8 20

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8_19

8_21

Contents

Image:8 20.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8_20's page at Knotilus!

Visit 8 20's page at the original Knot Atlas!

[edit 8 20 Quick Notes]

8_20 is also known as the pretzel knot P(3,-3,2).

Its complement contains no complete totally geodesic immersed surfaces.[citation needed]

[edit 8 20 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837
Gauss code 1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4
Dowker-Thistlethwaite code 4 8 -12 2 -14 -6 -16 -10
Conway Notation [3,21,2-]


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

Length is 8, width is 3,

Braid index is 3

Image:8 20_ML.gif Image:8 20_AP.gif
[{3, 8}, {2, 4}, {1, 3}, {11, 9}, {8, 10}, {9, 5}, {4, 6}, {5, 7}, {6, 11}, {10, 2}, {7, 1}]

[edit Notes on presentations of 8 20]

Knot 8_20.
Knot 8_20.
A graph, knot 8_20.
A graph, knot 8_20.
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 20"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837
In[5]:= GaussCode[K]
Out[5]= 1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4
In[6]:= DTCode[K]
Out[6]= 4 8 -12 2 -14 -6 -16 -10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3,21,2-]
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,−2,−1,−1,−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:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.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:8 20_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 8}, {2, 4}, {1, 3}, {11, 9}, {8, 10}, {9, 5}, {4, 6}, {5, 7}, {6, 11}, {10, 2}, {7, 1}]
In[14]:= Draw[ap]
Image:8 20_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-2]
Hyperbolic Volume 4.1249
A-Polynomial See Data:8 20/A-polynomial

[edit Notes for 8 20's three dimensional invariants]
8_20 ribbon diagram from A. Kawauchi's text.

Ribbon diagram for 8_20
Ribbon diagram for 8_20

[edit] Four dimensional invariants

Smooth 4 genus 0
Topological 4 genus 0
Concordance genus 0
Rasmussen s-Invariant 0

[edit Notes for 8 20's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t2−2t + 3−2t−1 + t−2
Conway polynomial z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 9, 0 }
Jones polynomial −q + 2−q−1 + 2q−2−q−3 + q−4−q−5
HOMFLY-PT polynomial (db, data sources) −z2a4−2a4 + z4a2 + 4z2a2 + 4a2−z2−1
Kauffman polynomial (db, data sources) a4z6 + a2z6 + a5z5 + 2a3z5 + az5−4a4z4−4a2z4−4a5z3−7a3z3−3az3 + 4a4z2 + 6a2z2 + 2z2 + 3a5z + 5a3z + 3az + za−1−2a4−4a2−1
The A2 invariant −q16−q14−q12 + 2q8 + 2q6 + 2q4 + q2−q−4
The G2 invariant q80 + q76−q74−2q68 + q66−q64−q62−q60−2q58−3q52−q50−q48 + q44−2q42 + q40 + q38 + 2q36 + q34 + 2q30 + 2q28 + 3q26 + 2q22 + 3q20 + q18 + q16 + q14 + 3q10−2q6 + q4 + 1−q−2−2q−4−q−12−q−14−q−20 + q−24
Further Quantum Invariants
Further quantum knot invariants for 8_20.

A1 Invariants.

