9 12

9_11

9_13

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 12 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 10 16 14 2 18 8 6 12
Conway Notation	[4212]

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

[edit Notes on presentations of 9 12]

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 12"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 16 14 2 18 8 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4212]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 8.83664 A-Polynomial See Data:9 12/A-polynomial

[edit Notes for 9 12'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 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+9 t-13+9 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 35, -2 }
Jones polynomial	[math]\displaystyle{ q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -6 q^{-4} +5 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^8+2 z^2 a^6+2 a^6-z^4 a^4-z^2 a^4-a^4-z^4 a^2-z^2 a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+2 z^7 a^7-5 z^5 a^7+3 z^3 a^7-z a^7+z^8 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6+4 z^7 a^5-11 z^5 a^5+13 z^3 a^5-4 z a^5+z^8 a^4-z^4 a^4+3 z^2 a^4-a^4+2 z^7 a^3-3 z^5 a^3+4 z^3 a^3-2 z a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{26}-q^{24}+q^{22}+q^{18}+2 q^{16}-q^{14}-q^{10}+q^6-q^4+2 q^2+ q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-2 q^{118}-2 q^{116}+7 q^{114}-10 q^{112}+10 q^{110}-10 q^{108}+4 q^{106}+4 q^{104}-15 q^{102}+20 q^{100}-20 q^{98}+14 q^{96}-q^{94}-12 q^{92}+19 q^{90}-18 q^{88}+15 q^{86}-3 q^{84}-10 q^{82}+14 q^{80}-11 q^{78}+2 q^{76}+10 q^{74}-16 q^{72}+18 q^{70}-8 q^{68}-4 q^{66}+17 q^{64}-26 q^{62}+29 q^{60}-21 q^{58}+6 q^{56}+11 q^{54}-23 q^{52}+30 q^{50}-25 q^{48}+13 q^{46}-13 q^{42}+16 q^{40}-14 q^{38}+3 q^{36}+8 q^{34}-13 q^{32}+10 q^{30}-2 q^{28}-9 q^{26}+17 q^{24}-19 q^{22}+15 q^{20}-6 q^{18}-5 q^{16}+14 q^{14}-16 q^{12}+17 q^{10}-10 q^8+5 q^6+q^4-7 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_12.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{17}+q^{15}-q^{13}+2 q^{11}-q^9+q^5-q^3+2 q- q^{-1} + q^{-3} }[/math]
2	[math]\displaystyle{ q^{48}-q^{46}-q^{44}+3 q^{42}-2 q^{40}-4 q^{38}+5 q^{36}+q^{34}-6 q^{32}+5 q^{30}+3 q^{28}-7 q^{26}+2 q^{24}+3 q^{22}-3 q^{20}-2 q^{18}+2 q^{16}+4 q^{14}-5 q^{12}+7 q^8-5 q^6-2 q^4+7 q^2-2-2 q^{-2} +3 q^{-4} - q^{-6} - q^{-8} + q^{-10} }[/math]
