7 6

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7_5

7_7

Contents

Image:7 6.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,7,-2,1,-3,6,-7,2,-4,5,-6,3,-5,4/goTop.html 7_6's page] at Knotilus!

Visit 7 6's page at the original Knot Atlas!

[edit 7 6 Quick Notes]
[edit 7 6 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X3849 X5,12,6,13 X9,1,10,14 X13,11,14,10 X11,6,12,7 X7283
Gauss code -1, 7, -2, 1, -3, 6, -7, 2, -4, 5, -6, 3, -5, 4
Dowker-Thistlethwaite code 4 8 12 2 14 6 10
Conway Notation [2212]


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

Length is 7, width is 4,

Braid index is 4

Image:7 6_ML.gif Image:7 6_AP.gif
[{10, 2}, {1, 8}, {6, 9}, {8, 10}, {7, 3}, {2, 6}, {4, 7}, {3, 5}, {9, 4}, {5, 1}]

[edit Notes on presentations of 7 6]

Knot 7_6.
Knot 7_6.
A graph, knot 7_6.
A graph, knot 7_6.
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 6"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3849 X5,12,6,13 X9,1,10,14 X13,11,14,10 X11,6,12,7 X7283
In[5]:= GaussCode[K]
Out[5]= -1, 7, -2, 1, -3, 6, -7, 2, -4, 5, -6, 3, -5, 4
In[6]:= DTCode[K]
Out[6]= 4 8 12 2 14 6 10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [2212]
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,−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, 7, 4 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.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 6_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 2}, {1, 8}, {6, 9}, {8, 10}, {7, 3}, {2, 6}, {4, 7}, {3, 5}, {9, 4}, {5, 1}]
In[14]:= Draw[ap]
Image:7 6_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][-1]
Hyperbolic Volume 7.08493
A-Polynomial See Data:7 6/A-polynomial

[edit Notes for 7 6's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus ConcordanceGenus(Knot(7,6))
Rasmussen s-Invariant -2

[edit Notes for 7 6's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t2−t−2 + 5t + 5t−1−7
Conway polynomial −z4 + z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 19, -2 }
Jones polynomial q−2 + 3q−1−3q−2 + 4q−3−3q−4 + 2q−5−q−6
HOMFLY-PT polynomial (db, data sources) −a6 + 2a4z2 + 2a4−a2z4−2a2z2−a2 + z2 + 1
Kauffman polynomial (db, data sources) a7z3−a7z + 2a6z4−2a6z2 + a6 + 2a5z5−a5z3 + a4z6 + 2a4z4−4a4z2 + 2a4 + 4a3z5−6a3z3 + 2a3z + a2z6 + a2z4−4a2z2 + a2 + 2az5−4az3 + az + z4−2z2 + 1
The A2 invariant −q20−q18 + q16 + q12 + q10 + q6−q4 + q2 + q−4
The G2 invariant q100−q98 + 2q96−2q94−3q88 + 5q86−6q84 + 4q82−4q80−q78 + 5q76−8q74 + 8q72−5q70 + q68 + 2q66−5q64 + 4q62−2q58 + 6q56−5q54 + 2q52 + 6q50−9q48 + 11q46−9q44 + 5q42 + 2q40−6q38 + 10q36−9q34 + 8q32−3q30−3q28 + 5q26−5q24 + 3q22−3q18 + 5q16−3q14−q12 + 5q10−8q8 + 8q6−5q4−q2 + 5−6q−2 + 8q−4−4q−6 + 2q−8 + q−10−3q−12 + 3q−14−q−16 + q−18
Further Quantum Invariants
Further quantum knot invariants for 7_6.

A1 Invariants.

