8 20

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8 19.gif

8_19

8 21.gif

8_21

8 20.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 20 at Knotilus!

[edit 8 20 Quick Notes]

8_20 is also known as the pretzel knot P(3,-3,2).

Its complement contains no complete totally geodesic immersed surfaces.[citation needed]

This appears to be the Ashley/oysterman stopper knot of practical knot tying.

[edit 8 20 Further Notes and Views]


The Oysterman's stopper[1]

Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837
Gauss code 1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4
Dowker-Thistlethwaite code 4 8 -12 2 -14 -6 -16 -10
Conway Notation [3,21,2-]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif

Length is 8, width is 3,

Braid index is 3

8 20 ML.gif 8 20 AP.gif
[{3, 8}, {2, 4}, {1, 3}, {11, 9}, {8, 10}, {9, 5}, {4, 6}, {5, 7}, {6, 11}, {10, 2}, {7, 1}]

[edit Notes on presentations of 8 20]

Knot 8_20.
A graph, knot 8_20.
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 20"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837
In[5]:= GaussCode[K]
Out[5]= 1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4
In[6]:= DTCode[K]
Out[6]= 4 8 -12 2 -14 -6 -16 -10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3,21,2-]
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,-2,-1,-1,-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]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.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 20 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 8}, {2, 4}, {1, 3}, {11, 9}, {8, 10}, {9, 5}, {4, 6}, {5, 7}, {6, 11}, {10, 2}, {7, 1}]
In[14]:= Draw[ap]
8 20 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-2]
Hyperbolic Volume 4.1249
A-Polynomial See Data:8 20/A-polynomial

[edit Notes for 8 20's three dimensional invariants]
8_20 ribbon diagram from A. Kawauchi's text.

