9 8

9_7

9_9

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 8 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X5,14,6,15 X9,1,10,18 X11,17,12,16 X15,13,16,12 X17,11,18,10 X13,6,14,7 X7283
Gauss code	-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -5, 6, -8, 3, -6, 5, -7, 4
Dowker-Thistlethwaite code	4 8 14 2 18 16 6 12 10
Conway Notation	[2412]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 5, Braid index is 5		[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {4, 9}, {3, 5}, {6, 4}, {5, 7}, {11, 6}, {7, 1}]

[edit Notes on presentations of 9 8]

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["9 8"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X3849 X5,14,6,15 X9,1,10,18 X11,17,12,16 X15,13,16,12 X17,11,18,10 X13,6,14,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -5, 6, -8, 3, -6, 5, -7, 4

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 18 16 6 12 10

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[2412]

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}(5,\{-1,-1,2,-1,2,3,-2,-4,3,-4\}) }[/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]=

{ 5, 10, 5 }

In[11]:=

Show[BraidPlot[br]]

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."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {4, 9}, {3, 5}, {6, 4}, {5, 7}, {11, 6}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-3] Hyperbolic Volume 8.19235 A-Polynomial See Data:9 8/A-polynomial

[edit Notes for 9 8's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant -2

[edit Notes for 9 8's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 1-2 z^4 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 31, -2 }
Jones polynomial	[math]\displaystyle{ q^3-2 q^2+3 q-4+5 q^{-1} -5 q^{-2} +5 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^6+2 z^2 a^4+2 a^4-z^4 a^2-z^2 a^2-z^4-2 z^2-1+z^2 a^{-2} + a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+4 a z^7+2 z^7 a^{-1} +2 a^4 z^6+z^6 a^{-2} -z^6+2 a^5 z^5-3 a^3 z^5-13 a z^5-8 z^5 a^{-1} +2 a^6 z^4-4 a^2 z^4-4 z^4 a^{-2} -6 z^4+a^7 z^3+2 a^3 z^3+11 a z^3+8 z^3 a^{-1} -2 a^6 z^2-3 a^4 z^2+2 a^2 z^2+4 z^2 a^{-2} +7 z^2-a^7 z-a^5 z-a^3 z-3 a z-2 z a^{-1} +a^6+2 a^4- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{20}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6-q^4- q^{-2} + q^{-4} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+4 q^{86}-5 q^{84}+4 q^{82}-4 q^{80}+3 q^{76}-5 q^{74}+6 q^{72}-7 q^{70}+5 q^{68}-4 q^{66}+3 q^{62}-5 q^{60}+9 q^{58}-6 q^{56}+5 q^{54}-q^{52}-2 q^{50}+7 q^{48}-5 q^{46}+4 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-3 q^{36}-3 q^{34}+11 q^{32}-13 q^{30}+10 q^{28}-5 q^{26}-5 q^{24}+14 q^{22}-17 q^{20}+14 q^{18}-10 q^{16}+7 q^{12}-12 q^{10}+12 q^8-9 q^6+3 q^4+3 q^2-7+7 q^{-2} -3 q^{-4} -2 q^{-6} +8 q^{-8} -10 q^{-10} +7 q^{-12} -8 q^{-16} +14 q^{-18} -15 q^{-20} +10 q^{-22} -2 q^{-24} -6 q^{-26} +11 q^{-28} -11 q^{-30} +10 q^{-32} -3 q^{-34} - q^{-36} +3 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_8.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+q^{11}-q^9+2 q^7+q- q^{-1} + q^{-3} - q^{-5} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{36}-q^{34}-q^{32}+2 q^{30}-2 q^{28}+3 q^{24}-4 q^{22}+4 q^{18}-3 q^{16}-q^{14}+3 q^{12}+q^{10}-2 q^8+3 q^4-q^2-2+4 q^{-2} -4 q^{-6} +3 q^{-8} +2 q^{-10} -4 q^{-12} + q^{-14} +3 q^{-16} -2 q^{-18} - q^{-20} + q^{-22} }[/math]