Weight Invariant
1 −q13 + q11−q9 + q7 + q5 + q−q−1 + q−3
2 q36−q34−q32 + 3q30−2q28−3q26 + 4q24−q22−3q20 + 2q18 + q16−q12 + 2q10 + q8−3q6 + 2q4 + 3q2−3 + q−2 + 3q−4−2q−6−q−8 + q−10
3 −q69 + q67 + q65−q63−2q61 + 2q59 + 5q57−3q55−7q53 + 2q51 + 9q49−q47−11q45−q43 + 11q41 + 2q39−7q37−4q35 + 5q33 + 3q31−2q29−4q27−2q25 + 3q23 + 4q21−3q19−6q17 + 5q15 + 8q13−2q11−8q9 + 2q7 + 10q5 + q3−9q−2q−1 + 7q−3 + 4q−5−5q−7−5q−9 + 3q−11 + 4q−13−2q−17−q−19 + q−21
4 q112−q110−q108 + q106 + 2q102−4q100−3q98 + 4q96 + 4q94 + 7q92−10q90−12q88 + 4q86 + 12q84 + 18q82−13q80−25q78−6q76 + 15q74 + 32q72−5q70−28q68−15q66 + 8q64 + 31q62 + 5q60−16q58−16q56−3q54 + 17q52 + 9q50−3q48−10q46−8q44 + 9q40 + 9q38−5q36−13q34−10q32 + 12q30 + 19q28−q26−16q24−18q22 + 14q20 + 26q18 + 4q16−16q14−26q12 + 7q10 + 25q8 + 13q6−7q4−29q2−4 + 16q−2 + 17q−4 + 6q−6−19q−8−11q−10 + q−12 + 11q−14 + 12q−16−6q−18−7q−20−5q−22 + q−24 + 6q−26 + q−28−2q−32−q−34 + q−36
5 −q165 + q163 + q161−q159 + 2q151 + 2q149−4q147−6q145−2q143 + 5q141 + 11q139 + 7q137−6q135−21q133−17q131 + 9q129 + 31q127 + 31q125−q123−41q121−52q119−11q117 + 47q115 + 68q113 + 30q111−43q109−81q107−49q105 + 31q103 + 85q101 + 61q99−16q97−78q95−65q93−q91 + 60q89 + 68q87 + 12q85−42q83−52q81−21q79 + 21q77 + 43q75 + 25q73−8q71−27q69−26q67−8q65 + 15q63 + 26q61 + 18q59−7q57−29q55−29q53−2q51 + 34q49 + 39q47 + 4q45−39q43−48q41−10q39 + 44q37 + 61q35 + 15q33−47q31−67q29−26q27 + 46q25 + 75q23 + 36q21−35q19−77q17−47q15 + 23q13 + 73q11 + 58q9−3q7−58q5−64q3−14q + 42q−1 + 60q−3 + 28q−5−19q−7−48q−9−38q−11 + q−13 + 32q−15 + 35q−17 + 12q−19−14q−21−26q−23−19q−25 + 3q−27 + 16q−29 + 15q−31 + 4q−33−5q−35−9q−37−7q−39 + q−41 + 4q−43 + 3q−45 + q−47−2q−51−q−53 + q−55
6 q228−q226−q224 + q222−2q216 + 2q214−q212−2q210 + 6q208 + 4q206−8q202−4q200−8q198−3q196 + 19q194 + 21q192 + 9q190−19q188−24q186−38q184−19q182 + 41q180 + 67q178 + 57q176−9q174−58q172−110q170−86q168 + 32q166 + 126q164 + 155q162 + 67q160−49q158−188q156−200q154−51q152 + 124q150 + 237q148 + 178q146 + 35q144−185q142−273q140−156q138 + 43q136 + 222q134 + 233q132 + 128q130−98q128−235q126−194q124−46q122 + 123q120 + 186q118 + 156q116−4q114−128q112−149q110−83q108 + 25q106 + 93q104 + 116q102 + 47q100−29q98−77q96−75q94−33q92 + 20q90 + 65q88 + 65q86 + 35q84−26q82−66q80−71q78−24q76 + 39q74 + 88q72 + 80q70−6q68−83q66−118q64−54q62 + 43q60 + 133q58 + 132q56 + 7q54−115q52−173q50−92q48 + 42q46 + 178q44 + 191q42 + 43q40−123q38−220q36−149q34−q32 + 184q30 + 239q28 + 111q26−69q24−213q22−200q20−88q18 + 116q16 + 230q14 + 175q12 + 32q10−130q8−192q6−166q4−q2 + 137 + 170q−2 + 114q−4−6q−6−101q−8−160q−10−84q−12 + 16q−14 + 86q−16 + 108q−18 + 69q−20 + 3q−22−78q−24−76q−26−47q−28 + 39q−32 + 56q−34 + 43q−36−8q−38−23q−40−32q−42−23q−44−6q−46 + 14q−48 + 23q−50 + 9q−52 + 4q−54−5q−56−8q−58−9q−60−q−62 + 4q−64 + q−66 + 3q−68 + q−70−2q−74−q−76 + q−78