Ribbon diagram for 8_20

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 0 }[/math]
Topological 4 genus [math]\displaystyle{ 0 }[/math]
Concordance genus [math]\displaystyle{ 0 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 8 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^2-2 t+3-2 t^{-1} + t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 9, 0 }
Jones polynomial [math]\displaystyle{ -q+2- q^{-1} +2 q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+4 a^2-z^2-1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^4 z^6+a^2 z^6+a^5 z^5+2 a^3 z^5+a z^5-4 a^4 z^4-4 a^2 z^4-4 a^5 z^3-7 a^3 z^3-3 a z^3+4 a^4 z^2+6 a^2 z^2+2 z^2+3 a^5 z+5 a^3 z+3 a z+z a^{-1} -2 a^4-4 a^2-1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{16}-q^{14}-q^{12}+2 q^8+2 q^6+2 q^4+q^2- q^{-4} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}+q^{76}-q^{74}-2 q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2 q^{58}-3 q^{52}-q^{50}-q^{48}+q^{44}-2 q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+2 q^{30}+2 q^{28}+3 q^{26}+2 q^{22}+3 q^{20}+q^{18}+q^{16}+q^{14}+3 q^{10}-2 q^6+q^4+1- q^{-2} -2 q^{-4} - q^{-12} - q^{-14} - q^{-20} + q^{-24} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_20.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+q^5+q^3+q+ q^{-1} - q^{-3} }[/math]
2 [math]\displaystyle{ q^{32}-q^{28}-q^{22}-q^{20}+q^{12}+q^{10}+q^6+q^4+q^2+ q^{-2} - q^{-6} }[/math]
3 [math]\displaystyle{ -q^{63}+q^{59}+q^{57}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{35}-2 q^{33}+q^{29}+q^{27}-q^{25}+q^{21}-q^{17}+q^{15}+q^{13}+q^5+2 q^3+2 q+2 q^{-7} - q^{-9} -2 q^{-11} - q^{-13} + q^{-17} }[/math]
4 [math]\displaystyle{ q^{104}-q^{100}-q^{98}-q^{96}+q^{94}+q^{92}+q^{90}-2 q^{86}-q^{84}+q^{80}+2 q^{78}+q^{76}-q^{72}-q^{70}+q^{68}+2 q^{66}+q^{64}-q^{62}-3 q^{60}-2 q^{58}+2 q^{54}+q^{52}-2 q^{50}-2 q^{48}+2 q^{44}+2 q^{42}-q^{40}-2 q^{38}+q^{34}+q^{32}-q^{30}-q^{28}-q^{20}-q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^8+q^4+3 q^2+4+2 q^{-2} - q^{-4} -4 q^{-6} -2 q^{-8} +3 q^{-10} +2 q^{-12} - q^{-14} -4 q^{-16} -3 q^{-18} + q^{-20} +2 q^{-22} +2 q^{-24} - q^{-28} }[/math]
