7 4

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7_3

7_5

Contents

Image:7 4.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-5,6,-7,2,-1,3,-4,7,-6,5,-2,4,-3/goTop.html 7_4's page] at Knotilus!

Visit 7 4's page at the original Knot Atlas!

[edit 7 4 Quick Notes]

Simplest version of Endless knot symbol.

[edit 7 4 Further Notes and Views]
Celtic or pseudo-Celtic knot
Celtic or pseudo-Celtic knot
a knot seen at the Castle of Kornik [1]
a knot seen at the Castle of Kornik [1]
Susan Williams' medallion [2], the "Endless knot" of Buddhism [3]
Susan Williams' medallion [2], the "Endless knot" of Buddhism [3]
Ornamental "Endless knot"
Ornamental "Endless knot"
A 7-4 knot reduced from TakaraMusubi with 9 crossings [4]
A 7-4 knot reduced from TakaraMusubi with 9 crossings [4]
TakaraMusubi knot seen in Japanese symbols, or Kolam in South India [5]
TakaraMusubi knot seen in Japanese symbols, or Kolam in South India [5]
Buddhist Endless Knot
Buddhist Endless Knot
Ornamental Endless Knot
Ornamental Endless Knot
Albrecht Dürer knot, 16th-century
Albrecht Dürer knot, 16th-century
A laser cut by Tom Longtin [6]
A laser cut by Tom Longtin [6]
Unicursal hexagram of occultism
Unicursal hexagram of occultism

[edit] Knot presentations

Planar diagram presentation X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9
Gauss code 1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3
Dowker-Thistlethwaite code 6 10 12 14 4 2 8
Conway Notation [313]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif

Length is 9, width is 4,

Braid index is 4

Image:7 4_ML.gif Image:7 4_AP.gif
[{3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {1, 8}, {9, 2}, {4, 1}]

[edit Notes on presentations of 7 4]

Knot 7_4.
Knot 7_4.
A graph, knot 7_4.
A graph, knot 7_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["7 4"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9
In[5]:= GaussCode[K]
Out[5]= 1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3
In[6]:= DTCode[K]
Out[6]= 6 10 12 14 4 2 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [313]
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]= BR(4,{1,1,2,−1,2,2,3,−2,3})
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, 9, 4 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.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."
Image:7 4_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {1, 8}, {9, 2}, {4, 1}]
In[14]:= Draw[ap]
Image:7 4_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 1
Bridge index 2
Super bridge index {3,4}
Nakanishi index 1
Maximal Thurston-Bennequin number Failed to parse (unknown error\text): \text{$\$$Failed}
Hyperbolic Volume 5.13794
A-Polynomial See Data:7 4/A-polynomial

[edit Notes for 7 4's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus ConcordanceGenus(Knot(7,4))
Rasmussen s-Invariant -2

[edit Notes for 7 4's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 4t + 4t−1−7
Conway polynomial 4z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 15, 2 }
Jones polynomial −q8 + q7−2q6 + 3q5−2q4 + 3q3−2q2 + q
HOMFLY-PT polynomial (db, data sources) −a−8 + z2a−6 + 2z2a−4 + 2a−4 + z2a−2
Kauffman polynomial (db, data sources) z6a−6 + z6a−8 + 2z5a−5 + 3z5a−7 + z5a−9 + 3z4a−4−3z4a−8 + 2z3a−3−2z3a−5−8z3a−7−4z3a−9 + z2a−2−4z2a−4−3z2a−6 + 2z2a−8 + 4za−7 + 4za−9 + 2a−4−a−8
The A2 invariant q−2−q−4 + q−8 + q−10 + 2q−12 + q−14 + q−16−q−20−q−24−q−26
The G2 invariant q−10−q−12 + q−14−q−16−q−22 + 4q−24−2q−26 + 2q−28−q−30 + q−34−2q−36 + 3q−38−2q−40 + q−44 + 2q−48 + q−50−q−52 + 2q−54−q−56 + 3q−58 + q−60−2q−62 + 6q−64−3q−66 + 4q−68 + 2q−70−3q−72 + 4q−74−3q−76 + 3q−78−q−80−q−82 + q−84−2q−86 + 2q−88−3q−92−q−94−2q−96−4q−102 + 2q−104−3q−106 + q−108 + q−110−4q−112 + 3q−114−2q−116 + q−118−2q−122 + 2q−124 + q−128
Further Quantum Invariants
Further quantum knot invariants for 7_4.