3	[math]\displaystyle{ -q^{93}+q^{91}+q^{89}-q^{87}-2 q^{85}+2 q^{83}+5 q^{81}-2 q^{79}-8 q^{77}+10 q^{73}+3 q^{71}-12 q^{69}-10 q^{67}+12 q^{65}+15 q^{63}-6 q^{61}-19 q^{59}+3 q^{57}+21 q^{55}+2 q^{53}-21 q^{51}-7 q^{49}+19 q^{47}+7 q^{45}-14 q^{43}-9 q^{41}+9 q^{39}+10 q^{37}-5 q^{35}-10 q^{33}-3 q^{31}+11 q^{29}+10 q^{27}-10 q^{25}-16 q^{23}+9 q^{21}+20 q^{19}-5 q^{17}-21 q^{15}+q^{13}+20 q^{11}+4 q^9-15 q^7-5 q^5+11 q^3+5 q-5 q^{-1} -4 q^{-3} +3 q^{-5} +2 q^{-7} -2 q^{-9} +2 q^{-13} - q^{-17} - q^{-19} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{152}-q^{150}-q^{148}+q^{146}+2 q^{142}-4 q^{140}-3 q^{138}+4 q^{136}+3 q^{134}+8 q^{132}-8 q^{130}-13 q^{128}+2 q^{126}+8 q^{124}+23 q^{122}-4 q^{120}-25 q^{118}-17 q^{116}-4 q^{114}+40 q^{112}+23 q^{110}-12 q^{108}-36 q^{106}-41 q^{104}+28 q^{102}+50 q^{100}+33 q^{98}-24 q^{96}-77 q^{94}-14 q^{92}+44 q^{90}+69 q^{88}+12 q^{86}-80 q^{84}-49 q^{82}+16 q^{80}+74 q^{78}+38 q^{76}-56 q^{74}-51 q^{72}-5 q^{70}+52 q^{68}+39 q^{66}-25 q^{64}-39 q^{62}-15 q^{60}+29 q^{58}+32 q^{56}+3 q^{54}-30 q^{52}-29 q^{50}+31 q^{46}+43 q^{44}-17 q^{42}-50 q^{40}-36 q^{38}+24 q^{36}+79 q^{34}+13 q^{32}-50 q^{30}-68 q^{28}-5 q^{26}+84 q^{24}+42 q^{22}-18 q^{20}-67 q^{18}-38 q^{16}+48 q^{14}+42 q^{12}+18 q^{10}-33 q^8-40 q^6+10 q^4+14 q^2+24- q^{-2} -18 q^{-4} -2 q^{-6} -5 q^{-8} +10 q^{-10} +4 q^{-12} -3 q^{-14} +3 q^{-16} -6 q^{-18} +4 q^{-26} - q^{-28} - q^{-32} - q^{-34} + q^{-36} }[/math]
5	[math]\displaystyle{ -q^{225}+q^{223}+q^{221}-q^{219}+2 q^{211}+2 q^{209}-4 q^{207}-6 q^{205}-q^{203}+4 q^{201}+9 q^{199}+8 q^{197}-4 q^{195}-19 q^{193}-17 q^{191}+3 q^{189}+23 q^{187}+33 q^{185}+13 q^{183}-24 q^{181}-50 q^{179}-36 q^{177}+12 q^{175}+56 q^{173}+66 q^{171}+23 q^{169}-47 q^{167}-91 q^{165}-71 q^{163}+6 q^{161}+91 q^{159}+119 q^{157}+58 q^{155}-56 q^{153}-149 q^{151}-138 q^{149}-4 q^{147}+147 q^{145}+202 q^{143}+99 q^{141}-106 q^{139}-247 q^{137}-186 q^{135}+40 q^{133}+245 q^{131}+256 q^{129}+44 q^{127}-220 q^{125}-299 q^{123}-113 q^{121}+170 q^{119}+300 q^{117}+164 q^{115}-109 q^{113}-278 q^{111}-188 q^{109}+65 q^{107}+231 q^{105}+181 q^{103}-19 q^{101}-182 q^{99}-164 q^{97}-3 q^{95}+138 q^{93}+134 q^{91}+17 q^{89}-96 q^{87}-114 q^{85}-35 q^{83}+68 q^{81}+106 q^{79}+51 q^{77}-34 q^{75}-103 q^{73}-93 q^{71}+q^{69}+108 q^{67}+140 q^{65}+48 q^{63}-107 q^{61}-193 q^{59}-114 q^{57}+89 q^{55}+238 q^{53}+187 q^{51}-47 q^{49}-263 q^{47}-256 q^{45}-19 q^{43}+249 q^{41}+307 q^{39}+93 q^{37}-201 q^{35}-316 q^{33}-160 q^{31}+122 q^{29}+287 q^{27}+207 q^{25}-35 q^{23}-224 q^{21}-210 q^{19}-36 q^{17}+139 q^{15}+184 q^{13}+84 q^{11}-65 q^9-135 q^7-90 q^5+7 q^3+80 q+81 q^{-1} +26 q^{-3} -39 q^{-5} -57 q^{-7} -32 q^{-9} +8 q^{-11} +33 q^{-13} +28 q^{-15} +7 q^{-17} -15 q^{-19} -20 q^{-21} -8 q^{-23} +3 q^{-25} +9 q^{-27} +9 q^{-29} +4 q^{-31} -5 q^{-33} -6 q^{-35} -2 q^{-37} -2 q^{-39} +2 q^{-41} +4 q^{-43} + q^{-45} - q^{-47} - q^{-51} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{26}-q^{24}+q^{22}+q^{18}+2 q^{16}-q^{14}-q^{10}+q^6-q^4+2 q^2+ q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{68}-2 q^{66}+4 q^{64}-8 q^{62}+17 q^{60}-24 q^{58}+32 q^{56}-48 q^{54}+57 q^{52}-64 q^{50}+62 q^{48}-58 q^{46}+46 q^{44}-18 q^{42}-6 