5 [math]\displaystyle{ -q^{155}+q^{151}+q^{149}+q^{147}-q^{143}-2 q^{141}-q^{139}+q^{135}+2 q^{133}+2 q^{131}-2 q^{127}-2 q^{125}-2 q^{123}-q^{121}+q^{119}+2 q^{117}+2 q^{115}+q^{113}-q^{111}-3 q^{109}-2 q^{107}+3 q^{103}+4 q^{101}+3 q^{99}-3 q^{95}-4 q^{93}-2 q^{91}+2 q^{89}+4 q^{87}+3 q^{85}-q^{83}-4 q^{81}-5 q^{79}-2 q^{77}+3 q^{75}+4 q^{73}+2 q^{71}-q^{69}-4 q^{67}-2 q^{65}+q^{63}+4 q^{61}+3 q^{59}-3 q^{55}-3 q^{53}+2 q^{49}+2 q^{47}-3 q^{43}-2 q^{41}+q^{37}-2 q^{33}-2 q^{31}+q^{27}+q^{25}+q^{15}+2 q^{13}+3 q^{11}+3 q^9+2 q^7-q^5-2 q^3+3 q^{-1} +5 q^{-3} +5 q^{-5} - q^{-7} -7 q^{-9} -7 q^{-11} -2 q^{-13} +3 q^{-15} +7 q^{-17} +3 q^{-19} -3 q^{-21} -6 q^{-23} -4 q^{-25} + q^{-27} +3 q^{-29} +3 q^{-31} + q^{-33} - q^{-35} - q^{-37} }[/math]
6 [math]\displaystyle{ q^{216}-q^{212}-q^{210}-q^{208}+2 q^{202}+2 q^{200}+q^{198}-q^{194}-2 q^{192}-3 q^{190}-q^{188}+2 q^{184}+3 q^{182}+3 q^{180}+2 q^{178}-q^{176}-2 q^{174}-3 q^{172}-3 q^{170}-2 q^{168}+3 q^{164}+4 q^{162}+3 q^{160}+q^{158}-2 q^{156}-5 q^{154}-6 q^{152}-3 q^{150}+q^{148}+4 q^{146}+6 q^{144}+6 q^{142}+2 q^{140}-4 q^{138}-6 q^{136}-6 q^{134}-3 q^{132}+3 q^{130}+8 q^{128}+8 q^{126}+4 q^{124}-q^{122}-7 q^{120}-9 q^{118}-5 q^{116}+q^{114}+6 q^{112}+7 q^{110}+5 q^{108}-2 q^{106}-8 q^{104}-8 q^{102}-4 q^{100}+2 q^{98}+7 q^{96}+9 q^{94}+4 q^{92}-3 q^{90}-7 q^{88}-6 q^{86}-q^{84}+4 q^{82}+8 q^{80}+5 q^{78}-2 q^{76}-6 q^{74}-5 q^{72}-q^{70}+3 q^{68}+6 q^{66}+3 q^{64}-3 q^{62}-5 q^{60}-4 q^{58}-q^{56}+2 q^{54}+2 q^{52}-3 q^{48}-2 q^{46}+q^{42}+q^{40}-q^{36}-2 q^{34}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+2 q^{24}+q^{22}-q^{18}-2 q^{16}-q^{14}+q^{12}+4 q^{10}+6 q^8+6 q^6+2 q^4-3 q^2-6-5 q^{-2} -2 q^{-4} +6 q^{-6} +11 q^{-8} +9 q^{-10} + q^{-12} -8 q^{-14} -12 q^{-16} -13 q^{-18} -2 q^{-20} +10 q^{-22} +12 q^{-24} +7 q^{-26} - q^{-28} -7 q^{-30} -11 q^{-32} -6 q^{-34} + q^{-36} +5 q^{-38} +5 q^{-40} +4 q^{-42} +2 q^{-44} -2 q^{-46} - q^{-48} - q^{-50} - q^{-52} - q^{-54} + q^{-58} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{16}-q^{14}-q^{12}+2 q^8+2 q^6+2 q^4+q^2- q^{-4} }[/math]
1,1 [math]\displaystyle{ q^{44}+2 q^{40}-2 q^{38}+2 q^{36}-2 q^{34}-2 q^{30}-4 q^{28}-4 q^{24}+2 q^{22}-3 q^{20}+4 q^{18}+4 q^{14}+q^{12}+2 q^{10}+4 q^8+5 q^4+2- q^{-4} -2 q^{-6} -2 q^{-8} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{42}+q^{40}+q^{38}-q^{32}-2 q^{30}-4 q^{28}-4 q^{26}-3 q^{24}-q^{22}+2 q^{20}+3 q^{18}+5 q^{16}+3 q^{14}+3 q^{12}+q^{10}+q^8+q^4- q^{-8} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{34}+q^{30}-q^{26}-2 q^{24}-4 q^{22}-4 q^{20}-3 q^{18}+2 q^{14}+6 q^{12}+6 q^{10}+6 q^8+3 q^6+q^4-q^2-2-2 q^{-2} - q^{-4} - q^{-6} + q^{-10} }[/math]