A1 Invariants.

Weight Invariant
1 q−1−q−3 + q−5 + q−7 + q−9 + q−11−q−13−q−17
2 q−2−q−4 + 3q−8−q−10 + 2q−14−q−16 + q−20 + 2q−22 + q−28−q−30−3q−32−2q−38 + q−40 + q−42−q−44 + q−48
3 q−3−q−5 + 2q−9 + q−11−q−13−q−15 + 3q−17 + q−19−3q−21 + q−23 + 4q−25 + 2q−27−3q−29−q−31 + q−33 + 2q−35−q−39−q−41 + q−43 + 3q−45−2q−47−3q−49 + 2q−53−2q−55−3q−57−2q−59 + 2q−61−2q−65−q−67 + 3q−69 + 3q−71−q−73−2q−75 + q−77 + 3q−79−2q−83−q−85 + q−87 + q−89−q−93
4 q−4−q−6 + 2q−10 + q−14−2q−16 + q−18 + 3q−20−2q−22 + q−24−2q−26 + 4q−28 + 6q−30−3q−32−4q−34−5q−36 + 5q−38 + 9q−40−4q−44−6q−46 + 6q−50 + 3q−52−3q−56−3q−58−q−60 + 2q−62 + 4q−64−q−66−4q−68−5q−70 + q−72 + 4q−74 + 2q−76−4q−78−6q−80 + q−82 + 4q−84 + 2q−86−5q−88−5q−90 + 2q−92 + 3q−94 + 4q−96−2q−98−5q−100 + 2q−102 + 3q−104 + 7q−106 + q−108−5q−110−3q−112−q−114 + 6q−116 + 4q−118−2q−120−4q−122−5q−124 + 2q−126 + 4q−128 + 2q−130−q−132−5q−134−q−136 + q−138 + 2q−140 + 2q−142−q−144−q−146−q−148 + q−152
5 q−5−q−7 + 2q−11 + q−21 + q−23−2q−27 + q−29 + 5q−31 + 5q−33−2q−35−6q−37−7q−39 + 11q−43 + 12q−45 + 3q−47−11q−49−15q−51−5q−53 + 9q−55 + 18q−57 + 12q−59−7q−61−16q−63−12q−65−q−67 + 11q−69 + 13q−71 + 3q−73−7q−75−8q−77−7q−79−q−81 + 6q−83 + 7q−85 + 4q−87−q−89−7q−91−8q−93−4q−95 + 5q−97 + 9q−99 + 2q−101−6q−103−10q−105−5q−107 + 5q−109 + 9q−111 + q−113−4q−115−9q−117−3q−119 + 8q−121 + 10q−123 + 2q−125−6q−127−10q−129−3q−131 + 9q−133 + 13q−135 + 3q−137−6q−139−11q−141−7q−143 + 7q−145 + 13q−147 + 9q−149−9q−153−12q−155−4q−157 + 6q−159 + 11q−161 + 6q−163−2q−165−10q−167−11q−169−4q−171 + 6q−173 + 10q−175 + 7q−177−q−179−8q−181−9q−183−3q−185 + 4q−187 + 8q−189 + 6q−191−5q−195−6q−197−3q−199 + 2q−201 + 5q−203 + 3q−205 + q−207−q−209−3q−211−2q−213 + q−217 + q−219 + q−221−q−225
6 q−6−q−8 + 2q−12−q−18 + 2q−20−q−24 + 3q−26−2q−28−q−30 + 2q−32 + 7q−34 + 2q−36−3q−38−3q−40−10q−42−4q−44 + 8q−46 + 21q−48 + 10q−50−5q−52−13q−54−25q−56−11q−58 + 13q−60 + 34q−62 + 25q−64 + q−66−19q−68−40q−70−28q−72 + 4q−74 + 35q−76 + 39q−78 + 19q−80−5q−82−36q−84−38q−86−17q−88 + 15q−90 + 32q−92 + 28q−94 + 13q−96−14q−98−26q−100−26q−102−8q−104 + 7q−106 + 15q−108 + 18q−110 + 8q−112−2q−114−12q−116−14q−118−12q−120−2q−122 + 10q−124 + 17q−126 + 12q−128−12q−132−19q−134−9q−136 + 3q−138 + 15q−140 + 13q−142−12q−146−18q−148−3q−150 + 8q−152 + 18q−154 + 12q−156−6q−158−16q−160−14q−162 + 2q−164 + 14q−166 + 24q−168 + 12q−170−9q−172−23q−174−19q−176−2q−178 + 14q−180 + 29q−182 + 20q−184−4q−186−26q−188−27q−190−12q−192 + 6q−194 + 29q−196 + 28q−198 + 10q−200−17q−202−27q−204−23q−206−11q−208 + 15q−210 + 27q−212 + 23q−214 + q−216−13q−218−23q−220−25q−222−5q−224 + 12q−226 + 24q−228 + 18q−230 + 9q−232−8q−234−23q−236−18q−238−9q−240 + 8q−242 + 15q−244 + 21q−246 + 12q−248−5q−250−13q−252−17q−254−10q−256−2q−258 + 12q−260 + 15q−262 + 9q−264 + 2q−266−6q−268−10q−270−12q−272−2q−274 + 4q−276 + 7q−278 + 7q−280 + 4q−282−6q−286−4q−288−3q−290−q−292 + q−294 + 3q−296 + 3q−298−q−304−q−306−q−308 + q−312

