8 2

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8_1

8_3

Contents

Image:8 2.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8_2's page at Knotilus!

Visit 8 2's page at the original Knot Atlas!

[edit 8 2 Quick Notes]
[edit 8 2 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9
Gauss code -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6
Dowker-Thistlethwaite code 4 10 12 14 16 2 6 8
Conway Notation [512]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 2]

Knot 8_2.
Knot 8_2.
A graph, knot 8_2.
A graph, knot 8_2.
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["8 2"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6
In[6]:= DTCode[K]
Out[6]= 4 10 12 14 16 2 6 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [512]
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,−1,2,−1,2})
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:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image: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:8 2_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {3, 5}, {4, 6}, {5, 7}, {6, 9}, {7, 1}]
In[14]:= Draw[ap]
Image:8 2_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index {4,5}
Nakanishi index 1
Maximal Thurston-Bennequin number [-11][1]
Hyperbolic Volume 4.93524
A-Polynomial See Data:8 2/A-polynomial

[edit Notes for 8 2's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 2
Topological 4 genus 2
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for 8 2's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t3 + 3t2−3t + 3−3t−1 + 3t−2−t−3
Conway polynomial −z6−3z4 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 17, -4 }
Jones polynomial 1−q−1 + 2q−2−2q−3 + 3q−4−3q−5 + 2q−6−2q−7 + q−8
HOMFLY-PT polynomial (db, data sources) z4a6 + 3z2a6 + a6−z6a4−5z4a4−7z2a4−3a4 + z4a2 + 4z2a2 + 3a2
Kauffman polynomial (db, data sources) z2a10 + 2z3a9−za9 + 2z4a8−z2a8 + 2z5a7−2z3a7−za7 + 2z6a6−5z4a6 + 3z2a6−a6 + z7a5−2z5a5−z3a5 + za5 + 3z6a4−12z4a4 + 12z2a4−3a4 + z7a3−4z5a3 + 3z3a3 + za3 + z6a2−5z4a2 + 7z2a2−3a2
The A2 invariant q24−q18−q16−q12 + q10 + q6 + q4 + q2 + 1
The G2 invariant q134−q132 + q130−q128−q126−q122 + 2q120−2q118 + q116 + q110 + q106−q104 + q102 + q96 + 2q92−q84 + q82−2q78 + q76−q74 + q70−4q68 + 2q66−3q64−3q58 + 3q56−2q54 + q52−q50−q48 + q46−2q44 + q42−q38 + q36 + q32 + 2q30−2q28 + 3q26−q24 + q22 + 3q20−3q18 + 4q16 + q12 + q10−q8 + 2q6 + q2
Further Quantum Invariants
Further quantum knot invariants for 8_2.

A1 Invariants.

