9 4

9_3

9_5

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 4 at Knotilus!

[edit 9 4 Quick Notes]

[edit 9 4 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₆₂₇ X_3,12,4,13 X_7,18,8,1 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9 X_5,14,6,15 X_11,2,12,3 X_13,4,14,5
Gauss code	-1, 8, -2, 9, -7, 1, -3, 6, -4, 5, -8, 2, -9, 7, -5, 4, -6, 3
Dowker-Thistlethwaite code	6 12 14 18 16 2 4 10 8
Conway Notation	[54]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 9 4]

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 4"];

In[4]:= PD[K]

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

Out[4]= X₁₆₂₇ X_3,12,4,13 X_7,18,8,1 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9 X_5,14,6,15 X_11,2,12,3 X_13,4,14,5

In[5]:= GaussCode[K]

Out[5]= -1, 8, -2, 9, -7, 1, -3, 6, -4, 5, -8, 2, -9, 7, -5, 4, -6, 3

In[6]:= DTCode[K]

Out[6]= 6 12 14 18 16 2 4 10 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [54]

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}(4,\{-1,-1,-1,-1,-1,-2,1,-2,-3,2,-3\}) }[/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]= { 4, 11, 4 }

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[{11, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 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{ \{4,7\} }[/math]
Nakanishi index	1
Maximal Thurston-Bennequin number	[-14][3]
Hyperbolic Volume	5.55652
A-Polynomial	See Data:9 4/A-polynomial

[edit Notes for 9 4'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{ 2 }[/math]
Rasmussen s-Invariant	-4