A2 Invariants.

Weight Invariant
1,0 q−2−q−4 + q−8 + q−10 + 2q−12 + q−14 + q−16−q−20−q−24−q−26
1,1 q−4−2q−6 + 2q−8−2q−10 + 7q−12−4q−14 + 4q−16−4q−18 + 7q−20 + 4q−24 + 4q−26 + q−28 + 6q−30−8q−32 + 8q−34−15q−36 + 8q−38−12q−40 + 6q−42−7q−44 + 4q−46 + 2q−48−2q−50 + 3q−52−6q−54 + 6q−56−8q−58 + 4q−60−4q−62 + 4q−64 + q−68
2,0 q−4−q−6−q−8 + 2q−10 + 2q−12−q−16 + q−18 + 2q−20 + q−24 + 2q−26 + 2q−28 + 2q−30 + 3q−32 + q−34 + q−36−q−40−4q−42−4q−44−2q−46−q−48−2q−50−q−52 + q−54 + q−56−q−60 + q−62 + q−64 + q−66
3,0 q−6−q−8−q−10 + q−12 + 3q−14 + 2q−16−4q−18−2q−20 + 3q−22 + 5q−24 + 3q−26−4q−28−q−30 + 2q−32 + 6q−34 + 4q−36−q−38−q−40 + 2q−42 + 3q−44 + 2q−46 + q−48 + 2q−50 + 2q−52 + q−54 + q−56 + 2q−58 + q−60−4q−62−5q−64−4q−66−q−68−4q−70−7q−72−7q−74−4q−76 + q−78−2q−82−2q−84 + q−86 + 6q−88 + 5q−90 + 2q−92 + 4q−98 + 3q−100 + q−102−2q−104−3q−106 + q−110 + 2q−112−q−116−q−118−q−120

A3 Invariants.

Weight Invariant
0,1,0 q−4−q−6−q−8 + 2q−10 + 3q−16 + q−18 + q−20 + 2q−22 + 2q−24 + q−26 + 2q−28 + 2q−30 + 2q−32−2q−34−q−36−2q−38−5q−40−3q−42−q−44−q−46 + q−48 + 2q−50 + q−54
1,0,0 q−3−q−5 + q−11 + q−13 + 2q−15 + 2q−17 + q−19 + q−21−q−27−q−31−q−33−q−35

A4 Invariants.