Weight Invariant
1 −q11 + q5 + q3 + q + q−1−q−3
2 q32−q28−q22−q20 + q12 + q10 + q6 + q4 + q2 + q−2−q−6
3 −q63 + q59 + q57−q53 + q49 + q47−q43−q41−q35−2q33 + q29 + q27−q25 + q21−q17 + q15 + q13 + q5 + 2q3 + 2q + 2q−7−q−9−2q−11−q−13 + q−17
4 q104−q100−q98−q96 + q94 + q92 + q90−2q86−q84 + q80 + 2q78 + q76−q72−q70 + q68 + 2q66 + q64−q62−3q60−2q58 + 2q54 + q52−2q50−2q48 + 2q44 + 2q42−q40−2q38 + q34 + q32−q30−q28−q20−q18 + q16 + q14 + q12 + q10 + q8 + q4 + 3q2 + 4 + 2q−2−q−4−4q−6−2q−8 + 3q−10 + 2q−12−q−14−4q−16−3q−18 + q−20 + 2q−22 + 2q−24−q−28
5 −q155 + q151 + q149 + q147−q143−2q141−q139 + q135 + 2q133 + 2q131−2q127−2q125−2q123−q121 + q119 + 2q117 + 2q115 + q113−q111−3q109−2q107 + 3q103 + 4q101 + 3q99−3q95−4q93−2q91 + 2q89 + 4q87 + 3q85−q83−4q81−5q79−2q77 + 3q75 + 4q73 + 2q71−q69−4q67−2q65 + q63 + 4q61 + 3q59−3q55−3q53 + 2q49 + 2q47−3q43−2q41 + q37−2q33−2q31 + q27 + q25 + q15 + 2q13 + 3q11 + 3q9 + 2q7−q5−2q3 + 3q−1 + 5q−3 + 5q−5−q−7−7q−9−7q−11−2q−13 + 3q−15 + 7q−17 + 3q−19−3q−21−6q−23−4q−25 + q−27 + 3q−29 + 3q−31 + q−33−q−35−q−37
6 q216−q212−q210−q208 + 2q202 + 2q200 + q198−q194−2q192−3q190−q188 + 2q184 + 3q182 + 3q180 + 2q178−q176−2q174−3q172−3q170−2q168 + 3q164 + 4q162 + 3q160 + q158−2q156−5q154−6q152−3q150 + q148 + 4q146 + 6q144 + 6q142 + 2q140−4q138−6q136−6q134−3q132 + 3q130 + 8q128 + 8q126 + 4q124−q122−7q120−9q118−5q116 + q114 + 6q112 + 7q110 + 5q108−2q106−8q104−8q102−4q100 + 2q98 + 7q96 + 9q94 + 4q92−3q90−7q88−6q86−q84 + 4q82 + 8q80 + 5q78−2q76−6q74−5q72−q70 + 3q68 + 6q66 + 3q64−3q62−5q60−4q58−q56 + 2q54 + 2q52−3q48−2q46 + q42 + q40−q36−2q34−q32 + q30 + 2q28 + 2q26 + 2q24 + q22−q18−2q16−q14 + q12 + 4q10 + 6q8 + 6q6 + 2q4−3q2−6−5q−2−2q−4 + 6q−6 + 11q−8 + 9q−10 + q−12−8q−14−12q−16−13q−18−2q−20 + 10q−22 + 12q−24 + 7q−26−q−28−7q−30−11q−32−6q−34 + q−36 + 5q−38 + 5q−40 + 4q−42 + 2q−44−2q−46−q−48−q−50−q−52−q−54 + q−58

A2 Invariants.

Weight Invariant
1,0 −q16−q14−q12 + 2q8 + 2q6 + 2q4 + q2−q−4
1,1 q44 + 2q40−2q38 + 2q36−2q34−2q30−4q28−4q24 + 2q22−3q20 + 4q18 + 4q14 + q12 + 2q10 + 4q8 + 5q4 + 2−q−4−2q−6−2q−8 + q−12
2,0 q42 + q40 + q38−q32−2q30−4q28−4q26−3q24−q22 + 2q20 + 3q18 + 5q16 + 3q14 + 3q12 + q10 + q8 + q4−q−8

A3 Invariants.

Weight Invariant
0,1,0 q34 + q30−q26−2q24−4q22−4q20−3q18 + 2q14 + 6q12 + 6q10 + 6q8 + 3q6 + q4−q2−2−2q−2−q−4−q−6 + q−10
1,0,0 −q21−q19−2q17−q15 + 2q11 + 3q9 + 3q7 + 2q5 + q3−q−1−q−5
1,0,1 q56 + 2q52 + q48−q44 + q42−2q40 + q38−4q36−2q34−6q32−5q30−5q28−6q26−2q22 + 6q20 + 5q18 + 10q16 + 9q14 + 9q12 + 8q10 + 2q8 + 3q6−2q4−3−q−2−q−4−2q−6−q−8−q−10 + q−16

A4 Invariants.