[edit Notes for 9 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^2-5 t+5-5 t^{-1} +3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 21, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} - q^{-3} +2 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-2 a^{10}+z^4 a^8+3 z^2 a^8+2 a^8+z^4 a^6+2 z^2 a^6+z^4 a^4+3 z^2 a^4+a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-4 z^3 a^{13}+3 z a^{13}+z^6 a^{12}-3 z^4 a^{12}+z^2 a^{12}+z^7 a^{11}-3 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+z^8 a^{10}-5 z^6 a^{10}+11 z^4 a^{10}-10 z^2 a^{10}+2 a^{10}+2 z^7 a^9-8 z^5 a^9+12 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+11 z^4 a^8-7 z^2 a^8+2 a^8+z^7 a^7-3 z^5 a^7+4 z^3 a^7+z^6 a^6-2 z^4 a^6+z^2 a^6+z^5 a^5-2 z^3 a^5+z^4 a^4-3 z^2 a^4+a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{10}+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{176}+q^{172}-q^{170}+q^{168}-q^{164}+2 q^{162}-2 q^{160}+2 q^{158}-2 q^{156}+q^{152}-2 q^{150}+3 q^{148}-5 q^{146}+2 q^{144}-q^{142}-3 q^{140}+2 q^{138}-5 q^{136}+q^{134}+q^{132}-3 q^{130}-3 q^{126}-q^{124}+3 q^{122}-5 q^{120}+2 q^{118}-q^{116}-q^{114}+5 q^{112}-3 q^{110}+4 q^{108}-q^{106}+3 q^{104}+2 q^{102}-3 q^{100}+6 q^{98}-3 q^{96}+4 q^{94}+q^{92}-2 q^{90}+3 q^{88}-q^{86}+q^{82}-3 q^{80}+q^{78}-2 q^{74}+4 q^{72}-3 q^{70}+2 q^{68}-q^{64}+q^{62}-2 q^{60}+3 q^{58}-q^{56}+q^{54}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+q^{34}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_4.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}-q^{19}+q^{17}+q^{13}+q^{11}-q^9+q^7+q^3 }[/math]
2	[math]\displaystyle{ q^{64}+q^{58}-q^{56}-2 q^{54}+q^{52}-2 q^{48}+q^{46}+q^{44}-2 q^{42}+q^{38}-q^{36}+q^{30}-2 q^{28}+3 q^{24}-q^{22}+2 q^{18}+q^{12}+q^6 }[/math]
3	[math]\displaystyle{ -q^{123}+q^{113}+q^{111}-q^{107}+q^{105}+2 q^{103}-3 q^{99}-2 q^{97}+2 q^{95}+2 q^{93}-3 q^{89}+3 q^{85}+2 q^{83}-3 q^{81}-2 q^{79}+2 q^{77}+3 q^{75}-2 q^{73}-2 q^{71}-2 q^{65}-q^{63}-q^{61}+2 q^{57}-2 q^{55}-2 q^{53}+q^{51}+4 q^{49}-4 q^{45}+3 q^{41}+2 q^{39}-q^{37}-q^{35}+q^{31}+q^{29}+q^{21}+q^{19}+q^{17}+q^9 }[/math]
4	[math]\displaystyle{ q^{200}-q^{192}-q^{188}+q^{184}-q^{182}-2 q^{178}-q^{176}+3 q^{174}+2 q^{172}+3 q^{170}-2 q^{168}-4 q^{166}-q^{164}+q^{162}+6 q^{160}+q^{158}-3 q^{156}-4 q^{154}-5 q^{152}+3 q^{150}+4 q^{148}+4 q^{146}-8 q^{142}-3 q^{140}+2 q^{138}+7 q^{136}+6 q^{134}-5 q^{132}-6 q^{130}-2 q^{128}+7 q^{126}+8 q^{124}-2 q^{122}-5 q^{120}-3 q^{118}+3 q^{116}+4 q^{114}-q^{112}-2 q^{110}-2 q^{108}-2 q^{102}-q^{100}-q^{98}-q^{96}+q^{94}+4 q^{92}-q^{90}-3 q^{88}-4 q^{86}-q^{84}+8 q^{82}+2 q^{80}-3 q^{78}-9 q^{76}-5 q^{74}+9 q^{72}+6 q^{70}+2 q^{68}-6 q^{66}-8 q^{64}+q^{62}+4 q^{60}+6 q^{58}+q^{56}-5 q^{54}-2 q^{52}-q^{50}+4 q^{48}+4 q^{46}-q^{42}-3 q^{40}+q^{38}+2 q^{36}+q^{34}+q^{32}-2 q^{30}+q^{26}+q^{24}+2 q^{22}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{295}+q^{287}+q^{285}-q^{277}+2 q^{273}+q^{271}-q^{267}-3 q^{265}-3 q^{263}+3 q^{259}+3 q^{257}+2 q^{255}-2 q^{253}-5 q^{251}-5 q^{249}+5 q^{245}+8 q^{243}+5 q^{241}-q^{239}-6 q^{237}-8 q^{235}-4 q^{233}+3 q^{231}+7 q^{229}+8 q^{227}+3 q^{225}-3 q^{223}-10 q^{221}-11 q^{219}-3 q^{217}+6 q^{215}+13 q^{213}+11 q^{211}-q^{209}-15 q^{207}-16 q^{205}-6 q^{203}+8 q^{201}+19 q^{199}+14 q^{197}-4 q^{195}-18 q^{193}-16 q^{191}-q^{189}+16 q^{187}+19 q^{185}+4 q^{183}-11 q^{181}-16 q^{179}-6 q^{177}+10 q^{175}+12 q^{173}+5 q^{171}-3 q^{169}-8 q^{167}-4 q^{165}+3 q^{163}+5 q^{161}+q^{159}-q^{157}-q^{155}-q^{153}+q^{151}+q^{149}-q^{147}-q^{143}-q^{141}-q^{139}+2 q^{137}+6 q^{135}+2 q^{133}-2 q^{131}-7 q^{129}-11 q^{127}-2 q^{125}+11 q^{123}+14 q^{121}+3 q^{119}-12 q^{117}-22 q^{115}-11 q^{113}+10 q^{111}+25 q^{109}+16 q^{107}-7 q^{105}-21 q^{103}-20 q^{101}-3 q^{99}+16 q^{97}+20 q^{95}+8 q^{93}-8 q^{91}-16 q^{89}-12 q^{87}+10 q^{83}+11 q^{81}+4 q^{79}-4 q^{77}-7 q^{75}-5 q^{73}+4 q^{69}+5 q^{67}+2 q^{65}-q^{63}-3 q^{61}-2 q^{59}+q^{57}+2 q^{55}+3 q^{53}+q^{51}-2 q^{49}-q^{47}+q^{43}+2 q^{41}+2 q^{39}-q^{37}-q^{35}+q^{29}+2 q^{27}+q^{25}+q^{15} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{10}+q^6 }[/math]
