7 5

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7_6

Contents

Image:7 5.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 5'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,-3,5,-4,6,-7,2,-6,3,-5,4/goTop.html 7_5's page] at Knotilus!

Visit 7 5's page at the original Knot Atlas!

[edit 7 5 Quick Notes]
[edit 7 5 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X5,12,6,13 X7,14,8,1 X13,6,14,7 X11,8,12,9 X9,2,10,3
Gauss code -1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
Dowker-Thistlethwaite code 4 10 12 14 2 8 6
Conway Notation [322]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 7 5]

Knot 7_5.
Knot 7_5.
A graph, knot 7_5.
A graph, knot 7_5.
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 5"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3,10,4,11 X5,12,6,13 X7,14,8,1 X13,6,14,7 X11,8,12,9 X9,2,10,3
In[5]:= GaussCode[K]
Out[5]= -1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
In[6]:= DTCode[K]
Out[6]= 4 10 12 14 2 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [322]
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(3,{−1,−1,−1,−1,−2,1,−2,−2})
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]= { 3, 8, 3 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.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 5_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{9, 2}, {1, 7}, {6, 8}, {7, 9}, {8, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 1}]
In[14]:= Draw[ap]
Image:7 5_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number Failed to parse (unknown error\text): \text{$\$$Failed}
Hyperbolic Volume 6.44354
A-Polynomial See Data:7 5/A-polynomial

[edit Notes for 7 5's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 2
Topological 4 genus 2
Concordance genus ConcordanceGenus(Knot(7,5))
Rasmussen s-Invariant -4

[edit Notes for 7 5's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 2t2−4t + 5−4t−1 + 2t−2
Conway polynomial 2z4 + 4z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 17, -4 }
Jones polynomial −q−9 + 2q−8−3q−7 + 3q−6−3q−5 + 3q−4−q−3 + q−2
HOMFLY-PT polynomial (db, data sources) a^8 \left(-z^2\right)-a^8+a^6 z^4+2 a^6 z^2+a^4 z^4+3 a^4 z^2+2 a^4
Kauffman polynomial (db, data sources) a11z3−a11z + 2a10z4−2a10z2 + 2a9z5−2a9z3 + a9z + a8z6 + a8z2−a8 + 3a7z5−4a7z3 + a7z + a6z6−a6z4 + a5z5−a5z3−a5z + a4z4−3a4z2 + 2a4
The A2 invariant −q28−q22−q18 + q16 + q14 + q12 + 2q10 + q6
The G2 invariant q148−q146 + 2q144−2q142 + q138−2q136 + 5q134−5q132 + 4q130−2q128−3q126 + 4q124−6q122 + 5q120−3q118−q116 + 3q114−3q112 + q110 + 2q108−5q106 + 4q104−3q102−2q100 + 5q98−7q96 + 8q94−7q92 + 2q90 + 2q88−6q86 + 6q84−7q82 + 4q80−2q76 + 3q74−3q72 + 2q70 + 3q68−5q66 + 3q64−2q60 + 7q58−5q56 + 5q54−q52 + 4q48−4q46 + 5q44−q42 + q40 + q38−q36 + 2q34 + q30
Further Quantum Invariants
Further quantum knot invariants for 7_5.

A1 Invariants.