Weight Invariant
0,1,0,0 q−6−q−8−q−10 + q−12 + q−14 + 2q−20 + 2q−22 + q−26 + 3q−28 + 3q−30 + 2q−32 + 5q−34 + 4q−36 + 4q−38 + 3q−40 + 3q−42−3q−46−3q−48−5q−50−8q−52−7q−54−3q−56−2q−58−q−60 + q−62 + 3q−64 + 2q−66 + q−68 + q−70 + q−72
1,0,0,0 q−4−q−6 + q−14 + q−16 + 2q−18 + 2q−20 + 2q−22 + q−24 + q−26−q−34−q−38−q−40−q−42−q−44

B2 Invariants.

Weight Invariant
0,1 q−4−q−6 + q−8−2q−10 + 2q−12 + q−16 + q−18 + q−20 + 2q−22−2q−24 + 3q−26−2q−28 + 2q−30−2q−32 + 2q−34−q−36 + q−40−q−42 + q−44−q−46 + q−48−2q−50−q−54
1,0 q−6−q−10−q−12 + 2q−16 + q−18−q−20 + q−24 + 2q−26 + 2q−34 + q−36 + 2q−42 + 2q−44 + q−46 + q−50 + q−52−2q−56−q−58−q−62−3q−64−3q−66−q−68−q−74−q−76 + q−78 + 2q−80 + q−88

D4 Invariants.

Weight Invariant
1,0,0,0 q−6−q−8−q−12 + 2q−14−q−16 + q−18 + 2q−22 + 2q−24 + 2q−26 + 2q−28 + q−30 + 3q−32 + 3q−36−q−38 + 3q−40 + 3q−44−q−46 + q−48−2q−50−2q−52−3q−54−4q−56−2q−58−3q−60−q−62−q−64 + 2q−66 + 2q−70 + q−74

G2 Invariants.

Weight Invariant
1,0 q−10−q−12 + q−14−q−16−q−22 + 4q−24−2q−26 + 2q−28−q−30 + q−34−2q−36 + 3q−38−2q−40 + q−44 + 2q−48 + q−50−q−52 + 2q−54−q−56 + 3q−58 + q−60−2q−62 + 6q−64−3q−66 + 4q−68 + 2q−70−3q−72 + 4q−74−3q−76 + 3q−78−q−80−q−82 + q−84−2q−86 + 2q−88−3q−92−q−94−2q−96−4q−102 + 2q−104−3q−106 + q−108 + q−110−4q−112 + 3q−114−2q−116 + q−118−2q−122 + 2q−124 + q−128

.

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["7 4"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 4t + 4t−1−7
In[5]:= Conway[K][z]
Out[5]= 4z2 + 1
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]= {1}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 15, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + q7−2q6 + 3q5−2q4 + 3q3−2q2 + q
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −a−8 + z2a−6 + 2z2a−4 + 2a−4 + z2a−2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z6a−6 + z6a−8 + 2z5a−5 + 3z5a−7 + z5a−9 + 3z4a−4−3z4a−8 + 2z3a−3−2z3a−5−8z3a−7−4z3a−9 + z2a−2−4z2a−4−3z2a−6 + 2z2a−8 + 4za−7 + 4za−9 + 2a−4−a−8

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_2,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

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["7 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]= { 4t + 4t−1−7, −q8 + q7−2q6 + 3q5−2q4 + 3q3−2q2 + q }
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]= {9_2,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

[edit] Vassiliev invariants

V2 and V3: (4, 8)
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
16 64 128 \frac{1016}{3} \frac{184}{3} 1024 \frac{5824}{3} \frac{1024}{3} 320 \frac{2048}{3} 2048 \frac{16256}{3} \frac{2944}{3} \frac{168062}{15} -\frac{1176}{5} \frac{233288}{45} \frac{898}{9} \frac{11102}{15}

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.