Weight Invariant
1 q17−q15−q11 + q7 + q3 + q−1
2 q46−q44−q42 + q40−q38 + q36 + q34−q32 + q28−q24−q18−q16 + q14 + 2q8 + q6−q4 + q2 + 1−q−2 + q−6
3 q87−q85−q83 + q79 + q77−q75−q71 + q69 + q67−2q63 + 2q59−q55−q53 + q49 + 2q47−q45−q43 + 2q39−q37−2q35 + q31−q29−q27 + q23 + q21−q19−q17 + q15 + 2q13 + q11−q9−q7 + 2q5 + 2q3−2q−1 + 2q−5 + q−7−q−9−q−11 + q−15
4 q140−q138−q136 + 3q130−q128−q126−2q124−q122 + 5q120 + q118−q116−4q114−2q112 + 5q110 + 4q108−q106−6q104−4q102 + 4q100 + 5q98 + 2q96−4q94−6q92 + 3q88 + 4q86−2q82−q80−2q78 + q76 + 4q74 + 2q72−q70−3q68−q66 + 3q64 + 2q62−2q60−3q58−q56 + 3q54 + 2q52−3q50−3q48 + 4q44 + 3q42−3q40−4q38−2q36 + 2q34 + 3q32−q30−q28−2q26 + 3q22 + 2q20 + 2q18−q16−2q14 + 4q8 + 2q6−q4−2q2−3 + 2q−2 + 3q−4 + 2q−6−4q−10−q−12 + q−14 + 2q−16 + 2q−18−q−20−q−22−q−24 + q−28
5 q205−q203−q201 + 2q195 + q193−q191−3q189−q187 + q185 + 4q183 + 3q181−q179−5q177−5q175 + 3q173 + 6q171 + 4q169−2q167−9q165−7q163 + 5q161 + 11q159 + 5q157−5q155−11q153−9q151 + 5q149 + 14q147 + 10q145−2q143−12q141−10q139−q137 + 9q135 + 12q133 + 5q131−5q129−9q127−8q125−2q123 + 4q121 + 5q119 + 4q117−5q113−5q111−3q109 + 2q107 + 6q105 + 6q103−q101−5q99−3q97 + q95 + 5q93 + 4q91−3q89−5q87−q85 + 3q83 + 6q81 + 3q79−4q77−8q75−3q73 + 4q71 + 8q69 + 5q67−3q65−10q63−7q61 + q59 + 8q57 + 7q55−8q51−9q49−3q47 + 7q45 + 9q43 + 4q41−3q39−8q37−5q35 + 5q31 + 6q29 + 2q27−q25−3q23−3q21−q19 + 2q17 + 4q15 + 3q13 + 3q11−3q7−4q5−2q3 + q + 4q−1 + 5q−3 + 2q−5−2q−7−5q−9−4q−11 + 3q−15 + 5q−17 + 3q−19−q−21−4q−23−3q−25−q−27 + q−29 + 3q−31 + 2q−33−q−37−q−39−q−41 + q−45
6 q282−q280−q278 + 2q272 + q268−3q266−2q264 + q262 + q260 + 4q258 + 2q256 + q254−7q252−4q250 + 2q246 + 6q244 + 3q242−q240−9q238−3q236 + 2q234 + 4q232 + 6q230−6q226−10q224 + 2q222 + 11q220 + 10q218 + 6q216−8q214−17q212−14q210 + 4q208 + 19q206 + 19q204 + 8q202−10q200−24q198−24q196−5q194 + 14q192 + 24q190 + 20q188 + 4q186−14q184−24q182−15q180−q178 + 11q176 + 18q174 + 15q172 + 5q170−7q168−11q166−11q164−7q162 + q160 + 8q158 + 12q156 + 8q154−8q150−12q148−8q146 + 9q142 + 9q140 + q138−6q136−8q134−4q132 + 3q130 + 10q128 + 5q126−5q124−9q122−7q120 + q118 + 9q116 + 14q114 + 8q112−8q110−14q108−10q106 + 11q102 + 16q100 + 11q98−5q96−15q94−14q92−7q90 + 6q88 + 16q86 + 16q84 + 2q82−12q80−17q78−15q76−2q74 + 13q72 + 21q70 + 13q68−2q66−14q64−21q62−14q60 + 2q58 + 17q56 + 19q54 + 10q52−3q50−17q48−19q46−10q44 + 6q42 + 14q40 + 15q38 + 10q36−3q34−10q32−12q30−5q28 + q26 + 6q24 + 10q22 + 6q20 + 3q18−3q16−3q14−5q12−5q10−q8 + q6 + 5q4 + 4q2 + 7 + 3q−2−3q−4−5q−6−7q−8−4q−10−2q−12 + 6q−14 + 8q−16 + 6q−18 + 2q−20−3q−22−6q−24−9q−26−2q−28 + 2q−30 + 5q−32 + 6q−34 + 4q−36 + q−38−5q−40−4q−42−3q−44−q−46 + q−48 + 3q−50 + 3q−52−q−58−q−60−q−62 + q−66

A2 Invariants.

Weight Invariant
1,0 q24−q18−q16−q12 + q10 + q6 + q4 + q2 + 1
1,1 q68−2q66 + 2q64−2q62 + 3q60−4q58 + 2q56 + 2q52−2q50−2q46 + q44−2q42 + 2q40 + 2q38 + 6q34−6q32 + 6q30−11q28 + 8q26−10q24 + 6q22−6q20 + 4q18−2q14 + 5q12−4q10 + 8q8−6q6 + 6q4−2q2 + 4 + q−4
2,0 q60−q54−q52−q44 + q42 + q40 + q38 + 2q34 + q32−q30−q28−2q26−2q24−2q22 + q16 + 2q14 + 2q12 + q6 + q4 + 1 + q−2 + q−4
3,0 q108−q102−q100 + q94−2q92−q90 + 3q86 + q84−q82 + 2q78 + 4q76−3q72−3q70 + q66−3q62−q60 + 2q56 + q50 + q48−q44 + q42−q38−2q36 + 2q32 + q30−2q28−3q26−q24 + 2q22 + q20−q16 + 2q14 + 4q12 + 3q10−q6 + 2q2 + 1−q−4 + q−8 + q−10 + q−12

A3 Invariants.