1,0,0 [math]\displaystyle{ -q^{21}-q^{19}-2 q^{17}-q^{15}+2 q^{11}+3 q^9+3 q^7+2 q^5+q^3- q^{-1} - q^{-5} }[/math]
1,0,1 [math]\displaystyle{ q^{56}+2 q^{52}+q^{48}-q^{44}+q^{42}-2 q^{40}+q^{38}-4 q^{36}-2 q^{34}-6 q^{32}-5 q^{30}-5 q^{28}-6 q^{26}-2 q^{22}+6 q^{20}+5 q^{18}+10 q^{16}+9 q^{14}+9 q^{12}+8 q^{10}+2 q^8+3 q^6-2 q^4-3- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} - q^{-10} + q^{-16} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{36}-q^{34}-3 q^{32}-6 q^{30}-9 q^{28}-9 q^{26}-7 q^{24}-3 q^{22}+q^{20}+8 q^{18}+12 q^{16}+13 q^{14}+12 q^{12}+9 q^{10}+3 q^8-q^6-4 q^4-5 q^2-5-3 q^{-2} - q^{-4} + q^{-10} + q^{-12} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{26}-q^{24}-2 q^{22}-2 q^{20}-q^{18}+2 q^{14}+3 q^{12}+4 q^{10}+3 q^8+2 q^6+q^4-1- q^{-2} - q^{-6} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}-q^{30}-q^{26}+q^{18}+2 q^{14}+2 q^{10}+q^6+q^4+q^2+ q^{-4} - q^{-6} - q^{-10} }[/math]
1,0 [math]\displaystyle{ q^{56}+q^{48}-q^{44}-q^{42}-q^{38}-2 q^{36}-2 q^{34}-2 q^{32}-q^{30}-q^{28}+2 q^{22}+2 q^{20}+3 q^{18}+2 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^6-1- q^{-2} - q^{-4} - q^{-8} - q^{-10} + q^{-16} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{46}+q^{42}+q^{38}-q^{36}-2 q^{34}-3 q^{32}-4 q^{30}-4 q^{28}-4 q^{26}-2 q^{24}-q^{22}+3 q^{20}+4 q^{18}+7 q^{16}+6 q^{14}+7 q^{12}+4 q^{10}+3 q^8+q^6-q^4-2 q^2-2-2 q^{-2} -2 q^{-4} - q^{-8} + q^{-14} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}+q^{76}-q^{74}-2 q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2 q^{58}-3 q^{52}-q^{50}-q^{48}+q^{44}-2 q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+2 q^{30}+2 q^{28}+3 q^{26}+2 q^{22}+3 q^{20}+q^{18}+q^{16}+q^{14}+3 q^{10}-2 q^6+q^4+1- q^{-2} -2 q^{-4} - q^{-12} - q^{-14} - q^{-20} + q^{-24} }[/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 20"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^2-2 t+3-2 t^{-1} + t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^4+2 z^2+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]= { 9, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q+2- q^{-1} +2 q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+4 a^2-z^2-1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^4 z^6+a^2 z^6+a^5 z^5+2 a^3 z^5+a z^5-4 a^4 z^4-4 a^2 z^4-4 a^5 z^3-7 a^3 z^3-3 a z^3+4 a^4 z^2+6 a^2 z^2+2 z^2+3 a^5 z+5 a^3 z+3 a z+z a^{-1} -2 a^4-4 a^2-1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_140, K11n73, K11n74,}