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 2 is the signature of 7 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    <table border=1> <tr align=center> <td width=16.6667%><table cellpadding=0 cellspacing=0> <tr><td>\</td><td> </td><td>r</td></tr> <tr><td> </td><td> \ </td><td> </td></tr> <tr><td>j</td><td> </td><td>\</td></tr> </table></td> <td width=8.33333%>0</td><td width=8.33333%>1</td><td width=8.33333%>2</td><td width=8.33333%>3</td><td width=8.33333%>4</td><td width=8.33333%>5</td><td width=8.33333%>6</td><td width=8.33333%>7</td><td width=16.6667%>χ</td></tr> <tr align=center><td>17</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>-1</td></tr> <tr align=center><td>15</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr> <tr align=center><td>13</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td>-1</td></tr> <tr align=center><td>11</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>9</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>7</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>5</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>3</td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> </table>
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = 0 {\mathbb Z} {\mathbb Z}
r = 1 {\mathbb Z}^{2}
r = 2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 5 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 6 {\mathbb Z}
r = 7 {\mathbb Z}_2 {\mathbb Z}

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the n + 1-dimensional representation of sl(2))

 n  Jn
2 q23−q22−q21 + 3q20−q19−4q18 + 5q17−q16−7q15 + 7q14 + q13−8q12 + 7q11 + 3q10−9q9 + 6q8 + 2q7−6q6 + 4q5 + q4−2q3 + q2
3 −q45 + q44 + q43−3q41 + 3q39 + 3q38−5q37−3q36 + 4q35 + 7q34−5q33−7q32 + 3q31 + 9q30−3q29−11q28 + 2q27 + 10q26 + q25−13q24−q23 + 11q22 + 6q21−15q20−3q19 + 11q18 + 7q17−13q16−4q15 + 9q14 + 5q13−8q12−2q11 + 6q10 + q9−4q8 + 2q6 + q5−2q4 + q3
4 q74−q73−q72 + 4q69−q68−2q67−2q66−4q65 + 8q64 + 2q63−4q61−11q60 + 9q59 + 4q58 + 6q57−2q56−18q55 + 7q54 + 2q53 + 12q52 + 4q51−22q50 + 6q49−5q48 + 15q47 + 10q46−23q45 + 5q44−12q43 + 15q42 + 17q41−21q40 + 2q39−19q38 + 17q37 + 23q36−19q35−q34−25q33 + 18q32 + 26q31−14q30−3q29−28q28 + 16q27 + 26q26−11q25−25q23 + 10q22 + 20q21−9q20 + 4q19−16q18 + 6q17 + 10q16−8q15 + 5q14−7q13 + 4q12 + 4q11−5q10 + 2q9−2q8 + 2q7 + q6−2q5 + q4
5 −q110 + q109 + q108−q105−3q104 + 3q102 + 2q101 + 2q100 + q99−6q98−5q97 + 3q95 + 7q94 + 7q93−4q92−9q91−7q90−3q89 + 8q88 + 14q87 + 4q86−6q85−11q84−13q83 + q82 + 15q81 + 12q80 + 2q79−6q78−18q77−9q76 + 7q75 + 15q74 + 11q73 + 3q72−14q71−15q70−7q69 + 11q68 + 16q67 + 12q66−4q65−19q64−19q63 + 4q62 + 20q61 + 20q60 + 4q59−21q58−30q57−2q56 + 25q55 + 25q54 + 12q53−25q52−40q51−7q50 + 29q49 + 33q48 + 19q47−29q46−49q45−11q44 + 30q43 + 39q42 + 24q41−26q40−50q39−18q38 + 24q37 + 38q36 + 25q35−16q34−40q33−20q32 + 12q31 + 27q30 + 21q29−7q28−21q27−14q26 + 3q25 + 13q24 + 11q23−3q22−7q21−5q20 + 2q19 + 2q18 + 4q17−2q16−3q15 + 2q14 + 2q13−2q12 + q11−2q9 + 2q8 + q7−2q6 + q5
