q+4 q^{-1} +5 q^{-3} +2 q^{-5} -2 q^{-7} -5 q^{-9} -4 q^{-11} +3 q^{-15} +5 q^{-17} +3 q^{-19} - q^{-21} -4 q^{-23} -3 q^{-25} - q^{-27} + q^{-29} +3 q^{-31} +2 q^{-33} - q^{-37} - q^{-39} - q^{-41} + q^{-45} }[/math]
6 [math]\displaystyle{ q^{282}-q^{280}-q^{278}+2 q^{272}+q^{268}-3 q^{266}-2 q^{264}+q^{262}+q^{260}+4 q^{258}+2 q^{256}+q^{254}-7 q^{252}-4 q^{250}+2 q^{246}+6 q^{244}+3 q^{242}-q^{240}-9 q^{238}-3 q^{236}+2 q^{234}+4 q^{232}+6 q^{230}-6 q^{226}-10 q^{224}+2 q^{222}+11 q^{220}+10 q^{218}+6 q^{216}-8 q^{214}-17 q^{212}-14 q^{210}+4 q^{208}+19 q^{206}+19 q^{204}+8 q^{202}-10 q^{200}-24 q^{198}-24 q^{196}-5 q^{194}+14 q^{192}+24 q^{190}+20 q^{188}+4 q^{186}-14 q^{184}-24 q^{182}-15 q^{180}-q^{178}+11 q^{176}+18 q^{174}+15 q^{172}+5 q^{170}-7 q^{168}-11 q^{166}-11 q^{164}-7 q^{162}+q^{160}+8 q^{158}+12 q^{156}+8 q^{154}-8 q^{150}-12 q^{148}-8 q^{146}+9 q^{142}+9 q^{140}+q^{138}-6 q^{136}-8 q^{134}-4 q^{132}+3 q^{130}+10 q^{128}+5 q^{126}-5 q^{124}-9 q^{122}-7 q^{120}+q^{118}+9 q^{116}+14 q^{114}+8 q^{112}-8 q^{110}-14 q^{108}-10 q^{106}+11 q^{102}+16 q^{100}+11 q^{98}-5 q^{96}-15 q^{94}-14 q^{92}-7 q^{90}+6 q^{88}+16 q^{86}+16 q^{84}+2 q^{82}-12 q^{80}-17 q^{78}-15 q^{76}-2 q^{74}+13 q^{72}+21 q^{70}+13 q^{68}-2 q^{66}-14 q^{64}-21 q^{62}-14 q^{60}+2 q^{58}+17 q^{56}+19 q^{54}+10 q^{52}-3 q^{50}-17 q^{48}-19 q^{46}-10 q^{44}+6 q^{42}+14 q^{40}+15 q^{38}+10 q^{36}-3 q^{34}-10 q^{32}-12 q^{30}-5 q^{28}+q^{26}+6 q^{24}+10 q^{22}+6 q^{20}+3 q^{18}-3 q^{16}-3 q^{14}-5 q^{12}-5 q^{10}-q^8+q^6+5 q^4+4 q^2+7+3 q^{-2} -3 q^{-4} -5 q^{-6} -7 q^{-8} -4 q^{-10} -2 q^{-12} +6 q^{-14} +8 q^{-16} +6 q^{-18} +2 q^{-20} -3 q^{-22} -6 q^{-24} -9 q^{-26} -2 q^{-28} +2 q^{-30} +5 q^{-32} +6 q^{-34} +4 q^{-36} + q^{-38} -5 q^{-40} -4 q^{-42} -3 q^{-44} - q^{-46} + q^{-48} +3 q^{-50} +3 q^{-52} - q^{-58} - q^{-60} - q^{-62} + q^{-66} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^6+q^4+q^2+1 }[/math]
1,1 [math]\displaystyle{ q^{68}-2 q^{66}+2 q^{64}-2 q^{62}+3 q^{60}-4 q^{58}+2 q^{56}+2 q^{52}-2 q^{50}-2 q^{46}+q^{44}-2 q^{42}+2 q^{40}+2 q^{38}+6 q^{34}-6 q^{32}+6 q^{30}-11 q^{28}+8 q^{26}-10 q^{24}+6 q^{22}-6 q^{20}+4 q^{18}-2 q^{14}+5 q^{12}-4 q^{10}+8 q^8-6 q^6+6 q^4-2 q^2+4+ q^{-4} }[/math]
2,0 [math]\displaystyle{ q^{60}-q^{54}-q^{52}-q^{44}+q^{42}+q^{40}+q^{38}+2 q^{34}+q^{32}-q^{30}-q^{28}-2 q^{26}-2 q^{24}-2 q^{22}+q^{16}+2 q^{14}+2 q^{12}+q^6+q^4+1+ q^{-2} + q^{-4} }[/math]