Weight Invariant
1 −q19 + q17−q15 + 2q7 + q3
2 q52−q50−q48 + 3q46−q44−3q42 + 3q40−2q36 + q34−q30−2q28 + q26 + q24−3q22 + q20 + 3q18−2q16 + q14 + 3q12 + q6
3 −q99 + q97 + q95−q93−2q91 + q89 + 5q87−q85−7q83−q81 + 7q79 + 3q77−7q75−3q73 + 7q71 + 4q69−4q67−3q65 + 2q63 + 2q61−3q57−2q55 + q53 + 4q51−2q49−6q47 + 7q43−q41−7q39−2q37 + 6q35 + 3q33−6q31−3q29 + 4q27 + 5q25−q23−2q21 + q19 + 3q17 + q15 + q9
4 q160−q158−q156 + q154 + 2q150−3q148−3q146 + 2q144 + 3q142 + 9q140−5q138−12q136−4q134 + 5q132 + 19q130−15q126−13q124 + 24q120 + 8q118−12q116−16q114−5q112 + 17q110 + 9q108−4q106−10q104−6q102 + 7q100 + 7q98 + 3q96−2q94−5q92−3q90 + 4q88 + 9q86 + q84−7q82−12q80 + 2q78 + 13q76 + 6q74−7q72−20q70 + q68 + 16q66 + 10q64−4q62−22q60−5q58 + 11q56 + 13q54 + 5q52−17q50−10q48 + 2q46 + 10q44 + 10q42−6q40−7q38−4q36 + 3q34 + 8q32 + q30−2q26 + 3q22 + q20 + q18 + q12
5 −q235 + q233 + q231−q229 + q221 + 2q219−2q217−5q215−3q213 + q211 + 8q209 + 10q207 + 4q205−12q203−21q201−9q199 + 12q197 + 29q195 + 22q193−7q191−37q189−36q187 + 38q183 + 46q181 + 12q179−34q177−53q175−23q173 + 29q171 + 51q169 + 29q167−18q165−45q163−30q161 + 8q159 + 35q157 + 27q155−2q153−23q151−24q149−5q147 + 13q145 + 17q143 + 8q141−3q139−12q137−13q135−2q133 + 10q131 + 16q129 + 12q127−7q125−21q123−14q121 + 7q119 + 27q117 + 23q115−7q113−34q111−26q109 + 4q107 + 36q105 + 37q103−3q101−41q99−41q97−4q95 + 37q93 + 45q91 + 13q89−34q87−47q85−22q83 + 20q81 + 41q79 + 29q77−6q75−36q73−32q71−4q69 + 21q67 + 29q65 + 14q63−11q61−23q59−16q57−q55 + 13q53 + 15q51 + 5q49−4q47−9q45−6q43 + q41 + 6q39 + 4q37 + 3q35−2q31 + 2q27 + q25 + q23 + q21 + q15
6 q324−q322−q320 + q318−2q312 + 2q310−2q306 + 4q304 + 3q302 + q300−5q298−2q296−8q294−8q292 + 8q290 + 16q288 + 17q286 + q284−4q282−31q280−39q278−6q276 + 28q274 + 55q272 + 41q270 + 19q268−49q266−90q264−61q262 + 4q260 + 78q258 + 100q256 + 84q254−26q252−117q250−125q248−58q246 + 54q244 + 123q242 + 142q240 + 32q238−89q236−140q234−104q232 + 2q230 + 92q228 + 142q226 + 68q224−34q222−100q220−98q218−33q216 + 37q214 + 93q212 + 63q210 + 6q208−44q206−60q204−38q202−2q200 + 39q198 + 40q196 + 24q194−4q192−24q190−32q188−23q186 + q184 + 22q182 + 35q180 + 21q178−5q176−37q174−39q172−21q170 + 23q168 + 58q166 + 45q164 + 4q162−54q160−66q158−43q156 + 33q154 + 92q152 + 74q150 + 11q148−73q146−100q144−76q142 + 30q140 + 115q138 + 109q136 + 38q134−67q132−119q130−115q128−4q126 + 102q124 + 127q122 + 79q120−22q118−98q116−134q114−57q112 + 42q110 + 101q108 + 100q106 + 39q104−33q102−105q100−82q98−24q96 + 37q94 + 73q92 + 65q90 + 28q88−40q86−56q84−47q82−17q80 + 18q78 + 41q76 + 40q74 + 6q72−10q70−25q68−24q66−13q64 + 7q62 + 19q60 + 10q58 + 8q56−q54−7q52−9q50−2q48 + 4q46 + 2q44 + 5q42 + 3q40 + q38−2q36 + 2q32 + q28 + q26 + q24 + q18

A2 Invariants.

Weight Invariant
1,0 −q28−q22−q18 + q16 + q14 + q12 + 2q10 + q6
1,1 q76−2q74 + 4q72−6q70 + 9q68−12q66 + 14q64−14q62 + 13q60−10q58 + 2q56 + 4q54−12q52 + 20q50−22q48 + 28q46−26q44 + 22q42−22q40 + 8q38−10q36−4q34 + 6q32−12q30 + 16q28−10q26 + 16q24−6q22 + 10q20−2q18 + 4q16 + q12
2,0 q70 + q62−2q58 + q54 + q48−3q44−3q42−2q40−3q38−2q36 + q34 + q32 + 3q28 + 3q26 + 2q24 + q22 + 3q20 + 2q18 + q12
3,0 −q126−q116 + 2q114 + 2q112 + 2q110−2q108−5q106−q104 + 2q102 + 3q100−2q98−3q96 + 2q94 + 6q92 + 4q90−2q88−2q86 + 2q84 + 5q82 + q80−3q78−q76 + q74−4q70−4q68−q66−2q64−6q62−6q60−3q58 + q56−3q54−5q52−2q50 + 5q48 + 6q46−q42 + 2q40 + 8q38 + 6q36 + q34−q32 + 2q30 + 4q28 + 3q26 + q18

A3 Invariants.