A2 Invariants.

Weight Invariant
1,0 −q20−q18 + q16 + q12 + q10 + q6−q4 + q2 + q−4
1,1 q52−2q50 + 4q48−6q46 + 11q44−14q42 + 16q40−22q38 + 19q36−18q34 + 10q32−2q30−7q28 + 22q26−28q24 + 36q22−39q20 + 36q18−34q16 + 26q14−16q12 + 8q10 + 6q8−10q6 + 19q4−20q2 + 22−18q−2 + 13q−4−10q−6 + 6q−8−2q−10 + q−12
2,0 q50 + q48−2q44−q42 + q40−q38−2q36 + 2q32−2q28 + q26 + q24−q22 + q20 + q18 + 2q12−q8 + q6 + 3q4−1 + 2q−2 + q−4−q−6−q−8 + q−12
3,0 −q90−q88 + q84 + 3q82−q78−q76 + 2q74 + 5q72−2q70−5q68−5q66 + 3q64 + 7q62−q60−7q58−6q56 + 4q54 + 8q52−5q48−2q46 + 5q44 + 3q42−3q40−4q38−q36−4q32−2q30 + q28 + 4q26 + q24−2q22 + 3q20 + 8q18 + 5q16−3q14−6q12 + 3q10 + 7q8 + 2q6−6q4−6q2 + 4 + 6q−2 + 2q−4−4q−6−4q−8 + 2q−10 + 3q−12 + 3q−14−q−16−2q−18−q−20 + q−24

A3 Invariants.

Weight Invariant
0,1,0 q42−q40 + 2q36−3q34−2q32 + 2q30−3q28−q26 + 3q24 + q20 + q18 + 2q16−q12 + 2q10 + q8−3q6 + 2q4 + 2q2−2 + 2q−2 + q−4−q−6 + q−8
1,0,0 −q27−q25−q23 + q21 + 2q17 + q15 + q13−q5 + q3 + q−1 + q−5

A4 Invariants.

Weight Invariant
0,1,0,0 q56 + q54−q50−3q44−4q42−2q36 + q34 + 4q32 + 2q30 + 2q26 + q24−2q22−q20 + 3q18−q16 + 4q12 + 2q10−2q8 + 2q4−1 + q−2 + 2q−4 + q−10
1,0,0,0 −q34−q32−q30−q28 + q26 + 2q22 + 2q20 + q18 + q16−q10−q6 + q4 + 1 + q−2 + q−6

B2 Invariants.

Weight Invariant
0,1 −q42 + q40−2q38 + 2q36−3q34 + 2q32−2q30 + q28 + q26−q24 + 4q22−3q20 + 5q18−4q16 + 4q14−3q12 + 2q10−q8−q6 + 2q4−2q2 + 2−2q−2 + 3q−4−q−6 + q−8
1,0 q68−q64−q62 + q60 + 2q58−3q54−2q52 + q50 + 2q48−q46−3q44 + 2q40 + q38−2q36 + 2q32 + 2q30−q28−q26 + q24 + 2q22−q18 + 2q14 + q12−2q10−2q8 + 2q6 + 3q4−2−q−2 + 2q−4 + 2q−6−q−8−q−10 + q−14

D4 Invariants.