Weight Invariant
0,1,0 q56−q54−q52 + q50−q48−q46 + q44 + q42 + 2q38 + q36−2q26−q24−q22−3q20−2q18 + 2q12 + 3q10 + 2q8 + 2q6 + 2q4 + 1
1,0,0 q31 + q27−q25−q21−q19−q17−q15 + 2q9 + q7 + 2q5 + q3 + q
1,0,1 q90−2q88 + q86 + q84−2q82 + 3q80−2q78−q76 + q74 + q70−q68 + q62−2q60 + 2q58−3q56−2q54 + 3q52−4q50 + 5q48−q46 + 4q44 + q40 + 3q38−2q36 + 4q34−5q32 + 2q30−5q28−q26−5q24−4q22−q20−3q18 + 4q16−2q14 + 6q12 + 2q10 + 5q8 + 6q6 + 2q4 + 4q2 + q−2

A4 Invariants.

Weight Invariant
0,1,0,0 q70−q66−2q60−2q58 + q52 + 2q50 + 4q48 + 2q46 + 2q44 + 2q42−q38−q34−4q32−3q30−4q28−4q26−4q24−2q22 + 2q18 + 3q16 + 4q14 + 4q12 + 4q10 + 3q8 + 2q6 + q4 + q2
1,0,0,0 q38 + q34−q26−q24−2q22−q20−2q18 + 2q12 + 2q10 + 2q8 + 2q6 + q4 + q2

B2 Invariants.

Weight Invariant
0,1 q56−q54 + q52−q50 + q48−q46 + q44−q42−q36 + 2q34−2q32 + 2q30−2q28 + 2q26−3q24 + q22−q20 + 2q12−q10 + 2q8 + 2q4 + 1
1,0 q90−q86−q84 + q80−q76−q74 + q70 + q68 + q62 + 2q60−q56 + q52−q48−q46−q40−q38−q32−2q30−q28 + q26 + q24−q20 + q18 + 2q16 + 2q14 + q8 + 2q6 + q−2

D4 Invariants.

Weight Invariant
1,0,0,0 q78−q76−q72 + q70−q68−q64 + q62 + q58 + q54 + q52 + 2q48−q46 + 2q44−q42 + 2q40−2q38 + q36−3q34−q32−4q30−2q28−3q26−2q24−q22 + 3q18 + 2q16 + 4q14 + 2q12 + 4q10 + q8 + 2q6 + q2

G2 Invariants.

Weight Invariant
1,0 q134−q132 + q130−q128−q126−q122 + 2q120−2q118 + q116 + q110 + q106−q104 + q102 + q96 + 2q92−q84 + q82−2q78 + q76−q74 + q70−4q68 + 2q66−3q64−3q58 + 3q56−2q54 + q52−q50−q48 + q46−2q44 + q42−q38 + q36 + q32 + 2q30−2q28 + 3q26−q24 + q22 + 3q20−3q18 + 4q16 + q12 + q10−q8 + 2q6 + q2

.

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["8 2"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t3 + 3t2−3t + 3−3t−1 + 3t−2−t−3
In[5]:= Conway[K][z]
Out[5]= −z6−3z4 + 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]= 1−q−1 + 2q−2−2q−3 + 3q−4−3q−5 + 2q−6−2q−7 + q−8
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z4a6 + 3z2a6 + a6−z6a4−5z4a4−7z2a4−3a4 + z4a2 + 4z2a2 + 3a2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z2a10 + 2z3a9−za9 + 2z4a8−z2a8 + 2z5a7−2z3a7−za7 + 2z6a6−5z4a6 + 3z2a6−a6 + z7a5−2z5a5−z3a5 + za5 + 3z6a4−12z4a4 + 12z2a4−3a4 + z7a3−4z5a3 + 3z3a3 + za3 + z6a2−5z4a2 + 7z2a2−3a2

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

Same Alexander/Conway Polynomial: {K11n6,}

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["8 2"];
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]= { −t3 + 3t2−3t + 3−3t−1 + 3t−2−t−3, 1−q−1 + 2q−2−2q−3 + 3q−4−3q−5 + 2q−6−2q−7 + q−8 }
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]= {K11n6,}
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: (0, 1)
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
0 8 0 0 24 0 -\frac{304}{3} -\frac{160}{3} −88 0 32 0 0 432 −104 472 80 32