Weight Invariant
0,1,0 q62−q60 + 2q56−2q54 + 3q50−2q48−q46−2q42−2q40−q38−q34−2q32 + q30 + q28−2q26 + 4q24 + 3q22 + q20 + 3q18 + 2q16 + q12
1,0,0 −q37−q33−q29−q25 + q21 + q19 + 2q17 + q15 + 2q13 + q9
1,0,1 q100−2q98 + 3q96−q94−3q92 + 8q90−10q88 + 6q86 + 3q84−12q82 + 18q80−15q78 + 4q76 + 6q74−17q72 + 17q70−10q68 + 3q66 + 9q64−2q62 + 3q60 + 6q58−6q56−7q54 + 5q52−26q50 + 6q48−14q46−12q44 + 13q42−17q40 + 17q38 + 3q36 + 3q34 + 15q32−q30 + 10q28 + 4q26 + 2q24 + 4q22 + q18

A4 Invariants.

Weight Invariant
0,1,0,0 q80−q76 + q74 + 2q72 + 2q66 + q64−2q62−2q60−q58−4q56−5q54−q52−2q50−3q48 + q44−2q42−q40 + 2q38 + 2q36 + q34 + 3q32 + 6q30 + 3q28 + 3q26 + 3q24 + 2q22 + q18
1,0,0,0 −q46−q42−q40−q36−q32 + q26 + q24 + 2q22 + 2q20 + q18 + 2q16 + q12

B2 Invariants.

Weight Invariant
0,1 −q62 + q60−2q58 + 2q56−2q54 + 2q52−q50 + q46−2q44 + 2q42−4q40 + 3q38−4q36 + 3q34−2q32 + q30 + q28 + 2q24−q22 + 3q20−q18 + 2q16 + q12
1,0 q100−q96−q94 + q92 + 2q90−2q86−q84 + 2q82 + 2q80−q78−2q76 + q72−q70−2q68−q66 + q64−q60−2q58 + q54−q52−2q50 + 2q46−q42−q40 + 3q38 + 3q36 + q34−q32 + q30 + 2q28 + 2q26 + q18

D4 Invariants.

Weight Invariant
1,0,0,0 q86−q84 + q82−q80 + 2q78−2q76 + q74−q72 + 2q70−q66−2q62 + q60−4q58−4q54 + 2q52−3q50 + 2q48−3q46 + q44 + 3q34 + q32 + 4q30 + q28 + 4q26 + q24 + 2q22 + q18

G2 Invariants.

Weight Invariant
1,0 q148−q146 + 2q144−2q142 + q138−2q136 + 5q134−5q132 + 4q130−2q128−3q126 + 4q124−6q122 + 5q120−3q118−q116 + 3q114−3q112 + q110 + 2q108−5q106 + 4q104−3q102−2q100 + 5q98−7q96 + 8q94−7q92 + 2q90 + 2q88−6q86 + 6q84−7q82 + 4q80−2q76 + 3q74−3q72 + 2q70 + 3q68−5q66 + 3q64−2q60 + 7q58−5q56 + 5q54−q52 + 4q48−4q46 + 5q44−q42 + q40 + q38−q36 + 2q34 + q30

.

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 5"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 2t2−4t + 5−4t−1 + 2t−2
In[5]:= Conway[K][z]
Out[5]= 2z4 + 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]= { 17, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q−9 + 2q−8−3q−7 + 3q−6−3q−5 + 3q−4−q−3 + q−2
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= a^8 \left(-z^2\right)-a^8+a^6 z^4+2 a^6 z^2+a^4 z^4+3 a^4 z^2+2 a^4
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a11z3−a11z + 2a10z4−2a10z2 + 2a9z5−2a9z3 + a9z + a8z6 + a8z2−a8 + 3a7z5−4a7z3 + a7z + a6z6−a6z4 + a5z5−a5z3−a5z + a4z4−3a4z2 + 2a4