3	[math]\displaystyle{ -q^{69}+q^{67}+q^{65}-2 q^{61}+2 q^{57}-q^{51}-q^{49}-q^{47}+4 q^{45}+2 q^{43}-6 q^{41}-5 q^{39}+8 q^{37}+6 q^{35}-5 q^{33}-10 q^{31}+2 q^{29}+9 q^{27}+2 q^{25}-5 q^{23}-6 q^{21}+3 q^{19}+8 q^{17}+q^{15}-9 q^{13}-2 q^{11}+8 q^9+4 q^7-8 q^5-4 q^3+8 q+7 q^{-1} -5 q^{-3} -7 q^{-5} +3 q^{-7} +9 q^{-9} -10 q^{-13} -4 q^{-15} +7 q^{-17} +8 q^{-19} -4 q^{-21} -9 q^{-23} + q^{-25} +8 q^{-27} +3 q^{-29} -6 q^{-31} -5 q^{-33} +3 q^{-35} +4 q^{-37} -2 q^{-41} - q^{-43} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{112}-q^{110}-q^{108}+4 q^{102}-2 q^{100}-2 q^{98}-2 q^{96}-2 q^{94}+9 q^{92}+q^{90}-q^{88}-7 q^{86}-9 q^{84}+12 q^{82}+10 q^{80}+5 q^{78}-14 q^{76}-24 q^{74}+9 q^{72}+23 q^{70}+21 q^{68}-15 q^{66}-44 q^{64}-6 q^{62}+29 q^{60}+42 q^{58}-46 q^{54}-25 q^{52}+9 q^{50}+40 q^{48}+23 q^{46}-20 q^{44}-27 q^{42}-17 q^{40}+12 q^{38}+24 q^{36}+10 q^{34}-6 q^{32}-25 q^{30}-14 q^{28}+14 q^{26}+25 q^{24}+7 q^{22}-21 q^{20}-22 q^{18}+9 q^{16}+31 q^{14}+9 q^{12}-22 q^{10}-27 q^8+7 q^6+36 q^4+12 q^2-19-33 q^{-2} -4 q^{-4} +34 q^{-6} +21 q^{-8} -5 q^{-10} -32 q^{-12} -20 q^{-14} +17 q^{-16} +22 q^{-18} +18 q^{-20} -13 q^{-22} -26 q^{-24} -7 q^{-26} +5 q^{-28} +26 q^{-30} +11 q^{-32} -10 q^{-34} -16 q^{-36} -19 q^{-38} +11 q^{-40} +18 q^{-42} +10 q^{-44} -2 q^{-46} -21 q^{-48} -7 q^{-50} +4 q^{-52} +12 q^{-54} +11 q^{-56} -7 q^{-58} -7 q^{-60} -5 q^{-62} + q^{-64} +6 q^{-66} + q^{-68} -2 q^{-72} - q^{-74} + q^{-76} }[/math]
5	[math]\displaystyle{ -q^{165}+q^{163}+q^{161}-2 q^{155}-2 q^{153}+2 q^{151}+4 q^{149}+q^{147}-6 q^{143}-7 q^{141}+q^{139}+9 q^{137}+10 q^{135}+q^{133}-11 q^{131}-17 q^{129}-6 q^{127}+18 q^{125}+28 q^{123}+8 q^{121}-24 q^{119}-37 q^{117}-18 q^{115}+31 q^{113}+62 q^{111}+26 q^{109}-44 q^{107}-84 q^{105}-45 q^{103}+49 q^{101}+113 q^{99}+72 q^{97}-51 q^{95}-138 q^{93}-107 q^{91}+36 q^{89}+156 q^{87}+136 q^{85}-q^{83}-145 q^{81}-163 q^{79}-38 q^{77}+119 q^{75}+163 q^{73}+73 q^{71}-63 q^{69}-142 q^{67}-102 q^{65}+12 q^{63}+99 q^{61}+100 q^{59}+32 q^{57}-45 q^{55}-87 q^{53}-63 q^{51}+3 q^{49}+61 q^{47}+72 q^{45}+31 q^{43}-37 q^{41}-79 q^{39}-46 q^{37}+29 q^{35}+75 q^{33}+53 q^{31}-24 q^{29}-82 q^{27}-54 q^{25}+31 q^{23}+89 q^{21}+60 q^{19}-33 q^{17}-100 q^{15}-70 q^{13}+34 q^{11}+111 q^9+87 q^7-24 q^5-118 q^3-103 q+4 q^{-1} +113 q^{-3} +123 q^{-5} +23 q^{-7} -97 q^{-9} -131 q^{-11} -53 q^{-13} +66 q^{-15} +128 q^{-17} +83 q^{-19} -27 q^{-21} -110 q^{-23} -98 q^{-25} -12 q^{-27} +71 q^{-29} +97 q^{-31} +48 q^{-33} -29 q^{-35} -77 q^{-37} -64 q^{-39} -11 q^{-41} +39 q^{-43} +61 q^{-45} +41 q^{-47} - q^{-49} -39 q^{-51} -48 q^{-53} -28 q^{-55} +4 q^{-57} +36 q^{-59} +43 q^{-61} +22 q^{-63} -13 q^{-65} -35 q^{-67} -34 q^{-69} -13 q^{-71} +18 q^{-73} +33 q^{-75} +25 q^{-77} -19 q^{-81} -24 