Weight Invariant
0,1,0,0 q44 + q42 + 2q40 + 2q38 + q36−q34−3q32−6q30−9q28−9q26−7q24−3q22 + q20 + 8q18 + 12q16 + 13q14 + 12q12 + 9q10 + 3q8−q6−4q4−5q2−5−3q−2−q−4 + q−10 + q−12
1,0,0,0 −q26−q24−2q22−2q20−q18 + 2q14 + 3q12 + 4q10 + 3q8 + 2q6 + q4−1−q−2−q−6

B2 Invariants.

Weight Invariant
0,1 −q34−q30−q26 + q18 + 2q14 + 2q10 + q6 + q4 + q2 + q−4−q−6−q−10
1,0 q56 + q48−q44−q42−q38−2q36−2q34−2q32−q30−q28 + 2q22 + 2q20 + 3q18 + 2q16 + 3q14 + 3q12 + 2q10 + q6−1−q−2−q−4−q−8−q−10 + q−16

D4 Invariants.

Weight Invariant
1,0,0,0 q46 + q42 + q38−q36−2q34−3q32−4q30−4q28−4q26−2q24−q22 + 3q20 + 4q18 + 7q16 + 6q14 + 7q12 + 4q10 + 3q8 + q6−q4−2q2−2−2q−2−2q−4−q−8 + q−14

G2 Invariants.

Weight Invariant
1,0 q80 + q76−q74−2q68 + q66−q64−q62−q60−2q58−3q52−q50−q48 + q44−2q42 + q40 + q38 + 2q36 + q34 + 2q30 + 2q28 + 3q26 + 2q22 + 3q20 + q18 + q16 + q14 + 3q10−2q6 + q4 + 1−q−2−2q−4−q−12−q−14−q−20 + q−24

.

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 20"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t2−2t + 3−2t−1 + t−2
In[5]:= Conway[K][z]
Out[5]= z4 + 2z2 + 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]= { 9, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q + 2−q−1 + 2q−2−q−3 + q−4−q−5
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z2a4−2a4 + z4a2 + 4z2a2 + 4a2−z2−1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a4z6 + a2z6 + a5z5 + 2a3z5 + az5−4a4z4−4a2z4−4a5z3−7a3z3−3az3 + 4a4z2 + 6a2z2 + 2z2 + 3a5z + 5a3z + 3az + za−1−2a4−4a2−1