Weight Invariant
1,0,0,0 q58−q56 + q54−q52 + 2q50−3q48−3q44 + q42−2q40−q38 + 4q32−q30 + 4q28−2q26 + 5q24−2q22 + 3q20−3q18 + 2q16−q14 + q12−q10−q8 + 2q6−q4 + 2q2−1 + 3q−2−q−4 + 2q−6−q−8 + q−10

G2 Invariants.

Weight Invariant
1,0 q100−q98 + 2q96−2q94−3q88 + 5q86−6q84 + 4q82−4q80−q78 + 5q76−8q74 + 8q72−5q70 + q68 + 2q66−5q64 + 4q62−2q58 + 6q56−5q54 + 2q52 + 6q50−9q48 + 11q46−9q44 + 5q42 + 2q40−6q38 + 10q36−9q34 + 8q32−3q30−3q28 + 5q26−5q24 + 3q22−3q18 + 5q16−3q14−q12 + 5q10−8q8 + 8q6−5q4−q2 + 5−6q−2 + 8q−4−4q−6 + 2q−8 + q−10−3q−12 + 3q−14−q−16 + q−18

.

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 6"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t2−t−2 + 5t + 5t−1−7
In[5]:= Conway[K][z]
Out[5]= −z4 + z2 + 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]= { 19, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q−2 + 3q−1−3q−2 + 4q−3−3q−4 + 2q−5−q−6
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −a6 + 2a4z2 + 2a4−a2z4−2a2z2−a2 + z2 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a7z3−a7z + 2a6z4−2a6z2 + a6 + 2a5z5−a5z3 + a4z6 + 2a4z4−4a4z2 + 2a4 + 4a3z5−6a3z3 + 2a3z + a2z6 + a2z4−4a2z2 + a2 + 2az5−4az3 + az + z4−2z2 + 1

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

Same Alexander/Conway Polynomial: {10_133,}

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 6"];
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]= { −t2−t−2 + 5t + 5t−1−7, q−2 + 3q−1−3q−2 + 4q−3−3q−4 + 2q−5−q−6 }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {10_133,}
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: (1, -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
4 −16 8 \frac{158}{3} \frac{34}{3} −64 -\frac{544}{3} -\frac{64}{3} −48 \frac{32}{3} 128 \frac{632}{3} \frac{136}{3} \frac{19471}{30} -\frac{474}{5} \frac{17342}{45} \frac{401}{18} \frac{1711}{30}