q^{-83} -15 q^{-85} +6 q^{-87} +17 q^{-89} +14 q^{-91} +3 q^{-93} -5 q^{-95} -9 q^{-97} -7 q^{-99} + q^{-101} +4 q^{-103} +3 q^{-105} + q^{-107} -2 q^{-111} - q^{-113} + q^{-115} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{20}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6-q^4- q^{-2} + q^{-4} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{52}-2 q^{50}+4 q^{48}-6 q^{46}+9 q^{44}-12 q^{42}+14 q^{40}-16 q^{38}+15 q^{36}-18 q^{34}+14 q^{32}-16 q^{30}+16 q^{28}-14 q^{26}+18 q^{24}-10 q^{22}+10 q^{20}+2 q^{18}-14 q^{16}+26 q^{14}-44 q^{12}+52 q^{10}-62 q^8+64 q^6-59 q^4+56 q^2-38+26 q^{-2} -7 q^{-4} -10 q^{-6} +24 q^{-8} -36 q^{-10} +41 q^{-12} -40 q^{-14} +36 q^{-16} -28 q^{-18} +19 q^{-20} -12 q^{-22} +6 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{50}+q^{48}-2 q^{44}-q^{42}-2 q^{38}-2 q^{36}+2 q^{34}+3 q^{32}-q^{30}-q^{28}+3 q^{26}+3 q^{24}-3 q^{22}+q^{18}-q^{16}-q^{14}-q^8+2 q^6+3 q^4+3 q^{-2} + q^{-4} -3 q^{-6} - q^{-8} + q^{-10} + q^{-12} - q^{-14} - q^{-16} +2 q^{-18} + q^{-20} - q^{-22} - q^{-24} + q^{-28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{42}-q^{40}+2 q^{36}-3 q^{34}-3 q^{32}+2 q^{30}-2 q^{28}-3 q^{26}+5 q^{24}+2 q^{22}+4 q^{18}+2 q^{16}-q^{14}-q^{12}-4 q^6+q^4+3 q^2-2+ q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-10} -2 q^{-12} + q^{-14} + q^{-16} - q^{-18} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ -q^{27}-q^{25}-q^{23}+q^{21}+2 q^{17}+q^{15}+2 q^{13}+q^9-q^5-q- q^{-3} + q^{-5} + q^{-9} + q^{-13} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}+q^{40}-2 q^{38}+2 q^{36}-3 q^{34}+3 q^{32}-4 q^{30}+4 q^{28}-3 q^{26}+3 q^{24}+4 q^{18}-4 q^{16}+7 q^{14}-7 q^{12}+8 q^{10}-8 q^8+6 q^6-5 q^4+3 q^2-2- q^{-2} +3 q^{-4} -4 q^{-6} +4 q^{-8} -4 q^{-10} +4 q^{-12} -3 q^{-14} +3 q^{-16} - q^{-18} + q^{-20} }[/math]
1,0	[math]\displaystyle{ q^{68}-q^{64}-q^{62}+q^{60}+2 q^{58}-3 q^{54}-3 q^{52}+3 q^{48}+q^{46}-3 q^{44}-3 q^{42}+q^{40}+5 q^{38}+q^{36}-2 q^{34}-q^{32}+4 q^{30}+3 q^{28}-q^{26}-3 q^{24}+q^{22}+3 q^{20}-3 q^{16}-q^{14}+2 q^{12}-3 q^8-q^6+3 q^4+3 q^2-1-4 q^{-2} +5 q^{-6} +3 q^{-8} -3 q^{-10} -4 q^{-12} + q^{-14} +4 q^{-16} + q^{-18} -3 q^{-20} -2 q^{-22} +2 q^{-24} +2 q^{-26} - q^{-28} - q^{-30} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+4 q^{86}-5 q^{84}+4 q^{82}-4 q^{80}+3 q^{76}-5 q^{74}+6 q^{72}-7 q^{70}+5 q^{68}-4 q^{66}+3 q^{62}-5 q^{60}+9 q^{58}-6 q^{56}+5 q^{54}-q^{52}-2 q^{50}+7 q^{48}-5 q^{46}+4 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-3 q^{36}-3 q^{34}+11 q^{32}-13 q^{30}+10 q^{28}-5 q^{26}-5 q^{24}+14 q^{22}-17 q^{20}+14 q^{18}-10 q^{16}+7 q^{12}-12 q^{10}+12 q^8-9 q^6+3 q^4+3 q^2-7+7 q^{-2} -3 q^{-4} -2 q^{-6} +8 q^{-8} -10 q^{-10} +7 q^{-12} -8 q^{-16} +14 q^{-18} -15 q^{-20} +10 q^{-22} -2 q^{-24} -6 q^{-26} +11 q^{-28} -11 q^{-30} +10 q^{-32} -3 q^{-34} - q^{-36} +3 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} }[/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["9 8"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-2 z^4 }[/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]=