1,1	[math]\displaystyle{ q^{92}+2 q^{88}-2 q^{86}+4 q^{84}-4 q^{82}+4 q^{80}-8 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}+3 q^{68}+2 q^{66}-2 q^{64}+8 q^{62}-12 q^{60}+12 q^{58}-14 q^{56}+10 q^{54}-15 q^{52}+10 q^{50}-10 q^{48}+6 q^{46}-3 q^{44}-2 q^{42}+2 q^{40}-4 q^{38}+7 q^{36}-4 q^{34}+6 q^{32}+7 q^{28}-2 q^{26}+4 q^{24}-2 q^{22}+4 q^{20}-2 q^{18}+2 q^{16}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{86}+q^{84}+2 q^{82}+q^{80}+q^{78}-q^{76}-2 q^{74}-2 q^{72}-2 q^{70}-2 q^{68}-2 q^{66}+q^{62}+q^{60}-q^{58}-q^{52}-q^{50}-q^{44}-q^{42}+q^{40}+q^{36}+3 q^{34}+2 q^{32}+q^{30}+2 q^{28}+2 q^{26}-q^{22}+q^{20}+q^{18}+q^{12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{74}+q^{70}+q^{68}-3 q^{60}-q^{58}-q^{56}-4 q^{54}-q^{52}-q^{48}-q^{46}+q^{44}+q^{40}+q^{38}+3 q^{36}+q^{34}+3 q^{30}-q^{26}+2 q^{24}+q^{22}+q^{18}+q^{16}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ -q^{45}-q^{43}-2 q^{41}-q^{39}-q^{37}+q^{35}+q^{33}+2 q^{31}+q^{29}+q^{27}+q^{21}+q^{17}+q^{13}+q^9 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{96}+q^{94}+q^{92}+2 q^{90}+3 q^{88}+q^{86}+q^{84}+q^{82}-q^{80}-5 q^{78}-6 q^{76}-5 q^{74}-6 q^{72}-6 q^{70}-2 q^{68}+2 q^{62}+2 q^{60}+q^{58}+2 q^{54}+2 q^{52}+2 q^{50}+2 q^{48}+4 q^{46}+2 q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+2 q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{56}-q^{54}-2 q^{52}-2 q^{50}-q^{48}-q^{46}+q^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{36}+q^{34}+q^{26}+q^{22}+q^{20}+q^{16}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{74}-q^{70}+q^{68}-2 q^{66}+2 q^{64}-2 q^{62}+q^{60}-q^{58}+q^{56}-q^{52}+2 q^{50}-3 q^{48}+3 q^{46}-3 q^{44}+4 q^{42}-3 q^{40}+3 q^{38}-q^{36}+q^{34}-q^{30}+2 q^{28}-q^{26}+2 q^{24}-q^{22}+2 q^{20}-q^{18}+q^{16}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{120}+q^{112}+q^{110}-q^{106}+q^{102}+q^{100}-2 q^{98}-3 q^{96}-q^{94}+q^{92}-3 q^{88}-2 q^{86}+q^{82}-q^{78}+q^{74}-q^{70}-q^{68}+q^{66}+q^{64}-q^{60}+q^{58}+2 q^{56}+q^{54}-q^{52}+3 q^{48}+2 q^{46}-q^{44}-q^{42}+2 q^{38}+q^{36}-q^{32}+q^{28}+q^{26}+q^{18} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{102}+q^{98}+2 q^{94}-q^{92}+q^{90}-2 q^{88}+q^{86}-2 q^{84}-2 q^{80}-q^{78}-2 q^{76}-3 q^{74}-q^{72}-3 q^{70}-4 q^{66}+2 q^{64}-2 q^{62}+4 q^{60}-q^{58}+4 q^{56}+4 q^{52}+q^{50}+2 q^{48}+2 q^{42}-q^{40}+q^{38}-q^{36}+2 q^{34}+2 q^{30}+2 q^{26}+q^{22}+q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{176}+q^{172}-q^{170}+q^{168}-q^{164}+2 q^{162}-2 q^{160}+2 q^{158}-2 q^{156}+q^{152}-2 q^{150}+3 q^{148}-5 q^{146}+2 q^{144}-q^{142}-3 q^{140}+2 q^{138}-5 q^{136}+q^{134}+q^{132}-3 q^{130}-3 q^{126}-q^{124}+3 q^{122}-5 q^{120}+2 q^{118}-q^{116}-q^{114}+5 q^{112}-3 q^{110}+4 q^{108}-q^{106}+3 q^{104}+2 q^{102}-3 q^{100}+6 q^{98}-3 q^{96}+4 q^{94}+q^{92}-2 q^{90}+3 q^{88}-q^{86}+q^{82}-3 q^{80}+q^{78}-2 q^{74}+4 q^{72}-3 q^{70}+2 q^{68}-q^{64}+q^{62}-2 q^{60}+3 q^{58}-q^{56}+q^{54}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+q^{34}+q^{30} }[/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.