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 6. 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%>-5</td><td width=8.33333%>-4</td><td width=8.33333%>-3</td><td width=8.33333%>-2</td><td width=8.33333%>-1</td><td width=8.33333%>0</td><td width=8.33333%>1</td><td width=8.33333%>2</td><td width=16.6667%>χ</td></tr> <tr align=center><td>3</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>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td>-1</td></tr> <tr align=center><td>-1</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>-3</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-5</td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-9</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-11</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-13</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 = −3 i = −1
r = −5 {\mathbb Z}
r = −4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r = 1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 2 {\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 q4−2q3−q2 + 6q−4−5q−1 + 12q−2−5q−3−10q−4 + 16q−5−4q−6−13q−7 + 17q−8−3q−9−12q−10 + 12q−11−q−12−7q−13 + 5q−14−2q−16 + q−17
3 q9−2q8−q7 + 2q6 + 5q5−3q4−9q3 + 2q2 + 14q−18q−1−5q−2 + 24q−3 + 9q−4−26q−5−15q−6 + 30q−7 + 19q−8−29q−9−26q−10 + 33q−11 + 26q−12−30q−13−31q−14 + 31q−15 + 28q−16−25q−17−29q−18 + 22q−19 + 25q−20−16q−21−20q−22 + 10q−23 + 15q−24−6q−25−10q−26 + 3q−27 + 6q−28−2q−29−2q−30 + 2q−32−q−33
4 q16−2q15−q14 + 2q13 + q12 + 6q11−7q10−7q9 + 2q7 + 24q6−8q5−17q4−12q3−6q2 + 49q + 3−18q−1−32q−2−31q−3 + 71q−4 + 23q−5−6q−6−50q−7−64q−8 + 81q−9 + 43q−10 + 16q−11−62q−12−96q−13 + 83q−14 + 58q−15 + 36q−16−69q−17−118q−18 + 80q−19 + 66q−20 + 50q−21−69q−22−127q−23 + 72q−24 + 64q−25 + 57q−26−57q−27−119q−28 + 52q−29 + 51q−30 + 57q−31−36q−32−93q−33 + 29q−34 + 28q−35 + 44q−36−13q−37−56q−38 + 12q−39 + 7q−40 + 25q−41−q−42−25q−43 + 6q−44−q−45 + 9q−46 + q−47−8q−48 + 3q−49−q−50 + 2q−51−2q−53 + q−54
5 q25−2q24−q23 + 2q22 + q21 + 2q20 + 2q19−5q18−9q17 + 5q15 + 11q14 + 13q13−4q12−22q11−22q10−2q9 + 23q8 + 39q7 + 19q6−25q5−53q4−41q3 + 13q2 + 68q + 66 + 7q−1−71q−2−97q−3−37q−4 + 74q−5 + 121q−6 + 68q−7−56q−8−147q−9−107q−10 + 44q−11 + 163q−12 + 139q−13−17q−14−176q−15−179q−16 + 3q−17 + 183q−18 + 201q−19 + 29q−20−193q−21−233q−22−35q−23 + 192q−24 + 244q−25 + 64q−26−198q−27−269q−28−62q−29 + 192q−30 + 266q−31 + 89q−32−190q−33−280q−34−85q−35 + 174q−36 + 265q−37 + 108q−38−157q−39−262q−40−107q−41 + 132q−42 + 234q−43 + 118q−44−103q−45−206q−46−115q−47 + 71q−48 + 170q−49 + 105q−50−41q−51−129q−52−91q−53 + 17q−54 + 90q−55 + 73q−56−3q−57−56q−58−53q−59−4q−60 + 32q−61 + 32q−62 + 8q−63−16q−64−21q−65−4q−66 + 10q−67 + 6q−68 + 4q−69−q−70−8q−71 + 4q−73−q−74 + q−76−2q−77 + 2q−79−q−80