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 20"];
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^2-2 t+3-2 t^{-1} + t^{-2} }[/math], [math]\displaystyle{ -q+2- q^{-1} +2 q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/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]= {10_140, K11n73, K11n74,}
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: (2, -2)
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{ 8 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{172}{3} }[/math] [math]\displaystyle{ \frac{20}{3} }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ -\frac{640}{3} }[/math] [math]\displaystyle{ -\frac{64}{3} }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1376}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ \frac{11911}{15} }[/math] [math]\displaystyle{ -\frac{524}{15} }[/math] [math]\displaystyle{ \frac{15484}{45} }[/math] [math]\displaystyle{ \frac{233}{9} }[/math] [math]\displaystyle{ \frac{631}{15} }[/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]0 is the signature of 8 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101χ
3      1-1
1     1 1
-1    12 1
-3   1   1
-5   1   1
-7 11    0
-9       0
-111      -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/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+1-2 q^{-1} +2 q^{-2} + q^{-3} -2 q^{-4} + q^{-5} +2 q^{-6} -2 q^{-7} +2 q^{-9} -2 q^{-10} - q^{-11} +2 q^{-12} - q^{-13} - q^{-14} + q^{-15} }[/math]
3 [math]\displaystyle{ q^7-q^6-q^5-q^4+2 q^3+2 q^2-3 q-1+2 q^{-1} +4 q^{-2} -3 q^{-3} -2 q^{-4} + q^{-5} +4 q^{-6} -3 q^{-7} - q^{-8} + q^{-9} +2 q^{-10} -2 q^{-11} + q^{-14} - q^{-17} - q^{-18} + q^{-19} + q^{-20} - q^{-21} -2 q^{-22} + q^{-23} +2 q^{-24} -2 q^{-26} + q^{-28} + q^{-29} - q^{-30} }[/math]
4 [math]\displaystyle{ -q^{12}+q^{11}+2 q^{10}-q^8-5 q^7+5 q^5+3 q^4-10 q^2-2 q+8+6 q^{-1} +2 q^{-2} -11 q^{-3} -4 q^{-4} +7 q^{-5} +7 q^{-6} +2 q^{-7} -11 q^{-8} -4 q^{-9} +7 q^{-10} +5 q^{-11} +2 q^{-12} -10 q^{-13} -4 q^{-14} +7 q^{-15} +4 q^{-16} +2 q^{-17} -8 q^{-18} -4 q^{-19} +6 q^{-20} +2 q^{-21} +3 q^{-22} -5 q^{-23} -4 q^{-24} +4 q^{-25} +3 q^{-27} -2 q^{-28} -3 q^{-29} +2 q^{-30} -2 q^{-31} +2 q^{-32} - q^{-34} +3 q^{-35} -3 q^{-36} +4 q^{-40} -2 q^{-41} - q^{-42} - q^{-43} - q^{-44} +3 q^{-45} - q^{-48} - q^{-49} + q^{-50} }[/math]
5 [math]\displaystyle{ -q^{16}+2 q^{14}+2 q^{13}-2 q^{11}-6 q^{10}-2 q^9+5 q^8+8 q^7+4 q^6-6 q^5-11 q^4-7 q^3+5 q^2+14 q+10-6 q^{-1} -13 q^{-2} -10 q^{-3} +3 q^{-4} +15 q^{-5} +13 q^{-6} -5 q^{-7} -13 q^{-8} -11 q^{-9} +2 q^{-10} +14 q^{-11} +13 q^{-12} -5 q^{-13} -13 q^{-14} -10 q^{-15} +2 q^{-16} +13 q^{-17} +11 q^{-18} -5 q^{-19} -11 q^{-20} -9 q^{-21} + q^{-22} +11 q^{-23} +10 q^{-24} -2 q^{-25} -9 q^{-26} -9 q^{-27} - q^{-28} +8 q^{-29} +10 q^{-30} + q^{-31} -6 q^{-32} -8 q^{-33} -4 q^{-34} +5 q^{-35} +8 q^{-36} +4 q^{-37} -3 q^{-38} -6 q^{-39} -5 q^{-40} +5 q^{-42} +5 q^{-43} -2 q^{-45} -4 q^{-46} -2 q^{-47} + q^{-48} +3 q^{-49} + q^{-50} + q^{-51} - q^{-52} - q^{-53} - q^{-56} + q^{-58} + q^{-59} + q^{-60} -2 q^{-62} -2 q^{-63} + q^{-65} + q^{-66} +2 q^{-67} -2 q^{-69} - q^{-70} + q^{-73} + q^{-74} - q^{-75} }[/math]