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 8 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012χ
1        11
-1         0
-3      21 1
-5     11  0
-7    21   1
-9   11    0
-11  12     -1
-13 11      0
-15 1       -1
-171        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −5 i = −3
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}\oplus{\mathbb Z}_2 {\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}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r = 1 {\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 q2−q−1 + 3q−1−q−2−3q−3 + 5q−4−5q−6 + 5q−7 + q−8−7q−9 + 5q−10 + 2q−11−7q−12 + 4q−13 + 3q−14−6q−15 + 3q−16 + 2q−17−4q−18 + 3q−19−2q−21 + q−22
3 q6−q5−q4 + 3q2−3−2q−1 + 5q−2 + 2q−3−3q−4−5q−5 + 5q−6 + 4q−7−2q−8−6q−9 + 3q−10 + 4q−11−6q−13 + 2q−14 + 3q−15−4q−17 + q−18 + q−19 + q−20−q−21−q−22 + q−24 + 2q−25−2q−26−q−27 + 2q−29−q−30 + q−31−2q−32 + q−34 + 2q−35−2q−36−2q−37 + 2q−38 + q−39−2q−41 + q−42
4 q12−q11−q10 + 4q7−q6−2q5−2q4−3q3 + 8q2 + q−1−3q−1−8q−2 + 9q−3 + 2q−4 + 2q−5−q−6−12q−7 + 9q−8 + 3q−10 + 2q−11−12q−12 + 10q−13−3q−14 + q−15 + 3q−16−12q−17 + 14q−18−4q−19−3q−20 + q−21−11q−22 + 20q−23−3q−24−7q−25−2q−26−11q−27 + 25q−28−2q−29−11q−30−4q−31−10q−32 + 29q−33−q−34−15q−35−6q−36−8q−37 + 32q−38 + q−39−18q−40−9q−41−7q−42 + 31q−43 + 3q−44−14q−45−10q−46−10q−47 + 25q−48 + 5q−49−8q−50−7q−51−11q−52 + 17q−53 + 3q−54−3q−55−2q−56−10q−57 + 10q−58 + q−59−6q−62 + 4q−63 + q−65−2q−67 + q−68
5 q20−q19−q18 + q15 + 3q14−3q12−2q11−2q10 + 6q8 + 4q7−q6−4q5−5q4−4q3 + 5q2 + 7q + 3−q−1−6q−2−7q−3 + 2q−4 + 5q−5 + 4q−6 + 2q−7−3q−8−7q−9 + 3q−10 + 3q−11 + q−12−3q−14−5q−15 + 6q−16 + 7q−17−5q−19−8q−20−8q−21 + 11q−22 + 14q−23 + 5q−24−7q−25−18q−26−14q−27 + 12q−28 + 22q−29 + 12q−30−6q−31−25q−32−22q−33 + 9q−34 + 29q−35 + 20q−36−3q−37−29q−38−29q−39 + 4q−40 + 33q−41 + 27q−42−32q−44−33q−45 + 35q−47 + 34q−48 + q−49−36q−50−37q−51−2q−52 + 39q−53 + 41q−54 + q−55−40q−56−42q−57−4q−58 + 39q−59 + 46q−60 + 5q−61−39q−62−43q−63−10q−64 + 33q−65 + 45q−66 + 9q−67−29q−68−36q−69−13q−70 + 23q−71 + 36q−72 + 7q−73−19q−74−24q−75−9q−76 + 14q−77 + 22q−78 + 5q−79−13q−80−14q−81−3q−82 + 8q−83 + 10q−84 + 3q−85−6q−86−8q−87−q−88 + 5q−89 + 2q−90 + 3q−91−2q−92−4q−93 + 2q−95 + q−97−2q−99 + q−100