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

Same Alexander/Conway Polynomial: {10_130,}

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 5"];
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]= { 2t2−4t + 5−4t−1 + 2t−2, −q−9 + 2q−8−3q−7 + 3q−6−3q−5 + 3q−4−q−3 + q−2 }
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_130,}
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{968}{3} \frac{136}{3} −1024 -\frac{5440}{3} -\frac{928}{3} −224 \frac{2048}{3} 2048 \frac{15488}{3} \frac{2176}{3} \frac{156422}{15} \frac{5912}{15} \frac{170888}{45} \frac{730}{9} \frac{7142}{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 = -4 is the signature of 7 5. 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%>-7</td><td width=8.33333%>-6</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=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>-5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>0</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td> </td><td>2</td></tr> <tr align=center><td>-9</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-11</td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-13</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-15</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>-17</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>-19</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 = −5 i = −3
r = −7 {\mathbb Z}
r = −6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z} {\mathbb Z}

[edit] The Coloured Jones Polynomials

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

 n  Jn
2 q−4−q−5 + 4q−7−3q−8−3q−9 + 9q−10−5q−11−7q−12 + 13q−13−5q−14−10q−15 + 14q−16−4q−17−9q−18 + 11q−19−2q−20−6q−21 + 5q−22−2q−24 + q−25
3 q−6−q−7 + q−9 + 3q−10−3q−11−3q−12 + 2q−13 + 9q−14−4q−15−10q−16−q−17 + 18q−18−q−19−18q−20−6q−21 + 24q−22 + 7q−23−25q−24−12q−25 + 28q−26 + 13q−27−28q−28−15q−29 + 27q−30 + 16q−31−26q−32−15q−33 + 22q−34 + 15q−35−18q−36−12q−37 + 12q−38 + 11q−39−8q−40−8q−41 + 4q−42 + 5q−43−2q−44−2q−45 + 2q−47−q−48
4 q−8−q−9 + q−11 + 3q−13−4q−14−2q−15 + 3q−16 + q−17 + 10q−18−9q−19−9q−20 + 2q−22 + 26q−23−9q−24−17q−25−12q−26−5q−27 + 48q−28−q−29−19q−30−28q−31−22q−32 + 66q−33 + 13q−34−13q−35−43q−36−43q−37 + 79q−38 + 26q−39−6q−40−54q−41−57q−42 + 84q−43 + 34q−44 + 2q−45−59q−46−64q−47 + 82q−48 + 37q−49 + 7q−50−55q−51−64q−52 + 69q−53 + 33q−54 + 13q−55−42q−56−56q−57 + 47q−58 + 22q−59 + 17q−60−22q−61−40q−62 + 23q−63 + 9q−64 + 15q−65−7q−66−21q−67 + 9q−68 + 7q−70−7q−72 + 3q−73−q−74 + 2q−75−2q−77 + q−78
5 q−10−q−11 + q−13 + 2q−16−3q−17−2q−18 + 3q−19 + 3q−20 + q−21 + 4q−22−8q−23−9q−24 + 8q−26 + 10q−27 + 14q−28−10q−29−23q−30−15q−31 + q−32 + 22q−33 + 39q−34 + 5q−35−31q−36−40q−37−27q−38 + 18q−39 + 69q−40 + 40q−41−19q−42−61q−43−69q−44−7q−45 + 82q−46 + 87q−47 + 13q−48−69q−49−110q−50−44q−51 + 82q−52 + 125q−53 + 53q−54−70q−55−142q−56−74q−57 + 74q−58 + 152q−59 + 83q−60−66q−61−162q−62−95q−63 + 67q−64 + 166q−65 + 102q−66−62q−67−168q−68−107q−69 + 56q−70 + 167q−71 + 111q−72−51q−73−159q−74−111q−75 + 38q−76 + 148q−77 + 112q−78−30q−79−130q−80−103q−81 + 11q−82 + 110q−83 + 97q−84−3q−85−83q−86−81q−87−11q−88 + 58q−89 + 67q−90 + 16q−91−37q−92−47q−93−19q−94 + 20q−95 + 31q−96 + 15q−97−7q−98−18q−99−12q−100 + 3q−101 + 10q−102 + 3q−103 + 2q−104−2q−105−6q−106 + q−107 + 3q−108−q−109 + q−111−2q−112 + 2q−114−q−115