{ 31, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^3-2 q^2+3 q-4+5 q^{-1} -5 q^{-2} +5 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -a^6+2 z^2 a^4+2 a^4-z^4 a^2-z^2 a^2-z^4-2 z^2-1+z^2 a^{-2} + a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+4 a z^7+2 z^7 a^{-1} +2 a^4 z^6+z^6 a^{-2} -z^6+2 a^5 z^5-3 a^3 z^5-13 a z^5-8 z^5 a^{-1} +2 a^6 z^4-4 a^2 z^4-4 z^4 a^{-2} -6 z^4+a^7 z^3+2 a^3 z^3+11 a z^3+8 z^3 a^{-1} -2 a^6 z^2-3 a^4 z^2+2 a^2 z^2+4 z^2 a^{-2} +7 z^2-a^7 z-a^5 z-a^3 z-3 a z-2 z a^{-1} +a^6+2 a^4- a^{-2} -1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_14, 10_131,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n60,}

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["9 8"];

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{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math], [math]\displaystyle{ q^3-2 q^2+3 q-4+5 q^{-1} -5 q^{-2} +5 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6} }[/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]=

{8_14, 10_131,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n60,}

Vassiliev invariants

V2 and V3:

(0, -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{ 0 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 344 }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ \frac{608}{3} }[/math] [math]\displaystyle{ \frac{104}{3} }[/math] [math]\displaystyle{ 24 }[/math]