q^{40}+38 q^{38}-65 q^{36}+90 q^{34}-112 q^{32}+116 q^{30}-122 q^{28}+110 q^{26}-90 q^{24}+68 q^{22}-37 q^{20}+10 q^{18}+20 q^{16}-36 q^{14}+47 q^{12}-54 q^{10}+56 q^8-48 q^6+42 q^4-36 q^2+30-20 q^{-2} +15 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{66}+q^{64}-2 q^{60}-q^{58}+q^{56}-q^{54}-3 q^{52}-q^{50}+4 q^{48}+3 q^{46}-2 q^{44}+4 q^{40}-5 q^{36}-3 q^{34}+2 q^{32}-2 q^{28}+2 q^{26}+q^{24}-q^{22}+3 q^{20}+2 q^{18}-3 q^{16}-q^{14}+4 q^{12}+q^{10}-4 q^8+6 q^4-3+ q^{-2} +2 q^{-4} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{54}-q^{52}+q^{48}-3 q^{46}+q^{44}+2 q^{42}-5 q^{40}+2 q^{38}+4 q^{36}-6 q^{34}+q^{32}+4 q^{30}-3 q^{28}+3 q^{24}+q^{22}-q^{20}-q^{18}+4 q^{16}-2 q^{14}-4 q^{12}+6 q^{10}-q^8-5 q^6+5 q^4+q^2-2+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ -q^{35}-q^{33}-q^{31}+q^{29}+2 q^{25}+q^{23}+2 q^{21}-q^{19}-q^{15}-q^{13}+q^7-q^5+2 q^3+ q^{-1} + q^{-5} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{54}+q^{52}-2 q^{50}+3 q^{48}-5 q^{46}+5 q^{44}-6 q^{42}+5 q^{40}-4 q^{38}+4 q^{36}-q^{32}+6 q^{30}-7 q^{28}+10 q^{26}-11 q^{24}+11 q^{22}-11 q^{20}+7 q^{18}-6 q^{16}+2 q^{14}-2 q^{10}+5 q^8-5 q^6+7 q^4-5 q^2+4-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{88}-q^{84}-q^{82}+q^{80}+2 q^{78}-q^{76}-4 q^{74}-q^{72}+4 q^{70}+4 q^{68}-3 q^{66}-6 q^{64}+6 q^{60}+4 q^{58}-5 q^{56}-5 q^{54}+q^{52}+5 q^{50}-4 q^{46}-q^{44}+4 q^{42}+2 q^{40}-3 q^{38}-2 q^{36}+3 q^{34}+4 q^{32}-2 q^{30}-4 q^{28}+q^{26}+5 q^{24}-5 q^{20}-2 q^{18}+5 q^{16}+5 q^{14}-3 q^{12}-6 q^{10}-q^8+6 q^6+3 q^4-2 q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-2 q^{118}-2 q^{116}+7 q^{114}-10 q^{112}+10 q^{110}-10 q^{108}+4 q^{106}+4 q^{104}-15 q^{102}+20 q^{100}-20 q^{98}+14 q^{96}-q^{94}-12 q^{92}+19 q^{90}-18 q^{88}+15 q^{86}-3 q^{84}-10 q^{82}+14 q^{80}-11 q^{78}+2 q^{76}+10 q^{74}-16 q^{72}+18 q^{70}-8 q^{68}-4 q^{66}+17 q^{64}-26 q^{62}+29 q^{60}-21 q^{58}+6 q^{56}+11 q^{54}-23 q^{52}+30 q^{50}-25 q^{48}+13 q^{46}-13 q^{42}+16 q^{40}-14 q^{38}+3 q^{36}+8 q^{34}-13 q^{32}+10 q^{30}-2 q^{28}-9 q^{26}+17 q^{24}-19 q^{22}+15 q^{20}-6 q^{18}-5 q^{16}+14 q^{14}-16 q^{12}+17 q^{10}-10 q^8+5 q^6+q^4-7 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/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 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^2+9 t-13+9 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^4+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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+2 z^7 a^7-5 z^5 a^7+3 z^3 a^7-z a^7+z^8 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6+4 z^7 a^5-11 z^5 a^5+13 z^3 a^5-4 z a^5+z^8 a^4-z^4 a^4+3 z^2 a^4-a^4+2 z^7 a^3-3 z^5 a^3+4 z^3 a^3-2 z a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n84,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n15,}