3,0 [math]\displaystyle{ q^{108}-q^{102}-q^{100}+q^{94}-2 q^{92}-q^{90}+3 q^{86}+q^{84}-q^{82}+2 q^{78}+4 q^{76}-3 q^{72}-3 q^{70}+q^{66}-3 q^{62}-q^{60}+2 q^{56}+q^{50}+q^{48}-q^{44}+q^{42}-q^{38}-2 q^{36}+2 q^{32}+q^{30}-2 q^{28}-3 q^{26}-q^{24}+2 q^{22}+q^{20}-q^{16}+2 q^{14}+4 q^{12}+3 q^{10}-q^6+2 q^2+1- q^{-4} + q^{-8} + q^{-10} + q^{-12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{56}-q^{54}-q^{52}+q^{50}-q^{48}-q^{46}+q^{44}+q^{42}+2 q^{38}+q^{36}-2 q^{26}-q^{24}-q^{22}-3 q^{20}-2 q^{18}+2 q^{12}+3 q^{10}+2 q^8+2 q^6+2 q^4+1 }[/math]
1,0,0 [math]\displaystyle{ q^{31}+q^{27}-q^{25}-q^{21}-q^{19}-q^{17}-q^{15}+2 q^9+q^7+2 q^5+q^3+q }[/math]
1,0,1 [math]\displaystyle{ q^{90}-2 q^{88}+q^{86}+q^{84}-2 q^{82}+3 q^{80}-2 q^{78}-q^{76}+q^{74}+q^{70}-q^{68}+q^{62}-2 q^{60}+2 q^{58}-3 q^{56}-2 q^{54}+3 q^{52}-4 q^{50}+5 q^{48}-q^{46}+4 q^{44}+q^{40}+3 q^{38}-2 q^{36}+4 q^{34}-5 q^{32}+2 q^{30}-5 q^{28}-q^{26}-5 q^{24}-4 q^{22}-q^{20}-3 q^{18}+4 q^{16}-2 q^{14}+6 q^{12}+2 q^{10}+5 q^8+6 q^6+2 q^4+4 q^2+ q^{-2} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{70}-q^{66}-2 q^{60}-2 q^{58}+q^{52}+2 q^{50}+4 q^{48}+2 q^{46}+2 q^{44}+2 q^{42}-q^{38}-q^{34}-4 q^{32}-3 q^{30}-4 q^{28}-4 q^{26}-4 q^{24}-2 q^{22}+2 q^{18}+3 q^{16}+4 q^{14}+4 q^{12}+4 q^{10}+3 q^8+2 q^6+q^4+q^2 }[/math]
1,0,0,0 [math]\displaystyle{ q^{38}+q^{34}-q^{26}-q^{24}-2 q^{22}-q^{20}-2 q^{18}+2 q^{12}+2 q^{10}+2 q^8+2 q^6+q^4+q^2 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{56}-q^{54}+q^{52}-q^{50}+q^{48}-q^{46}+q^{44}-q^{42}-q^{36}+2 q^{34}-2 q^{32}+2 q^{30}-2 q^{28}+2 q^{26}-3 q^{24}+q^{22}-q^{20}+2 q^{12}-q^{10}+2 q^8+2 q^4+1 }[/math]
1,0 [math]\displaystyle{ q^{90}-q^{86}-q^{84}+q^{80}-q^{76}-q^{74}+q^{70}+q^{68}+q^{62}+2 q^{60}-q^{56}+q^{52}-q^{48}-q^{46}-q^{40}-q^{38}-q^{32}-2 q^{30}-q^{28}+q^{26}+q^{24}-q^{20}+q^{18}+2 q^{16}+2 q^{14}+q^8+2 q^6+ q^{-2} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{78}-q^{76}-q^{72}+q^{70}-q^{68}-q^{64}+q^{62}+q^{58}+q^{54}+q^{52}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+2 q^{40}-2 q^{38}+q^{36}-3 q^{34}-q^{32}-4 q^{30}-2 q^{28}-3 q^{26}-2 q^{24}-q^{22}+3 q^{18}+2 q^{16}+4 q^{14}+2 q^{12}+4 q^{10}+q^8+2 q^6+q^2 }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2 q^{120}-2 q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2 q^{92}-q^{84}+q^{82}-2 q^{78}+q^{76}-q^{74}+q^{70}-4 q^{68}+2 q^{66}-3 q^{64}-3 q^{58}+3 q^{56}-2 q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2 q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2 q^{30}-2 q^{28}+3 q^{26}-q^{24}+q^{22}+3 q^{20}-3 q^{18}+4 q^{16}+q^{12}+q^{10}-q^8+2 q^6+q^2 }[/math]