V_2,1 through V_6,9 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]-2 is the signature of 9 8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 1   -1 3               2 1   1 1             2 1     -1 -1           3 2       1 -3         3 3         0 -5       2 2           0 -7     1 3             2 -9   1 2               -1 -11   1                 1 -13 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/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}\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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/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^{10}-2 q^9-q^8+6 q^7-4 q^6-6 q^5+12 q^4-3 q^3-13 q^2+16 q+1-19 q^{-1} +17 q^{-2} +5 q^{-3} -22 q^{-4} +15 q^{-5} +8 q^{-6} -20 q^{-7} +11 q^{-8} +6 q^{-9} -13 q^{-10} +7 q^{-11} +2 q^{-12} -6 q^{-13} +4 q^{-14} -2 q^{-16} + q^{-17} }[/math]
3	[math]\displaystyle{ q^{21}-2 q^{20}-q^{19}+2 q^{18}+5 q^{17}-3 q^{16}-9 q^{15}+q^{14}+14 q^{13}+2 q^{12}-16 q^{11}-9 q^{10}+19 q^9+14 q^8-17 q^7-20 q^6+13 q^5+24 q^4-8 q^3-26 q^2+3 q+26+4 q^{-1} -25 q^{-2} -9 q^{-3} +22 q^{-4} +16 q^{-5} -21 q^{-6} -19 q^{-7} +15 q^{-8} +26 q^{-9} -14 q^{-10} -24 q^{-11} +6 q^{-12} +27 q^{-13} -7 q^{-14} -17 q^{-15} - q^{-16} +15 q^{-17} -2 q^{-18} -6 q^{-19} + q^{-20} +2 q^{-21} -3 q^{-22} +2 q^{-23} +3 q^{-24} -3 q^{-25} -3 q^{-26} +2 q^{-27} +4 q^{-28} -3 q^{-29} - q^{-30} +2 q^{-32} - q^{-33} }[/math]
4	[math]\displaystyle{ q^{36}-2 q^{35}-q^{34}+2 q^{33}+q^{32}+6 q^{31}-7 q^{30}-7 q^{29}+q^{27}+24 q^{26}-6 q^{25}-15 q^{24}-11 q^{23}-13 q^{22}+43 q^{21}+6 q^{20}-7 q^{19}-18 q^{18}-43 q^{17}+46 q^{16}+12 q^{15}+14 q^{14}-3 q^{13}-64 q^{12}+34 q^{11}-7 q^{10}+27 q^9+28 q^8-60 q^7+29 q^6-44 q^5+15 q^4+55 q^3-34 q^2+42 q-82-14 q^{-1} +69 q^{-2} -3 q^{-3} +66 q^{-4} -111 q^{-5} -48 q^{-6} +74 q^{-7} +28 q^{-8} +88 q^{-9} -133 q^{-10} -79 q^{-11} +75 q^{-12} +56 q^{-13} +106 q^{-14} -144 q^{-15} -107 q^{-16} +64 q^{-17} +75 q^{-18} +122 q^{-19} -130 q^{-20} -119 q^{-21} +35 q^{-22} +65 q^{-23} +129 q^{-24} -87 q^{-25} -102 q^{-26} +4 q^{-27} +31 q^{-28} +108 q^{-29} -41 q^{-30} -60 q^{-31} -9 q^{-32} -4 q^{-33} +70 q^{-34} -12 q^{-35} -24 q^{-36} -7 q^{-37} -18 q^{-38} +37 q^{-39} -2 q^{-40} -5 q^{-41} -2 q^{-42} -16 q^{-43} +16 q^{-44} + q^{-46} -8 q^{-48} +5 q^{-49} + q^{-51} -2 q^{-53} + q^{-54} }[/math]