6 [math]\displaystyle{ q^{26}-q^{25}-q^{24}-q^{20}+5 q^{19}+q^{18}-4 q^{15}-7 q^{14}-6 q^{13}+9 q^{12}+7 q^{11}+8 q^{10}+5 q^9-6 q^8-19 q^7-17 q^6+10 q^5+11 q^4+17 q^3+13 q^2-4 q-24-25 q^{-1} +7 q^{-2} +10 q^{-3} +20 q^{-4} +18 q^{-5} -24 q^{-7} -27 q^{-8} +4 q^{-9} +8 q^{-10} +19 q^{-11} +19 q^{-12} + q^{-13} -23 q^{-14} -26 q^{-15} +4 q^{-16} +8 q^{-17} +18 q^{-18} +17 q^{-19} -22 q^{-21} -25 q^{-22} +5 q^{-23} +8 q^{-24} +17 q^{-25} +15 q^{-26} - q^{-27} -19 q^{-28} -23 q^{-29} +5 q^{-30} +5 q^{-31} +14 q^{-32} +14 q^{-33} + q^{-34} -13 q^{-35} -20 q^{-36} +2 q^{-37} + q^{-38} +10 q^{-39} +13 q^{-40} +5 q^{-41} -6 q^{-42} -17 q^{-43} -2 q^{-44} -4 q^{-45} +5 q^{-46} +12 q^{-47} +9 q^{-48} + q^{-49} -12 q^{-50} -4 q^{-51} -9 q^{-52} - q^{-53} +8 q^{-54} +9 q^{-55} +7 q^{-56} -5 q^{-57} -2 q^{-58} -10 q^{-59} -6 q^{-60} +2 q^{-61} +5 q^{-62} +9 q^{-63} + q^{-64} +3 q^{-65} -6 q^{-66} -6 q^{-67} -3 q^{-68} - q^{-69} +6 q^{-70} + q^{-71} +5 q^{-72} -2 q^{-74} -3 q^{-75} -3 q^{-76} +3 q^{-77} -3 q^{-78} +2 q^{-79} + q^{-80} + q^{-81} - q^{-83} +4 q^{-84} -4 q^{-85} - q^{-86} - q^{-87} +5 q^{-91} - q^{-92} - q^{-94} - q^{-95} -2 q^{-96} - q^{-97} +3 q^{-98} + q^{-100} - q^{-103} - q^{-104} + q^{-105} }[/math]
7 [math]\displaystyle{ -q^{35}+q^{34}+q^{33}+q^{32}-2 q^{30}-q^{29}-2 q^{28}-3 q^{27}+4 q^{25}+6 q^{24}+7 q^{23}-q^{21}-7 q^{20}-14 q^{19}-10 q^{18}+11 q^{16}+21 q^{15}+16 q^{14}+6 q^{13}-9 q^{12}-28 q^{11}-26 q^{10}-16 q^9+6 q^8+34 q^7+35 q^6+21 q^5-3 q^4-33 q^3-38 q^2-30 q-3+34 q^{-1} +44 q^{-2} +31 q^{-3} +6 q^{-4} -31 q^{-5} -39 q^{-6} -35 q^{-7} -10 q^{-8} +30 q^{-9} +42 q^{-10} +32 q^{-11} +10 q^{-12} -28 q^{-13} -38 q^{-14} -34 q^{-15} -11 q^{-16} +29 q^{-17} +40 q^{-18} +31 q^{-19} +9 q^{-20} -28 q^{-21} -38 q^{-22} -33 q^{-23} -9 q^{-24} +30 q^{-25} +40 q^{-26} +30 q^{-27} +6 q^{-28} -28 q^{-29} -37 q^{-30} -31 q^{-31} -7 q^{-32} +28 q^{-33} +38 q^{-34} +27 q^{-35} +5 q^{-36} -24 q^{-37} -34 q^{-38} -28 q^{-39} -7 q^{-40} +23 q^{-41} +33 q^{-42} +24 q^{-43} +6 q^{-44} -17 q^{-45} -28 q^{-46} -25 q^{-47} -9 q^{-48} +14 q^{-49} +26 q^{-50} +22 q^{-51} +10 q^{-52} -8 q^{-53} -20 q^{-54} -21 q^{-55} -13 q^{-56} +3 q^{-57} +16 q^{-58} +19 q^{-59} +13 q^{-60} +3 q^{-61} -10 q^{-62} -15 q^{-63} -14 q^{-64} -8 q^{-65} +4 q^{-66} +11 q^{-67} +14 q^{-68} +11 q^{-69} -5 q^{-71} -10 q^{-72} -13 q^{-73} -7 q^{-74} +7 q^{-76} +13 q^{-77} +6 q^{-78} +6 q^{-79} -10 q^{-81} -9 q^{-82} -8 q^{-83} -3 q^{-84} +5 q^{-85} +5 q^{-86} +10 q^{-87} +9 q^{-88} -2 q^{-89} -3 q^{-90} -6 q^{-91} -9 q^{-92} -3 q^{-93} -3 q^{-94} +5 q^{-95} +9 q^{-96} +2 q^{-97} +4 q^{-98} + q^{-99} -5 q^{-100} -3 q^{-101} -6 q^{-102} -2 q^{-103} +4 q^{-104} - q^{-105} +3 q^{-106} +3 q^{-107} +2 q^{-109} -2 q^{-110} -2 q^{-111} +2 q^{-112} -3 q^{-113} - q^{-114} -2 q^{-116} +3 q^{-117} + q^{-118} +3 q^{-120} - q^{-123} -4 q^{-124} - q^{-127} +2 q^{-128} + q^{-129} +2 q^{-130} + q^{-131} -2 q^{-132} - q^{-133} - q^{-135} + q^{-138} + q^{-139} - q^{-140} }[/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.

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