.

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]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["8 2"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-3 z^4+1 }[/math]
In[6]:= Alexander[K, 2][t]
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.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 17, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 1- q^{-1} +2 q^{-2} -2 q^{-3} +3 q^{-4} -3 q^{-5} +2 q^{-6} -2 q^{-7} + q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^6+3 z^2 a^6+a^6-z^6 a^4-5 z^4 a^4-7 z^2 a^4-3 a^4+z^4 a^2+4 z^2 a^2+3 a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^2 a^{10}+2 z^3 a^9-z a^9+2 z^4 a^8-z^2 a^8+2 z^5 a^7-2 z^3 a^7-z a^7+2 z^6 a^6-5 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-2 z^5 a^5-z^3 a^5+z a^5+3 z^6 a^4-12 z^4 a^4+12 z^2 a^4-3 a^4+z^7 a^3-4 z^5 a^3+3 z^3 a^3+z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n6,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

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`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["8 2"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { [math]\displaystyle{ -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ 1- q^{-1} +2 q^{-2} -2 q^{-3} +3 q^{-4} -3 q^{-5} +2 q^{-6} -2 q^{-7} + q^{-8} }[/math] }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {K11n6,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (0, 1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{304}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -88 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 432 }[/math] [math]\displaystyle{ -104 }[/math] [math]\displaystyle{ 472 }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ 32 }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-4 is the signature of 8 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012χ
1        11
-1         0
-3      21 1
-5     11  0
-7    21   1
-9   11    0
-11  12     -1
-13 11      0
-15 1       -1