6 q30−q29−q28 + q25 + 4q23−q22−3q21−2q20−2q19−q17 + 10q16 + 2q15−q14−3q13−5q12−4q11−8q10 + 13q9 + 5q8 + 4q7 + q6−3q5−6q4−16q3 + 11q2 + 2q + 6 + 3q−1 + 3q−2−2q−3−19q−4 + 12q−5−2q−6 + 4q−7−q−8 + 3q−9−19q−11 + 18q−12 + q−13 + 8q−14−5q−15−2q−16−6q−17−26q−18 + 20q−19 + 8q−20 + 21q−21−3q−23−14q−24−42q−25 + 11q−26 + 10q−27 + 35q−28 + 13q−29 + 6q−30−16q−31−57q−32−5q−33 + 3q−34 + 42q−35 + 25q−36 + 21q−37−8q−38−65q−39−20q−40−10q−41 + 40q−42 + 30q−43 + 35q−44 + 6q−45−65q−46−30q−47−23q−48 + 33q−49 + 29q−50 + 45q−51 + 22q−52−60q−53−35q−54−34q−55 + 23q−56 + 25q−57 + 51q−58 + 38q−59−54q−60−40q−61−42q−62 + 15q−63 + 23q−64 + 55q−65 + 48q−66−49q−67−47q−68−49q−69 + 11q−70 + 25q−71 + 62q−72 + 56q−73−49q−74−56q−75−57q−76 + 7q−77 + 29q−78 + 70q−79 + 64q−80−45q−81−60q−82−64q−83−q−84 + 25q−85 + 70q−86 + 68q−87−33q−88−50q−89−61q−90−8q−91 + 13q−92 + 56q−93 + 59q−94−23q−95−32q−96−45q−97−4q−98 + 3q−99 + 37q−100 + 40q−101−23q−102−18q−103−27q−104 + 7q−105 + 3q−106 + 22q−107 + 22q−108−26q−109−9q−110−13q−111 + 11q−112 + 4q−113 + 13q−114 + 10q−115−22q−116−3q−117−7q−118 + 9q−119 + 2q−120 + 8q−121 + 4q−122−14q−123 + q−124−4q−125 + 5q−126 + 4q−128 + q−129−6q−130 + 2q−131−2q−132 + 2q−133 + q−135−2q−137 + q−138
7 q42−q41−q40 + q37 + q35 + 3q34−q33−3q32−2q31−3q30 + q29 + q27 + 9q26 + 3q25−q24−3q23−8q22−3q21−4q20−4q19 + 11q18 + 8q17 + 6q16 + 5q15−9q14−4q13−8q12−13q11 + 6q10 + 5q9 + 8q8 + 13q7−4q6−4q4−16q3 + 4q2−2q + 2 + 14q−1−5q−2 + q−3−q−4−14q−5 + 10q−6 + 2q−7 + q−8 + 16q−9−9q−10−6q−11−8q−12−23q−13 + 12q−14 + 8q−15 + 12q−16 + 30q−17−q−18−7q−19−17q−20−43q−21−7q−22 + 2q−23 + 17q−24 + 50q−25 + 20q−26 + 10q−27−11q−28−58q−29−32q−30−22q−31 + 3q−32 + 55q−33 + 41q−34 + 36q−35 + 11q−36−52q−37−44q−38−46q−39−25q−40 + 40q−41 + 44q−42 + 54q−43 + 36q−44−30q−45−35q−46−54q−47−49q−48 + 14q−49 + 28q−50 + 54q−51 + 52q−52−5q−53−12q−54−44q−55−57q−56−9q−57 + q−58 + 37q−59 + 55q−60 + 16q−61 + 12q−62−22q−63−52q−64−25q−65−27q−66 + 13q−67 + 49q−68 + 30q−69 + 36q−70 + q−71−42q−72−36q−73−49q−74−11q−75 + 38q−76 + 41q−77 + 57q−78 + 21q−79−33q−80−43q−81−67q−82−31q−83 + 30q−84 + 50q−85 + 73q−86 + 33q−87−28q−88−52q−89−80q−90−40q−91 + 30q−92 + 62q−93 + 85q−94 + 38q−95−32q−96−68q−97−90q−98−41q−99 + 36q−100 + 76q−101 + 97q−102 + 45q−103−37q−104−84q−105−103q−106−49q−107 + 33q−108 + 84q−109 + 108q−110 + 59q−111−25q−112−80q−113−109q−114−66q−115 + 13q−116 + 67q−117 + 103q−118 + 70q−119 + 2q−120−50q−121−92q−122−70q−123−12q−124 + 35q−125 + 73q−126 + 59q−127 + 17q−128−14q−129−54q−130−49q−131−16q−132 + 4q−133 + 36q−134 + 30q−135 + 12q−136 + 6q−137−19q−138−16q−139−7q−140−11q−141 + 9q−142 + 6q−143−2q−144 + 11q−145 + 5q−147 + 4q−148−15q−149−q−150−7q−151−7q−152 + 11q−153 + 3q−154 + 10q−155 + 8q−156−10q−157−4q−158−8q−159−6q−160 + 8q−161 + 2q−162 + 5q−163 + 6q−164−6q−165−q−166−4q−167−4q−168 + 5q−169 + 2q−171 + 2q−172−3q−173−2q−176 + 2q−177 + q−179−2q−181 + q−182

[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.

8_1

8_3

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