In[3]:= K = Knot["9 4"];

In[4]:= Alexander[K][t]

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

Out[4]= [math]\displaystyle{ 3 t^2-5 t+5-5 t^{-1} +3 t^{-2} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ 3 z^4+7 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]= { 21, -4 }

In[8]:= Jones[K][q]

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

Out[8]= [math]\displaystyle{ q^{-2} - q^{-3} +2 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} - q^{-11} }[/math]

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= [math]\displaystyle{ -z^2 a^{10}-2 a^{10}+z^4 a^8+3 z^2 a^8+2 a^8+z^4 a^6+2 z^2 a^6+z^4 a^4+3 z^2 a^4+a^4 }[/math]

In[10]:= Kauffman[K][a, z]

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

Out[10]= [math]\displaystyle{ z^5 a^{13}-4 z^3 a^{13}+3 z a^{13}+z^6 a^{12}-3 z^4 a^{12}+z^2 a^{12}+z^7 a^{11}-3 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+z^8 a^{10}-5 z^6 a^{10}+11 z^4 a^{10}-10 z^2 a^{10}+2 a^{10}+2 z^7 a^9-8 z^5 a^9+12 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+11 z^4 a^8-7 z^2 a^8+2 a^8+z^7 a^7-3 z^5 a^7+4 z^3 a^7+z^6 a^6-2 z^4 a^6+z^2 a^6+z^5 a^5-2 z^3 a^5+z^4 a^4-3 z^2 a^4+a^4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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.

In[3]:= K = Knot["9 4"];

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{ 3 t^2-5 t+5-5 t^{-1} +3 t^{-2} }[/math], [math]\displaystyle{ q^{-2} - q^{-3} +2 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} - q^{-11} }[/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]= {}

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:

(7, -19)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 28 }[/math]

[math]\displaystyle{ -152 }[/math]

[math]\displaystyle{ 392 }[/math]

[math]\displaystyle{ \frac{3122}{3} }[/math]

[math]\displaystyle{ \frac{502}{3} }[/math]

[math]\displaystyle{ -4256 }[/math]

[math]\displaystyle{ -\frac{23696}{3} }[/math]

[math]\displaystyle{ -\frac{4160}{3} }[/math]

[math]\displaystyle{ -1144 }[/math]

[math]\displaystyle{ \frac{10976}{3} }[/math]

[math]\displaystyle{ 11552 }[/math]

[math]\displaystyle{ \frac{87416}{3} }[/math]

[math]\displaystyle{ \frac{14056}{3} }[/math]

[math]\displaystyle{ \frac{1820137}{30} }[/math]