-171        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^2-q-1+3 q^{-1} - q^{-2} -3 q^{-3} +5 q^{-4} -5 q^{-6} +5 q^{-7} + q^{-8} -7 q^{-9} +5 q^{-10} +2 q^{-11} -7 q^{-12} +4 q^{-13} +3 q^{-14} -6 q^{-15} +3 q^{-16} +2 q^{-17} -4 q^{-18} +3 q^{-19} -2 q^{-21} + q^{-22} }[/math]
3 [math]\displaystyle{ q^6-q^5-q^4+3 q^2-3-2 q^{-1} +5 q^{-2} +2 q^{-3} -3 q^{-4} -5 q^{-5} +5 q^{-6} +4 q^{-7} -2 q^{-8} -6 q^{-9} +3 q^{-10} +4 q^{-11} -6 q^{-13} +2 q^{-14} +3 q^{-15} -4 q^{-17} + q^{-18} + q^{-19} + q^{-20} - q^{-21} - q^{-22} + q^{-24} +2 q^{-25} -2 q^{-26} - q^{-27} +2 q^{-29} - q^{-30} + q^{-31} -2 q^{-32} + q^{-34} +2 q^{-35} -2 q^{-36} -2 q^{-37} +2 q^{-38} + q^{-39} -2 q^{-41} + q^{-42} }[/math]
4 [math]\displaystyle{ q^{12}-q^{11}-q^{10}+4 q^7-q^6-2 q^5-2 q^4-3 q^3+8 q^2+q-1-3 q^{-1} -8 q^{-2} +9 q^{-3} +2 q^{-4} +2 q^{-5} - q^{-6} -12 q^{-7} +9 q^{-8} +3 q^{-10} +2 q^{-11} -12 q^{-12} +10 q^{-13} -3 q^{-14} + q^{-15} +3 q^{-16} -12 q^{-17} +14 q^{-18} -4 q^{-19} -3 q^{-20} + q^{-21} -11 q^{-22} +20 q^{-23} -3 q^{-24} -7 q^{-25} -2 q^{-26} -11 q^{-27} +25 q^{-28} -2 q^{-29} -11 q^{-30} -4 q^{-31} -10 q^{-32} +29 q^{-33} - q^{-34} -15 q^{-35} -6 q^{-36} -8 q^{-37} +32 q^{-38} + q^{-39} -18 q^{-40} -9 q^{-41} -7 q^{-42} +31 q^{-43} +3 q^{-44} -14 q^{-45} -10 q^{-46} -10 q^{-47} +25 q^{-48} +5 q^{-49} -8 q^{-50} -7 q^{-51} -11 q^{-52} +17 q^{-53} +3 q^{-54} -3 q^{-55} -2 q^{-56} -10 q^{-57} +10 q^{-58} + q^{-59} -6 q^{-62} +4 q^{-63} + q^{-65} -2 q^{-67} + q^{-68} }[/math]
5 [math]\displaystyle{ q^{20}-q^{19}-q^{18}+q^{15}+3 q^{14}-3 q^{12}-2 q^{11}-2 q^{10}+6 q^8+4 q^7-q^6-4 q^5-5 q^4-4 q^3+5 q^2+7 q+3- q^{-1} -6 q^{-2} -7 q^{-3} +2 q^{-4} +5 q^{-5} +4 q^{-6} +2 q^{-7} -3 q^{-8} -7 q^{-9} +3 q^{-10} +3 q^{-11} + q^{-12} -3 q^{-14} -5 q^{-15} +6 q^{-16} +7 q^{-17} -5 q^{-19} -8 q^{-20} -8 q^{-21} +11 q^{-22} +14 q^{-23} +5 q^{-24} -7 q^{-25} -18 q^{-26} -14 q^{-27} +12 q^{-28} +22 q^{-29} +12 q^{-30} -6 q^{-31} -25 q^{-32} -22 q^{-33} +9 q^{-34} +29 q^{-35} +20 q^{-36} -3 q^{-37} -29 q^{-38} -29 q^{-39} +4 q^{-40} +33 q^{-41} +27 q^{-42} -32 q^{-44} -33 q^{-45} +35 q^{-47} +34 q^{-48} + q^{-49} -36 q^{-50} -37 q^{-51} -2 q^{-52} +39 q^{-53} +41 q^{-54} + q^{-55} -40 q^{-56} -42 q^{-57} -4 q^{-58} +39 q^{-59} +46 q^{-60} +5 q^{-61} -39 q^{-62} -43 q^{-63} -10 q^{-64} +33 q^{-65} +45 q^{-66} +9 q^{-67} -29 q^{-68} -36 q^{-69} -13 q^{-70} +23 q^{-71} +36 q^{-72} +7 q^{-73} -19 q^{-74} -24 q^{-75} -9 q^{-76} +14 q^{-77} +22 q^{-78} +5 q^{-79} -13 q^{-80} -14 q^{-81} -3 q^{-82} +8 q^{-83} +10 q^{-84} +3 q^{-85} -6 q^{-86} -8 q^{-87} - q^{-88} +5 q^{-89} +2 q^{-90} +3 q^{-91} -2 q^{-92} -4 q^{-93} +2 q^{-95} + q^{-97} -2 q^{-99} + q^{-100} }[/math]