6 q153−q152−q151 + q148 + 4q146−q145−3q144−2q143−2q142−2q140 + 10q139 + 3q138−2q136−5q135−6q134−12q133 + 12q132 + 7q131 + 8q130 + 5q129 + q128−9q127−26q126 + 4q125 + 12q123 + 13q122 + 18q121−32q119−3q118−17q117 + 3q116 + 8q115 + 33q114 + 17q113−23q112 + 3q111−29q110−14q109−12q108 + 35q107 + 27q106−9q105 + 25q104−25q103−26q102−38q101 + 23q100 + 23q99 + q98 + 52q97−7q96−25q95−61q94 + 5q93 + 8q92 + 2q91 + 74q90 + 17q89−16q88−76q87−11q86−9q85−2q84 + 88q83 + 38q82−4q81−86q80−23q79−25q78−4q77 + 98q76 + 56q75 + 2q74−96q73−34q72−40q71 + 2q70 + 110q69 + 69q68 + 4q67−108q66−46q65−50q64 + 9q63 + 122q62 + 81q61 + 9q60−115q59−58q58−60q57 + 7q56 + 124q55 + 91q54 + 19q53−105q52−61q51−68q50−8q49 + 106q48 + 91q47 + 31q46−78q45−46q44−64q43−25q42 + 74q41 + 70q40 + 33q39−47q38−22q37−44q36−29q35 + 43q34 + 38q33 + 21q32−26q31−2q30−20q29−20q28 + 22q27 + 14q26 + 7q25−14q24 + 6q23−5q22−9q21 + 9q20 + 2q19 + q18−7q17 + 6q16−4q14 + 4q13−q12−2q10 + 2q9 + q8−2q7 + q6
7 −q203 + q202 + q201−q198−q196−3q195 + q194 + 3q193 + 2q192 + 3q191−q190−9q187−4q186 + 2q184 + 8q183 + 3q182 + 7q181 + 8q180−10q179−10q178−10q177−10q176 + 5q175 + 2q174 + 13q173 + 24q172 + 3q171−q170−12q169−25q168−10q167−15q166 + 3q165 + 31q164 + 17q163 + 22q162 + 9q161−22q160−15q159−36q158−24q157 + 14q156 + 9q155 + 34q154 + 37q153 + 5q152 + 7q151−36q150−45q149−12q148−23q147 + 14q146 + 44q145 + 29q144 + 47q143−6q142−37q141−22q140−61q139−26q138 + 18q137 + 25q136 + 78q135 + 40q134−3q133−5q132−80q131−65q130−25q129−10q128 + 86q127 + 79q126 + 41q125 + 34q124−77q123−88q122−69q121−58q120 + 70q119 + 99q118 + 81q117 + 79q116−55q115−97q114−101q113−104q112 + 41q111 + 105q110 + 112q109 + 120q108−29q107−96q106−124q105−144q104 + 15q103 + 108q102 + 136q101 + 150q100−10q99−99q98−143q97−174q96 + q95 + 118q94 + 157q93 + 169q92−2q91−110q90−165q89−194q88−3q87 + 137q86 + 179q85 + 185q84 + 3q83−129q82−188q81−212q80−7q79 + 151q78 + 199q77 + 204q76 + 14q75−133q74−202q73−229q72−30q71 + 138q70 + 204q69 + 220q68 + 40q67−107q66−187q65−233q64−60q63 + 93q62 + 173q61 + 213q60 + 67q59−59q58−142q57−200q56−72q55 + 39q54 + 113q53 + 167q52 + 69q51−21q50−83q49−138q48−52q47 + 10q46 + 53q45 + 103q44 + 43q43−5q42−37q41−75q40−21q39 + 4q38 + 18q37 + 48q36 + 13q35−q34−10q33−34q32−3q31 + 5q30 + 4q29 + 15q28−q27 + q26 + 2q25−14q24 + 2q23 + 4q22 + 2q20−4q19 + 2q18 + 4q17−6q16 + 2q15 + 2q14−q13−2q11 + 2q10 + q9−2q8 + q7

[edit] 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.

[edit] 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.

7_3

7_5

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