5	[math]\displaystyle{ q^{55}-2 q^{54}-q^{53}+2 q^{52}+q^{51}+2 q^{50}+2 q^{49}-5 q^{48}-9 q^{47}+5 q^{45}+10 q^{44}+13 q^{43}-2 q^{42}-20 q^{41}-21 q^{40}-4 q^{39}+15 q^{38}+32 q^{37}+23 q^{36}-12 q^{35}-36 q^{34}-35 q^{33}-6 q^{32}+31 q^{31}+45 q^{30}+23 q^{29}-15 q^{28}-42 q^{27}-38 q^{26}-q^{25}+25 q^{24}+32 q^{23}+23 q^{22}-18 q^{20}-23 q^{19}-25 q^{18}-21 q^{17}+10 q^{16}+48 q^{15}+59 q^{14}+26 q^{13}-51 q^{12}-104 q^{11}-76 q^{10}+36 q^9+142 q^8+136 q^7-6 q^6-166 q^5-195 q^4-42 q^3+176 q^2+256 q+94-176 q^{-1} -304 q^{-2} -149 q^{-3} +161 q^{-4} +350 q^{-5} +205 q^{-6} -152 q^{-7} -381 q^{-8} -253 q^{-9} +131 q^{-10} +417 q^{-11} +298 q^{-12} -123 q^{-13} -439 q^{-14} -338 q^{-15} +103 q^{-16} +475 q^{-17} +375 q^{-18} -101 q^{-19} -485 q^{-20} -413 q^{-21} +70 q^{-22} +517 q^{-23} +443 q^{-24} -60 q^{-25} -496 q^{-26} -471 q^{-27} +4 q^{-28} +493 q^{-29} +485 q^{-30} +17 q^{-31} -428 q^{-32} -472 q^{-33} -83 q^{-34} +379 q^{-35} +445 q^{-36} +96 q^{-37} -292 q^{-38} -382 q^{-39} -127 q^{-40} +222 q^{-41} +320 q^{-42} +114 q^{-43} -148 q^{-44} -245 q^{-45} -107 q^{-46} +102 q^{-47} +177 q^{-48} +83 q^{-49} -61 q^{-50} -122 q^{-51} -66 q^{-52} +38 q^{-53} +83 q^{-54} +44 q^{-55} -21 q^{-56} -52 q^{-57} -30 q^{-58} +7 q^{-59} +34 q^{-60} +25 q^{-61} -8 q^{-62} -20 q^{-63} -10 q^{-64} -3 q^{-65} +10 q^{-66} +14 q^{-67} -2 q^{-68} -8 q^{-69} - q^{-70} -4 q^{-71} +2 q^{-72} +6 q^{-73} - q^{-74} -2 q^{-75} - q^{-77} +2 q^{-79} - q^{-80} }[/math]
6	[math]\displaystyle{ q^{78}-2 q^{77}-q^{76}+2 q^{75}+q^{74}+2 q^{73}-2 q^{72}+4 q^{71}-7 q^{70}-9 q^{69}+3 q^{68}+4 q^{67}+11 q^{66}+2 q^{65}+18 q^{64}-14 q^{63}-27 q^{62}-13 q^{61}-7 q^{60}+17 q^{59}+9 q^{58}+63 q^{57}+4 q^{56}-31 q^{55}-35 q^{54}-43 q^{53}-12 q^{52}-20 q^{51}+102 q^{50}+44 q^{49}+13 q^{48}-15 q^{47}-46 q^{46}-48 q^{45}-98 q^{44}+82 q^{43}+28 q^{42}+44 q^{41}+31 q^{40}+23 q^{39}+q^{38}-125 q^{37}+53 q^{36}-56 q^{35}-29 q^{34}-23 q^{33}+66 q^{32}+108 q^{31}-21 q^{30}+156 q^{29}-69 q^{28}-138 q^{27}-225 q^{26}-62 q^{25}+105 q^{24}+102 q^{23}+401 q^{22}+125 q^{21}-99 q^{20}-423 q^{19}-337 q^{18}-121 q^{17}+55 q^{16}+614 q^{15}+465 q^{14}+170 q^{13}-432 