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 12"];

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+9 t-13+9 t^{-1} -2 t^{-2} }[/math], [math]\displaystyle{ q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -6 q^{-4} +5 q^{-5} -3 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]=

{K11n84,}

In[6]:=

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

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

Out[6]=

{K11n15,}

Vassiliev invariants

V2 and V3:

(1, -3)

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{ 4 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{302}{3} }[/math] [math]\displaystyle{ \frac{58}{3} }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -400 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -120 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{1208}{3} }[/math] [math]\displaystyle{ \frac{232}{3} }[/math] [math]\displaystyle{ \frac{49471}{30} }[/math] [math]\displaystyle{ -\frac{5942}{15} }[/math] [math]\displaystyle{ \frac{49622}{45} }[/math] [math]\displaystyle{ \frac{1793}{18} }[/math] [math]\displaystyle{ \frac{4831}{30} }[/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 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 1   -1 -1               3 1   2 -3             3 2     -1 -5           3 2       1 -7         3 3         0 -9       2 3           -1 -11     1 3             2 -13   1 2               -1 -15   1                 1 -17 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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^4-2 q^3+5 q-7+14 q^{-2} -16 q^{-3} -3 q^{-4} +26 q^{-5} -23 q^{-6} -8 q^{-7} +35 q^{-8} -25 q^{-9} -12 q^{-10} +34 q^{-11} -19 q^{-12} -13 q^{-13} +25 q^{-14} -9 q^{-15} -11 q^{-16} +14 q^{-17} -2 q^{-18} -7 q^{-19} +5 q^{-20} -2 q^{-22} + q^{-23} }[/math]
3	[math]\displaystyle{ q^9-2 q^8+q^6+3 q^5-4 q^4-2 q^3+5 q^2+4 q-11-3 q^{-1} +15 q^{-2} +10 q^{-3} -27 q^{-4} -13 q^{-5} +34 q^{-6} +26 q^{-7} -46 q^{-8} -35 q^{-9} +50 q^{-10} +51 q^{-11} -57 q^{-12} -60 q^{-13} +56 q^{-14} +71 q^{-15} -56 q^{-16} -74 q^{-17} +49 q^{-18} +76 q^{-19} -41 q^{-20} -75 q^{-21} +31 q^{-22} +71 q^{-23} -20 q^{-24} -63 q^{-25} +5 q^{-26} +57 q^{-27} +3 q^{-28} -44 q^{-29} -13 q^{-30} +35 q^{-31} +16 q^{-32} -23 q^{-33} -16 q^{-34} +13 q^{-35} +14 q^{-36} -8 q^{-37} -9 q^{-38} +3 q^{-39} +6 q^{-40} -2 q^{-41} -2 q^{-42} +2 q^{-44} - q^{-45} }[/math]