[math]\displaystyle{ \frac{7526}{15} }[/math]

[math]\displaystyle{ \frac{1136714}{45} }[/math]

[math]\displaystyle{ \frac{9335}{18} }[/math]

[math]\displaystyle{ \frac{101737}{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]-4 is the signature of 9 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

1

0

-7

1

-9

2

1

-1

-11

2

1

-13

1

2

1

-15

2

0

-17

1

-19

1

2

-1

-21

0

-23

1

-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=-9 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-6 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\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^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/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} - q^{-5} +2 q^{-7} -2 q^{-8} +4 q^{-10} -4 q^{-11} - q^{-12} +8 q^{-13} -7 q^{-14} -3 q^{-15} +11 q^{-16} -8 q^{-17} -3 q^{-18} +10 q^{-19} -6 q^{-20} -4 q^{-21} +8 q^{-22} -3 q^{-23} -4 q^{-24} +5 q^{-25} - q^{-26} -3 q^{-27} +2 q^{-28} - q^{-30} + q^{-31} }[/math]
3	[math]\displaystyle{ q^{-6} - q^{-7} +2 q^{-10} - q^{-11} - q^{-13} +2 q^{-14} - q^{-15} + q^{-16} - q^{-17} + q^{-18} -2 q^{-19} + q^{-20} +2 q^{-21} +2 q^{-22} -5 q^{-23} -3 q^{-24} +6 q^{-25} +6 q^{-26} -8 q^{-27} -6 q^{-28} +6 q^{-29} +10 q^{-30} -10 q^{-31} -7 q^{-32} +6 q^{-33} +9 q^{-34} -8 q^{-35} -7 q^{-36} +4 q^{-37} +9 q^{-38} -3 q^{-39} -8 q^{-40} +8 q^{-42} +2 q^{-43} -7 q^{-44} -3 q^{-45} +5 q^{-46} +5 q^{-47} -5 q^{-48} -3 q^{-49} + q^{-50} +4 q^{-51} -2 q^{-52} - q^{-53} +2 q^{-55} - q^{-56} + q^{-59} - q^{-60} }[/math]
4	[math]\displaystyle{ q^{-8} - q^{-9} +3 q^{-13} -2 q^{-14} - q^{-16} -2 q^{-17} +6 q^{-18} -2 q^{-19} + q^{-20} -2 q^{-21} -6 q^{-22} +8 q^{-23} - q^{-24} +5 q^{-25} -2 q^{-26} -11 q^{-27} +7 q^{-28} -4 q^{-29} +11 q^{-30} +3 q^{-31} -13 q^{-32} +4 q^{-33} -13 q^{-34} +13 q^{-35} +11 q^{-36} -9 q^{-37} +7 q^{-38} -27 q^{-39} +9 q^{-40} +17 q^{-41} -4 q^{-42} +13 q^{-43} -36 q^{-44} +6 q^{-45} +18 q^{-46} -2 q^{-47} +18 q^{-48} -39 q^{-49} +4 q^{-50} +18 q^{-51} -2 q^{-52} +17 q^{-53} -37 q^{-54} +4 q^{-55} +16 q^{-56} -2 q^{-57} +18 q^{-58} -32 q^{-59} +3 q^{-60} +10 q^{-61} -4 q^{-62} +21 q^{-63} -22 q^{-64} +2 q^{-65} + q^{-66} -8 q^{-67} +22 q^{-68} -11 q^{-69} +3 q^{-70} -4 q^{-71} -13 q^{-72} +17 q^{-73} -3 q^{-74} +7 q^{-75} -4 q^{-76} -14 q^{-77} +9 q^{-78} -2 q^{-79} +8 q^{-80} -9 q^{-82} +4 q^{-83} -4 q^{-84} +5 q^{-85} +2 q^{-86} -4 q^{-87} +3 q^{-88} -3 q^{-89} + q^{-90} + q^{-91} -2 q^{-92} +2 q^{-93} - q^{-94} - q^{-97} + q^{-98} }[/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).

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Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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