q^{12}-571 q^{11}-483 q^{10}-223 q^9+629 q^8+765 q^7+569 q^6-215 q^5-622 q^4-805 q^3-624 q^2+431 q+904+938 q^{-1} +123 q^{-2} -493 q^{-3} -993 q^{-4} -1007 q^{-5} +126 q^{-6} +895 q^{-7} +1200 q^{-8} +455 q^{-9} -285 q^{-10} -1064 q^{-11} -1303 q^{-12} -171 q^{-13} +830 q^{-14} +1370 q^{-15} +721 q^{-16} -101 q^{-17} -1098 q^{-18} -1519 q^{-19} -405 q^{-20} +789 q^{-21} +1508 q^{-22} +926 q^{-23} +22 q^{-24} -1150 q^{-25} -1701 q^{-26} -591 q^{-27} +772 q^{-28} +1639 q^{-29} +1118 q^{-30} +135 q^{-31} -1179 q^{-32} -1854 q^{-33} -792 q^{-34} +682 q^{-35} +1690 q^{-36} +1297 q^{-37} +325 q^{-38} -1063 q^{-39} -1883 q^{-40} -992 q^{-41} +428 q^{-42} +1519 q^{-43} +1347 q^{-44} +556 q^{-45} -737 q^{-46} -1654 q^{-47} -1048 q^{-48} +99 q^{-49} +1105 q^{-50} +1136 q^{-51} +653 q^{-52} -342 q^{-53} -1185 q^{-54} -847 q^{-55} -108 q^{-56} +635 q^{-57} +727 q^{-58} +526 q^{-59} -82 q^{-60} -695 q^{-61} -496 q^{-62} -119 q^{-63} +307 q^{-64} +343 q^{-65} +296 q^{-66} + q^{-67} -361 q^{-68} -207 q^{-69} -46 q^{-70} +151 q^{-71} +118 q^{-72} +125 q^{-73} +2 q^{-74} -190 q^{-75} -59 q^{-76} -2 q^{-77} +85 q^{-78} +31 q^{-79} +48 q^{-80} - q^{-81} -106 q^{-82} -9 q^{-83} +4 q^{-84} +46 q^{-85} +7 q^{-86} +23 q^{-87} + q^{-88} -55 q^{-89} +2 q^{-90} -2 q^{-91} +21 q^{-92} +13 q^{-94} +2 q^{-95} -24 q^{-96} +4 q^{-97} -4 q^{-98} +8 q^{-99} - q^{-100} +5 q^{-101} + q^{-102} -8 q^{-103} +3 q^{-104} -2 q^{-105} +2 q^{-106} + q^{-108} -2 q^{-110} + q^{-111} }[/math]
7	[math]\displaystyle{ q^{105}-2 q^{104}-q^{103}+2 q^{102}+q^{101}+2 q^{100}-2 q^{99}+2 q^{97}-7 q^{96}-6 q^{95}+2 q^{94}+4 q^{93}+13 q^{92}+4 q^{91}+9 q^{89}-18 q^{88}-23 q^{87}-16 q^{86}-9 q^{85}+27 q^{84}+25 q^{83}+21 q^{82}+41 q^{81}-6 q^{80}-34 q^{79}-47 q^{78}-74 q^{77}-2 q^{76}+20 q^{75}+33 q^{74}+98 q^{73}+51 q^{72}+22 q^{71}-21 q^{70}-120 q^{69}-68 q^{68}-47 q^{67}-37 q^{66}+88 q^{65}+66 q^{64}+87 q^{63}+92 q^{62}-60 q^{61}-30 q^{60}-55 q^{59}-106 q^{58}+8 q^{57}-47 q^{56}-9 q^{55}+86 q^{54}-21 q^{53}+89 q^{52}+111 q^{51}+24 q^{50}+113 q^{49}-87 q^{48}-194 q^{47}-160 q^{46}-282 q^{45}-37 q^{44}+189 q^{43}+276 q^{42}+518 q^{41}+280 q^{40}-60 q^{39}-308 q^{38}-739 q^{37}-585 q^{36}-217 q^{35}+169 q^{34}+857 q^{33}+926 q^{32}+616 q^{31}+119 q^{30}-826 q^{29}-1166 