4	[math]\displaystyle{ q^{16}-2 q^{15}+q^{13}-q^{12}+6 q^{11}-6 q^{10}+q^8-7 q^7+15 q^6-12 q^5+7 q^4+7 q^3-22 q^2+18 q-28+24 q^{-1} +32 q^{-2} -32 q^{-3} +14 q^{-4} -78 q^{-5} +31 q^{-6} +83 q^{-7} -8 q^{-8} +20 q^{-9} -164 q^{-10} +2 q^{-11} +132 q^{-12} +52 q^{-13} +62 q^{-14} -253 q^{-15} -61 q^{-16} +150 q^{-17} +115 q^{-18} +128 q^{-19} -308 q^{-20} -121 q^{-21} +136 q^{-22} +148 q^{-23} +188 q^{-24} -320 q^{-25} -152 q^{-26} +107 q^{-27} +147 q^{-28} +221 q^{-29} -291 q^{-30} -155 q^{-31} +63 q^{-32} +123 q^{-33} +235 q^{-34} -227 q^{-35} -142 q^{-36} +6 q^{-37} +77 q^{-38} +230 q^{-39} -133 q^{-40} -106 q^{-41} -52 q^{-42} +12 q^{-43} +199 q^{-44} -41 q^{-45} -49 q^{-46} -77 q^{-47} -46 q^{-48} +136 q^{-49} +12 q^{-50} +8 q^{-51} -60 q^{-52} -68 q^{-53} +67 q^{-54} +17 q^{-55} +32 q^{-56} -25 q^{-57} -51 q^{-58} +23 q^{-59} +4 q^{-60} +24 q^{-61} -4 q^{-62} -24 q^{-63} +8 q^{-64} -2 q^{-65} +9 q^{-66} + q^{-67} -8 q^{-68} +3 q^{-69} - q^{-70} +2 q^{-71} -2 q^{-73} + q^{-74} }[/math]
5	[math]\displaystyle{ q^{25}-2 q^{24}+q^{22}-q^{21}+2 q^{20}+4 q^{19}-4 q^{18}-4 q^{17}+q^{16}-5 q^{15}+3 q^{14}+13 q^{13}+q^{12}-4 q^{11}-5 q^{10}-16 q^9-9 q^8+18 q^7+23 q^6+17 q^5-41 q^3-49 q^2-7 q+41+82 q^{-1} +55 q^{-2} -42 q^{-3} -122 q^{-4} -104 q^{-5} -4 q^{-6} +152 q^{-7} +204 q^{-8} +58 q^{-9} -167 q^{-10} -279 q^{-11} -178 q^{-12} +138 q^{-13} +393 q^{-14} +300 q^{-15} -87 q^{-16} -444 q^{-17} -460 q^{-18} -18 q^{-19} +508 q^{-20} +594 q^{-21} +127 q^{-22} -502 q^{-23} -728 q^{-24} -255 q^{-25} +501 q^{-26} +810 q^{-27} +361 q^{-28} -451 q^{-29} -877 q^{-30} -458 q^{-31} +422 q^{-32} +896 q^{-33} +516 q^{-34} -359 q^{-35} -909 q^{-36} -565 q^{-37} +328 q^{-38} +886 q^{-39} +585 q^{-40} -274 q^{-41} -854 q^{-42} -603 q^{-43} +225 q^{-44} +807 q^{-45} +603 q^{-46} -161 q^{-47} -737 q^{-48} -599 q^{-49} +84 q^{-50} +646 q^{-51} +585 q^{-52} +2 q^{-53} -537 q^{-54} -549 q^{-55} -82 q^{-56} +393 q^{-57} +495 q^{-58} +171 q^{-59} -264 q^{-60} -413 q^{-61} -212 q^{-62} +110 q^{-63} +309 q^{-64} +250 q^{-65} -201 q^{-67} -223 q^{-68} -95 q^{-69} +83 q^{-70} +189 q^{-71} +141 q^{-72} +4 q^{-73} -120 q^{-74} -150 q^{-75} -68 q^{-76} +55 q^{-77} +130 q^{-78} +97 q^{-79} -6 q^{-80} -89 q^{-81} -96 q^{-82} -30 q^{-83} +53 q^{-84} +77 q^{-85} +38 q^{-86} -19 q^{-87} -53 q^{-88} -40 q^{-89} +9 q^{-90} +29 q^{-91} +24 q^{-92} +7 q^{-93} -16 q^{-94} -20 q^{-95} - q^{-96} +9 q^{-97} +4 q^{-98} +5 q^{-99} - q^{-100} -8 q^{-101} +4 q^{-103} - q^{-104} + q^{-106} -2 q^{-107} +2 q^{-109} - q^{-110} }[/math]