6 q−12−q−13 + q−15−q−18 + 3q−19−3q−20−2q−21 + 4q−22 + 2q−23 + 2q−24−4q−25 + 5q−26−9q−27−9q−28 + 6q−29 + 8q−30 + 11q−31−2q−32 + 14q−33−21q−34−29q−35−5q−36 + 7q−37 + 26q−38 + 14q−39 + 48q−40−20q−41−52q−42−40q−43−23q−44 + 17q−45 + 30q−46 + 116q−47 + 17q−48−44q−49−76q−50−84q−51−41q−52 + 7q−53 + 188q−54 + 89q−55 + 17q−56−75q−57−143q−58−140q−59−70q−60 + 224q−61 + 165q−62 + 118q−63−27q−64−168q−65−246q−66−181q−67 + 220q−68 + 217q−69 + 223q−70 + 44q−71−162q−72−331q−73−287q−74 + 196q−75 + 244q−76 + 307q−77 + 107q−78−144q−79−390q−80−363q−81 + 173q−82 + 256q−83 + 365q−84 + 148q−85−131q−86−425q−87−407q−88 + 155q−89 + 258q−90 + 397q−91 + 174q−92−117q−93−438q−94−428q−95 + 131q−96 + 249q−97 + 405q−98 + 194q−99−89q−100−422q−101−429q−102 + 90q−103 + 213q−104 + 383q−105 + 210q−106−39q−107−365q−108−399q−109 + 34q−110 + 143q−111 + 318q−112 + 208q−113 + 27q−114−263q−115−325q−116−16q−117 + 53q−118 + 212q−119 + 172q−120 + 78q−121−142q−122−215q−123−35q−124−16q−125 + 100q−126 + 105q−127 + 86q−128−51q−129−105q−130−19q−131−38q−132 + 26q−133 + 40q−134 + 57q−135−10q−136−37q−137 + 3q−138−24q−139−q−140 + 6q−141 + 24q−142−2q−143−10q−144 + 8q−145−8q−146−2q−147−2q−148 + 8q−149−2q−150−4q−151 + 5q−152−2q−153−q−155 + 2q−156−2q−158 + q−159
7 q−14−q−15 + q−17−q−20 + 3q−22−3q−23−q−24 + 3q−25 + 2q−26 + 2q−27−3q−28−4q−29 + 5q−30−8q−31−4q−32 + 5q−33 + 8q−34 + 13q−35−q−36−8q−37 + 3q−38−22q−39−19q−40−2q−41 + 10q−42 + 38q−43 + 22q−44 + 7q−45 + 15q−46−41q−47−53q−48−41q−49−23q−50 + 45q−51 + 57q−52 + 61q−53 + 82q−54−17q−55−75q−56−102q−57−120q−58−22q−59 + 38q−60 + 113q−61 + 205q−62 + 97q−63−13q−64−115q−65−239q−66−175q−67−94q−68 + 71q−69 + 311q−70 + 270q−71 + 168q−72−10q−73−292q−74−341q−75−320q−76−105q−77 + 307q−78 + 422q−79 + 409q−80 + 214q−81−232q−82−441q−83−558q−84−370q−85 + 193q−86 + 483q−87 + 633q−88 + 482q−89−84q−90−463q−91−744q−92−628q−93 + 23q−94 + 471q−95 + 791q−96 + 727q−97 + 72q−98−439q−99−864q−100−830q−101−127q−102 + 433q−103 + 893q−104 + 899q−105 + 193q−106−415q−107−934q−108−962q−109−228q−110 + 408q−111 + 950q−112 + 1007q−113 + 264q−114−398q−115−972q−116−1039q−117−289q−118 + 390q−119 + 979q−120 + 1063q−121 + 313q−122−376q−123−979q−124−1079q−125−341q−126 + 355q−127 + 970q−128 + 1087q−129 + 365q−130−321q−131−940q−132−1081q−133−402q−134 + 270q−135 + 900q−136 + 1064q−137 + 428q−138−211q−139−818q−140−1019q−141−466q−142 + 128q−143 + 731q−144 + 959q−145 + 471q−146−49q−147−597q−148−859q−149−480q−150−44q−151 + 472q−152 + 741q−153 + 450q−154 + 109q−155−323q−156−597q−157−406q−158−162q−159 + 193q−160 + 457q−161 + 336q−162 + 181q−163−89q−164−317q−165−252q−166−177q−167 + 13q−168 + 200q−169 + 173q−170 + 149q−171 + 30q−172−115q−173−105q−174−108q−175−41q−176 + 57q−177 + 44q−178 + 74q−179 + 45q−180−25q−181−22q−182−43q−183−22q−184 + 11q−185−7q−186 + 18q−187 + 26q−188−3q−189−12q−191−4q−192 + 7q−193−12q−194 + q−195 + 8q−196 + 2q−198−4q−199 + 5q−201−4q−202−2q−203 + 2q−204 + q−206−2q−207 + 2q−209−q−210

[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).

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