6 [math]\displaystyle{ q^{30}-q^{29}-q^{28}+q^{25}+4 q^{23}-q^{22}-3 q^{21}-2 q^{20}-2 q^{19}-q^{17}+10 q^{16}+2 q^{15}-q^{14}-3 q^{13}-5 q^{12}-4 q^{11}-8 q^{10}+13 q^9+5 q^8+4 q^7+q^6-3 q^5-6 q^4-16 q^3+11 q^2+2 q+6+3 q^{-1} +3 q^{-2} -2 q^{-3} -19 q^{-4} +12 q^{-5} -2 q^{-6} +4 q^{-7} - q^{-8} +3 q^{-9} -19 q^{-11} +18 q^{-12} + q^{-13} +8 q^{-14} -5 q^{-15} -2 q^{-16} -6 q^{-17} -26 q^{-18} +20 q^{-19} +8 q^{-20} +21 q^{-21} -3 q^{-23} -14 q^{-24} -42 q^{-25} +11 q^{-26} +10 q^{-27} +35 q^{-28} +13 q^{-29} +6 q^{-30} -16 q^{-31} -57 q^{-32} -5 q^{-33} +3 q^{-34} +42 q^{-35} +25 q^{-36} +21 q^{-37} -8 q^{-38} -65 q^{-39} -20 q^{-40} -10 q^{-41} +40 q^{-42} +30 q^{-43} +35 q^{-44} +6 q^{-45} -65 q^{-46} -30 q^{-47} -23 q^{-48} +33 q^{-49} +29 q^{-50} +45 q^{-51} +22 q^{-52} -60 q^{-53} -35 q^{-54} -34 q^{-55} +23 q^{-56} +25 q^{-57} +51 q^{-58} +38 q^{-59} -54 q^{-60} -40 q^{-61} -42 q^{-62} +15 q^{-63} +23 q^{-64} +55 q^{-65} +48 q^{-66} -49 q^{-67} -47 q^{-68} -49 q^{-69} +11 q^{-70} +25 q^{-71} +62 q^{-72} +56 q^{-73} -49 q^{-74} -56 q^{-75} -57 q^{-76} +7 q^{-77} +29 q^{-78} +70 q^{-79} +64 q^{-80} -45 q^{-81} -60 q^{-82} -64 q^{-83} - q^{-84} +25 q^{-85} +70 q^{-86} +68 q^{-87} -33 q^{-88} -50 q^{-89} -61 q^{-90} -8 q^{-91} +13 q^{-92} +56 q^{-93} +59 q^{-94} -23 q^{-95} -32 q^{-96} -45 q^{-97} -4 q^{-98} +3 q^{-99} +37 q^{-100} +40 q^{-101} -23 q^{-102} -18 q^{-103} -27 q^{-104} +7 q^{-105} +3 q^{-106} +22 q^{-107} +22 q^{-108} -26 q^{-109} -9 q^{-110} -13 q^{-111} +11 q^{-112} +4 q^{-113} +13 q^{-114} +10 q^{-115} -22 q^{-116} -3 q^{-117} -7 q^{-118} +9 q^{-119} +2 q^{-120} +8 q^{-121} +4 q^{-122} -14 q^{-123} + q^{-124} -4 q^{-125} +5 q^{-126} +4 q^{-128} + q^{-129} -6 q^{-130} +2 q^{-131} -2 q^{-132} +2 q^{-133} + q^{-135} -2 q^{-137} + q^{-138} }[/math]