6	[math]\displaystyle{ q^{36}-2 q^{35}+q^{33}-q^{32}+2 q^{31}+6 q^{29}-8 q^{28}-4 q^{27}+3 q^{26}-6 q^{25}+4 q^{24}+4 q^{23}+23 q^{22}-14 q^{21}-10 q^{20}+q^{19}-23 q^{18}-5 q^{17}+7 q^{16}+63 q^{15}-8 q^{14}-6 q^{13}+q^{12}-66 q^{11}-48 q^{10}-9 q^9+128 q^8+34 q^7+41 q^6+30 q^5-140 q^4-167 q^3-103 q^2+179 q+123+203 q^{-1} +184 q^{-2} -185 q^{-3} -384 q^{-4} -388 q^{-5} +74 q^{-6} +164 q^{-7} +514 q^{-8} +622 q^{-9} +3 q^{-10} -566 q^{-11} -904 q^{-12} -390 q^{-13} -102 q^{-14} +801 q^{-15} +1362 q^{-16} +649 q^{-17} -403 q^{-18} -1420 q^{-19} -1200 q^{-20} -893 q^{-21} +729 q^{-22} +2109 q^{-23} +1675 q^{-24} +284 q^{-25} -1581 q^{-26} -2006 q^{-27} -2038 q^{-28} +171 q^{-29} +2492 q^{-30} +2666 q^{-31} +1258 q^{-32} -1295 q^{-33} -2446 q^{-34} -3078 q^{-35} -598 q^{-36} +2439 q^{-37} +3266 q^{-38} +2081 q^{-39} -819 q^{-40} -2475 q^{-41} -3703 q^{-42} -1212 q^{-43} +2177 q^{-44} +3457 q^{-45} +2535 q^{-46} -430 q^{-47} -2300 q^{-48} -3934 q^{-49} -1547 q^{-50} +1910 q^{-51} +3419 q^{-52} +2693 q^{-53} -174 q^{-54} -2078 q^{-55} -3930 q^{-56} -1709 q^{-57} +1630 q^{-58} +3254 q^{-59} +2709 q^{-60} +79 q^{-61} -1770 q^{-62} -3764 q^{-63} -1836 q^{-64} +1206 q^{-65} +2904 q^{-66} +2635 q^{-67} +435 q^{-68} -1252 q^{-69} -3366 q^{-70} -1943 q^{-71} +568 q^{-72} +2263 q^{-73} +2379 q^{-74} +848 q^{-75} -500 q^{-76} -2642 q^{-77} -1884 q^{-78} -154 q^{-79} +1349 q^{-80} +1810 q^{-81} +1092 q^{-82} +303 q^{-83} -1635 q^{-84} -1485 q^{-85} -652 q^{-86} +403 q^{-87} +959 q^{-88} +938 q^{-89} +810 q^{-90} -626 q^{-91} -773 q^{-92} -670 q^{-93} -204 q^{-94} +121 q^{-95} +428 q^{-96} +794 q^{-97} +6 q^{-98} -73 q^{-99} -295 q^{-100} -283 q^{-101} -336 q^{-102} -82 q^{-103} +412 q^{-104} +124 q^{-105} +265 q^{-106} +83 q^{-107} -43 q^{-108} -333 q^{-109} -281 q^{-110} +63 q^{-111} -25 q^{-112} +223 q^{-113} +196 q^{-114} +145 q^{-115} -138 q^{-116} -201 q^{-117} -49 q^{-118} -120 q^{-119} +69 q^{-120} +115 q^{-121} +152 q^{-122} -17 q^{-123} -75 q^{-124} -19 q^{-125} -94 q^{-126} -6 q^{-127} +28 q^{-128} +82 q^{-129} +6 q^{-130} -20 q^{-131} +10 q^{-132} -40 q^{-133} -12 q^{-134} - q^{-135} +31 q^{-136} + q^{-137} -8 q^{-138} +11 q^{-139} -11 q^{-140} -4 q^{-141} -3 q^{-142} +10 q^{-143} - q^{-144} -5 q^{-145} +5 q^{-146} -2 q^{-147} - q^{-149} +2 q^{-150} -2 q^{-152} + q^{-153} }[/math]
7	[math]\displaystyle{ q^{49}-2 q^{48}+q^{46}-q^{45}+2 q^{44}+2 q^{42}+2 q^{41}-8 q^{40}-2 q^{39}+2 q^{38}-5 q^{37}+6 q^{36}+2 q^{35}+10 q^{34}+14 q^{33}-20 q^{32}-8 q^{31}-3 q^{30}-18 q^{29}+5 q^{28}+q^{27}+27 q^{26}+47 q^{25}-22 q^{24}-15 q^{23}-17 q^{22}-52 q^{21}-3 q^{20}-13 q^{19}+47 q^{18}+115 q^{17}+2 q^{16}-10 q^{15}-54 q^{14}-129 q^{13}-46 q^{12}-45 q^{11}+87 q^{10}+245 q^9+105 q^8+56 q^7-107 q^6-328 q^5-232 q^4-190 q^3+107 q^2+497 q+437+394 q^{-1} -21 q^{-2} -633 q^{-3} -751 q^{-4} -785 q^{-5} -213 q^{-6} +719 q^{-7} +1108 q^{-8} +1367 q^{-9} +734 q^{-10} -604 q^{-11} -1499 q^{-12} -2163 q^{-13} -1532 q^{-14} +211 q^{-15} +1679 q^{-16} +3035 q^{-17} +2740 q^{-18} +666 