q^{28}-1039 q^{27}-577 q^{26}+584 q^{25}+1262 q^{24}+1432 q^{23}+1110 q^{22}-168 q^{21}-1154 q^{20}-1689 q^{19}-1640 q^{18}-385 q^{17}+837 q^{16}+1754 q^{15}+2101 q^{14}+1008 q^{13}-356 q^{12}-1635 q^{11}-2413 q^{10}-1596 q^9-248 q^8+1313 q^7+2554 q^6+2137 q^5+897 q^4-861 q^3-2540 q^2-2550 q-1529+315 q^{-1} +2369 q^{-2} +2843 q^{-3} +2130 q^{-4} +271 q^{-5} -2121 q^{-6} -3031 q^{-7} -2635 q^{-8} -832 q^{-9} +1789 q^{-10} +3125 q^{-11} +3072 q^{-12} +1371 q^{-13} -1466 q^{-14} -3167 q^{-15} -3425 q^{-16} -1833 q^{-17} +1156 q^{-18} +3174 q^{-19} +3717 q^{-20} +2229 q^{-21} -889 q^{-22} -3187 q^{-23} -3968 q^{-24} -2553 q^{-25} +684 q^{-26} +3226 q^{-27} +4189 q^{-28} +2816 q^{-29} -538 q^{-30} -3284 q^{-31} -4403 q^{-32} -3058 q^{-33} +419 q^{-34} +3389 q^{-35} +4638 q^{-36} +3272 q^{-37} -328 q^{-38} -3462 q^{-39} -4848 q^{-40} -3538 q^{-41} +154 q^{-42} +3524 q^{-43} +5082 q^{-44} +3800 q^{-45} +52 q^{-46} -3449 q^{-47} -5191 q^{-48} -4102 q^{-49} -422 q^{-50} +3238 q^{-51} +5228 q^{-52} +4359 q^{-53} +816 q^{-54} -2844 q^{-55} -5022 q^{-56} -4488 q^{-57} -1295 q^{-58} +2267 q^{-59} +4638 q^{-60} +4470 q^{-61} +1682 q^{-62} -1637 q^{-63} -4014 q^{-64} -4185 q^{-65} -1941 q^{-66} +945 q^{-67} +3259 q^{-68} +3730 q^{-69} +2007 q^{-70} -391 q^{-71} -2460 q^{-72} -3070 q^{-73} -1872 q^{-74} -39 q^{-75} +1705 q^{-76} +2370 q^{-77} +1579 q^{-78} +275 q^{-79} -1070 q^{-80} -1693 q^{-81} -1208 q^{-82} -357 q^{-83} +616 q^{-84} +1104 q^{-85} +816 q^{-86} +339 q^{-87} -303 q^{-88} -659 q^{-89} -498 q^{-90} -258 q^{-91} +136 q^{-92} +358 q^{-93} +242 q^{-94} +156 q^{-95} -44 q^{-96} -161 q^{-97} -78 q^{-98} -92 q^{-99} +13 q^{-100} +72 q^{-101} -12 q^{-102} +26 q^{-103} -14 q^{-104} -15 q^{-105} +62 q^{-106} -2 q^{-107} +6 q^{-108} +3 q^{-109} -59 q^{-110} -11 q^{-111} -22 q^{-112} -6 q^{-113} +65 q^{-114} +20 q^{-115} +12 q^{-116} - q^{-117} -42 q^{-118} -5 q^{-119} -19 q^{-120} -13 q^{-121} +34 q^{-122} +13 q^{-123} +10 q^{-124} +3 q^{-125} -20 q^{-126} -8 q^{-128} -9 q^{-129} +14 q^{-130} +2 q^{-131} +4 q^{-132} +4 q^{-133} -8 q^{-134} + q^{-135} -3 q^{-136} -2 q^{-137} +5 q^{-138} - q^{-139} +2 q^{-141} -2 q^{-142} - q^{-144} +2 q^{-146} - q^{-147} }[/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.

9_7 9_9