7 [math]\displaystyle{ q^{42}-q^{41}-q^{40}+q^{37}+q^{35}+3 q^{34}-q^{33}-3 q^{32}-2 q^{31}-3 q^{30}+q^{29}+q^{27}+9 q^{26}+3 q^{25}-q^{24}-3 q^{23}-8 q^{22}-3 q^{21}-4 q^{20}-4 q^{19}+11 q^{18}+8 q^{17}+6 q^{16}+5 q^{15}-9 q^{14}-4 q^{13}-8 q^{12}-13 q^{11}+6 q^{10}+5 q^9+8 q^8+13 q^7-4 q^6-4 q^4-16 q^3+4 q^2-2 q+2+14 q^{-1} -5 q^{-2} + q^{-3} - q^{-4} -14 q^{-5} +10 q^{-6} +2 q^{-7} + q^{-8} +16 q^{-9} -9 q^{-10} -6 q^{-11} -8 q^{-12} -23 q^{-13} +12 q^{-14} +8 q^{-15} +12 q^{-16} +30 q^{-17} - q^{-18} -7 q^{-19} -17 q^{-20} -43 q^{-21} -7 q^{-22} +2 q^{-23} +17 q^{-24} +50 q^{-25} +20 q^{-26} +10 q^{-27} -11 q^{-28} -58 q^{-29} -32 q^{-30} -22 q^{-31} +3 q^{-32} +55 q^{-33} +41 q^{-34} +36 q^{-35} +11 q^{-36} -52 q^{-37} -44 q^{-38} -46 q^{-39} -25 q^{-40} +40 q^{-41} +44 q^{-42} +54 q^{-43} +36 q^{-44} -30 q^{-45} -35 q^{-46} -54 q^{-47} -49 q^{-48} +14 q^{-49} +28 q^{-50} +54 q^{-51} +52 q^{-52} -5 q^{-53} -12 q^{-54} -44 q^{-55} -57 q^{-56} -9 q^{-57} + q^{-58} +37 q^{-59} +55 q^{-60} +16 q^{-61} +12 q^{-62} -22 q^{-63} -52 q^{-64} -25 q^{-65} -27 q^{-66} +13 q^{-67} +49 q^{-68} +30 q^{-69} +36 q^{-70} + q^{-71} -42 q^{-72} -36 q^{-73} -49 q^{-74} -11 q^{-75} +38 q^{-76} +41 q^{-77} +57 q^{-78} +21 q^{-79} -33 q^{-80} -43 q^{-81} -67 q^{-82} -31 q^{-83} +30 q^{-84} +50 q^{-85} +73 q^{-86} +33 q^{-87} -28 q^{-88} -52 q^{-89} -80 q^{-90} -40 q^{-91} +30 q^{-92} +62 q^{-93} +85 q^{-94} +38 q^{-95} -32 q^{-96} -68 q^{-97} -90 q^{-98} -41 q^{-99} +36 q^{-100} +76 q^{-101} +97 q^{-102} +45 q^{-103} -37 q^{-104} -84 q^{-105} -103 q^{-106} -49 q^{-107} +33 q^{-108} +84 q^{-109} +108 q^{-110} +59 q^{-111} -25 q^{-112} -80 q^{-113} -109 q^{-114} -66 q^{-115} +13 q^{-116} +67 q^{-117} +103 q^{-118} +70 q^{-119} +2 q^{-120} -50 q^{-121} -92 q^{-122} -70 q^{-123} -12 q^{-124} +35 q^{-125} +73 q^{-126} +59 q^{-127} +17 q^{-128} -14 q^{-129} -54 q^{-130} -49 q^{-131} -16 q^{-132} +4 q^{-133} +36 q^{-134} +30 q^{-135} +12 q^{-136} +6 q^{-137} -19 q^{-138} -16 q^{-139} -7 q^{-140} -11 q^{-141} +9 q^{-142} +6 q^{-143} -2 q^{-144} +11 q^{-145} +5 q^{-147} +4 q^{-148} -15 q^{-149} - q^{-150} -7 q^{-151} -7 q^{-152} +11 q^{-153} +3 q^{-154} +10 q^{-155} +8 q^{-156} -10 q^{-157} -4 q^{-158} -8 q^{-159} -6 q^{-160} +8 q^{-161} +2 q^{-162} +5 q^{-163} +6 q^{-164} -6 q^{-165} - q^{-166} -4 q^{-167} -4 q^{-168} +5 q^{-169} +2 q^{-171} +2 q^{-172} -3 q^{-173} -2 q^{-176} +2 q^{-177} + q^{-179} -2 q^{-181} + q^{-182} }[/math]

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.

8 1.gif

8_1

8 3.gif

8_3

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