6 q36−2q35−q34 + 2q33 + q32 + 2q31−2q30 + 4q29−7q28−9q27 + 3q26 + 4q25 + 11q24 + 3q23 + 18q22−16q21−29q20−14q19−5q18 + 20q17 + 18q16 + 69q15−3q14−46q13−53q12−52q11−9q10 + 16q9 + 150q8 + 63q7−7q6−75q5−122q4−109q3−60q2 + 209q + 158 + 113q−1−19q−2−155q−3−247q−4−225q−5 + 183q−6 + 220q−7 + 275q−8 + 124q−9−100q−10−361q−11−429q−12 + 71q−13 + 207q−14 + 419q−15 + 304q−16 + 28q−17−416q−18−614q−19−77q−20 + 136q−21 + 516q−22 + 470q−23 + 176q−24−429q−25−750q−26−211q−27 + 55q−28 + 574q−29 + 592q−30 + 301q−31−428q−32−840q−33−308q−34−9q−35 + 609q−36 + 669q−37 + 387q−38−420q−39−889q−40−371q−41−56q−42 + 614q−43 + 709q−44 + 448q−45−390q−46−889q−47−416q−48−109q−49 + 572q−50 + 707q−51 + 496q−52−314q−53−820q−54−439q−55−177q−56 + 464q−57 + 641q−58 + 517q−59−190q−60−660q−61−409q−62−240q−63 + 295q−64 + 493q−65 + 476q−66−53q−67−434q−68−304q−69−251q−70 + 116q−71 + 294q−72 + 359q−73 + 35q−74−214q−75−161q−76−192q−77 + 3q−78 + 119q−79 + 210q−80 + 47q−81−75q−82−45q−83−104q−84−26q−85 + 25q−86 + 92q−87 + 23q−88−23q−89 + 4q−90−38q−91−16q−92−q−93 + 32q−94 + 4q−95−9q−96 + 9q−97−10q−98−4q−99−3q−100 + 10q−101−q−102−5q−103 + 5q−104−2q−105−q−107 + 2q−108−2q−110 + q−111
7 q49−2q48−q47 + 2q46 + q45 + 2q44−2q43 + 2q41−7q40−6q39 + 2q38 + 4q37 + 13q36 + 4q35 + q34 + 9q33−20q32−25q31−17q30−7q29 + 30q28 + 29q27 + 29q26 + 44q25−15q24−51q23−67q22−82q21 + 3q20 + 41q19 + 80q18 + 146q17 + 66q16−10q15−103q14−210q13−137q12−61q11 + 61q10 + 264q9 + 243q8 + 177q7 + 14q6−281q5−334q4−328q3−151q2 + 244q + 402 + 483q−1 + 344q−2−133q−3−423q−4−643q−5−568q−6−19q−7 + 374q−8 + 753q−9 + 804q−10 + 251q−11−270q−12−838q−13−1033q−14−485q−15 + 115q−16 + 846q−17 + 1231q−18 + 752q−19 + 88q−20−830q−21−1410q−22−986q−23−293q−24 + 764q−25 + 1525q−26 + 1227q−27 + 515q−28−695q−29−1640q−30−1409q−31−698q−32 + 593q−33 + 1702q−34 + 1594q−35 + 882q−36−533q−37−1773q−38−1707q−39−1014q−40 + 442q−41 + 1802q−42 + 1843q−43 + 1142q−44−411q−45−1852q−46−1903q−47−1215q−48 + 335q−49 + 1857q−50 + 1997q−51 + 1310q−52−320q−53−1889q−54−2017q−55−1348q−56 + 242q−57 + 1855q−58 + 2076q−59 + 1433q−60−210q−61−1843q−62−2062q−63−1459q−64 + 103q−65 + 1748q−66 + 2069q−67 + 1529q−68−21q−69−1657q−70−1998q−71−1541q−72−114q−73 + 1477q−74 + 1907q−75 + 1561q−76 + 239q−77−1278q−78−1748q−79−1517q−80−374q−81 + 1031q−82 + 1536q−83 + 1432q−84 + 486q−85−768q−86−1277q−87−1301q−88−551q−89 + 514q−90 + 994q−91 + 1110q−92 + 571q−93−285q−94−715q−95−892q−96−545q−97 + 117q−98 + 465q−99 + 671q−100 + 465q−101−9q−102−259q−103−464q−104−366q−105−50q−106 + 120q−107 + 296q−108 + 265q−109 + 61q−110−41q−111−170q−112−167q−113−48q−114−13q−115 + 89q−116 + 110q−117 + 34q−118 + 12q−119−49q−120−48q−121−11q−122−26q−123 + 14q−124 + 37q−125 + 10q−126 + 8q−127−15q−128−9q−129 + 7q−130−14q−131−q−132 + 10q−133 + 2q−134 + 3q−135−6q−136−q−137 + 6q−138−4q−139−2q−140 + 2q−141 + q−143−2q−144 + 2q−146−q−147

[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_5

7_7

Retrieved from "http://katlas.math.toronto.edu/wiki/7_6"
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