q^{-19} -1634 q^{-20} -3948 q^{-21} -4153 q^{-22} -1911 q^{-23} +1075 q^{-24} +4556 q^{-25} +5758 q^{-26} +3676 q^{-27} -81 q^{-28} -4912 q^{-29} -7247 q^{-30} -5570 q^{-31} -1417 q^{-32} +4701 q^{-33} +8508 q^{-34} +7619 q^{-35} +3201 q^{-36} -4138 q^{-37} -9366 q^{-38} -9411 q^{-39} -5108 q^{-40} +3157 q^{-41} +9801 q^{-42} +10930 q^{-43} +6913 q^{-44} -2041 q^{-45} -9850 q^{-46} -11991 q^{-47} -8469 q^{-48} +875 q^{-49} +9628 q^{-50} +12691 q^{-51} +9659 q^{-52} +131 q^{-53} -9240 q^{-54} -13006 q^{-55} -10516 q^{-56} -995 q^{-57} +8840 q^{-58} +13141 q^{-59} +11041 q^{-60} +1579 q^{-61} -8434 q^{-62} -13063 q^{-63} -11373 q^{-64} -2057 q^{-65} +8099 q^{-66} +12985 q^{-67} +11532 q^{-68} +2363 q^{-69} -7775 q^{-70} -12798 q^{-71} -11617 q^{-72} -2689 q^{-73} +7427 q^{-74} +12593 q^{-75} +11682 q^{-76} +3005 q^{-77} -7008 q^{-78} -12276 q^{-79} -11681 q^{-80} -3415 q^{-81} +6397 q^{-82} +11827 q^{-83} +11667 q^{-84} +3922 q^{-85} -5618 q^{-86} -11181 q^{-87} -11530 q^{-88} -4511 q^{-89} +4590 q^{-90} +10267 q^{-91} +11255 q^{-92} +5150 q^{-93} -3337 q^{-94} -9083 q^{-95} -10777 q^{-96} -5738 q^{-97} +1963 q^{-98} +7609 q^{-99} +9957 q^{-100} +6175 q^{-101} -463 q^{-102} -5890 q^{-103} -8898 q^{-104} -6365 q^{-105} -848 q^{-106} +4064 q^{-107} +7429 q^{-108} +6152 q^{-109} +2011 q^{-110} -2213 q^{-111} -5819 q^{-112} -5594 q^{-113} -2696 q^{-114} +633 q^{-115} +4038 q^{-116} +4596 q^{-117} +2962 q^{-118} +657 q^{-119} -2385 q^{-120} -3432 q^{-121} -2731 q^{-122} -1386 q^{-123} +993 q^{-124} +2130 q^{-125} +2124 q^{-126} +1672 q^{-127} -4 q^{-128} -999 q^{-129} -1338 q^{-130} -1489 q^{-131} -512 q^{-132} +135 q^{-133} +529 q^{-134} +1043 q^{-135} +632 q^{-136} +370 q^{-137} +107 q^{-138} -522 q^{-139} -459 q^{-140} -520 q^{-141} -489 q^{-142} +56 q^{-143} +163 q^{-144} +444 q^{-145} +608 q^{-146} +232 q^{-147} +118 q^{-148} -241 q^{-149} -546 q^{-150} -343 q^{-151} -278 q^{-152} +30 q^{-153} +369 q^{-154} +315 q^{-155} +358 q^{-156} +105 q^{-157} -224 q^{-158} -217 q^{-159} -294 q^{-160} -160 q^{-161} +65 q^{-162} +109 q^{-163} +245 q^{-164} +166 q^{-165} -26 q^{-166} -45 q^{-167} -134 q^{-168} -113 q^{-169} -24 q^{-170} -19 q^{-171} +92 q^{-172} +93 q^{-173} +11 q^{-174} +10 q^{-175} -42 q^{-176} -35 q^{-177} -10 q^{-178} -32 q^{-179} +18 q^{-180} +36 q^{-181} +6 q^{-182} +8 q^{-183} -14 q^{-184} -6 q^{-185} +6 q^{-186} -16 q^{-187} +10 q^{-189} +2 q^{-190} +3 q^{-191} -6 q^{-192} - q^{-193} +6 q^{-194} -4 q^{-195} -2 q^{-196} +2 q^{-197} + q^{-199} -2 q^{-200} +2 q^{-202} - q^{-203} }[/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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