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

Same Alexander/Conway Polynomial: {10_140, K11n73, K11n74,}

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 20"];
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]= { t2−2t + 3−2t−1 + t−2, −q + 2−q−1 + 2q−2−q−3 + q−4−q−5 }
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_140, K11n73, K11n74,}
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: (2, -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
8 −16 32 \frac{172}{3} \frac{20}{3} −128 -\frac{640}{3} -\frac{64}{3} −48 \frac{256}{3} 128 \frac{1376}{3} \frac{160}{3} \frac{11911}{15} -\frac{524}{15} \frac{15484}{45} \frac{233}{9} \frac{631}{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 = 0 is the signature of 8 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101χ
3      1-1
1     1 1
-1    12 1
-3   1   1
-5   1   1
-7 11    0
-9       0
-111      -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −5 {\mathbb Z}
r = −4 {\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}
r = −2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −1 {\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z}^{2} {\mathbb Z}
r = 1 {\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 + 1−2q−1 + 2q−2 + q−3−2q−4 + q−5 + 2q−6−2q−7 + 2q−9−2q−10−q−11 + 2q−12−q−13−q−14 + q−15
3 q7−q6−q5−q4 + 2q3 + 2q2−3q−1 + 2q−1 + 4q−2−3q−3−2q−4 + q−5 + 4q−6−3q−7−q−8 + q−9 + 2q−10−2q−11 + q−14−q−17−q−18 + q−19 + q−20−q−21−2q−22 + q−23 + 2q−24−2q−26 + q−28 + q−29−q−30
4 −q12 + q11 + 2q10−q8−5q7 + 5q5 + 3q4−10q2−2q + 8 + 6q−1 + 2q−2−11q−3−4q−4 + 7q−5 + 7q−6 + 2q−7−11q−8−4q−9 + 7q−10 + 5q−11 + 2q−12−10q−13−4q−14 + 7q−15 + 4q−16 + 2q−17−8q−18−4q−19 + 6q−20 + 2q−21 + 3q−22−5q−23−4q−24 + 4q−25 + 3q−27−2q−28−3q−29 + 2q−30−2q−31 + 2q−32−q−34 + 3q−35−3q−36 + 4q−40−2q−41−q−42−q−43−q−44 + 3q−45−q−48−q−49 + q−50
5 −q16 + 2q14 + 2q13−2q11−6q10−2q9 + 5q8 + 8q7 + 4q6−6q5−11q4−7q3 + 5q2 + 14q + 10−6q−1−13q−2−10q−3 + 3q−4 + 15q−5 + 13q−6−5q−7−13q−8−11q−9 + 2q−10 + 14q−11 + 13q−12−5q−13−13q−14−10q−15 + 2q−16 + 13q−17 + 11q−18−5q−19−11q−20−9q−21 + q−22 + 11q−23 + 10q−24−2q−25−9q−26−9q−27−q−28 + 8q−29 + 10q−30 + q−31−6q−32−8q−33−4q−34 + 5q−35 + 8q−36 + 4q−37−3q−38−6q−39−5q−40 + 5q−42 + 5q−43−2q−45−4q−46−2q−47 + q−48 + 3q−49 + q−50 + q−51−q−52−q−53−q−56 + q−58 + q−59 + q−60−2q−62−2q−63 + q−65 + q−66 + 2q−67−2q−69−q−70 + q−73 + q−74−q−75
6 q26−q25−q24−q20 + 5q19 + q18−4q15−7q14−6q13 + 9q12 + 7q11 + 8q10 + 5q9−6q8−19q7−17q6 + 10q5 + 11q4 + 17q3 + 13q2−4q−24−25q−1 + 7q−2 + 10q−3 + 20q−4 + 18q−5−24q−7−27q−8 + 4q−9 + 8q−10 + 19q−11 + 19q−12 + q−13−23q−14−26q−15 + 4q−16 + 8q−17 + 18q−18 + 17q−19−22q−21−25q−22 + 5q−23 + 8q−24 + 17q−25 + 15q−26−q−27−19q−28−23q−29 + 5q−30 + 5q−31 + 14q−32 + 14q−33 + q−34−13q−35−20q−36 + 2q−37 + q−38 + 10q−39 + 13q−40 + 5q−41−6q−42−17q−43−2q−44−4q−45 + 5q−46 + 12q−47 + 9q−48 + q−49−12q−50−4q−51−9q−52−q−53 + 8q−54 + 9q−55 + 7q−56−5q−57−2q−58−10q−59−6q−60 + 2q−61 + 5q−62 + 9q−63 + q−64 + 3q−65−6q−66−6q−67−3q−68−q−69 + 6q−70 + q−71 + 5q−72−2q−74−3q−75−3q−76 + 3q−77−3q−78 + 2q−79 + q−80 + q−81−q−83 + 4q−84−4q−85−q−86−q−87 + 5q−91−q−92−q−94−q−95−2q−96−q−97 + 3q−98 + q−100−q−103−q−104 + q−105
7 −q35 + q34 + q33 + q32−2q30−q29−2q28−3q27 + 4q25 + 6q24 + 7q23−q21−7q20−14q19−10q18 + 11q16 + 21q15 + 16q14 + 6q13−9q12−28q11−26q10−16q9 + 6q8 + 34q7 + 35q6 + 21q5−3q4−33q3−38q2−30q−3 + 34q−1 + 44q−2 + 31q−3 + 6q−4−31q−5−39q−6−35q−7−10q−8 + 30q−9 + 42q−10 + 32q−11 + 10q−12−28q−13−38q−14−34q−15−11q−16 + 29q−17 + 40q−18 + 31q−19 + 9q−20−28q−21−38q−22−33q−23−9q−24 + 30q−25 + 40q−26 + 30q−27 + 6q−28−28q−29−37q−30−31q−31−7q−32 + 28q−33 + 38q−34 + 27q−35 + 5q−36−24q−37−34q−38−28q−39−7q−40 + 23q−41 + 33q−42 + 24q−43 + 6q−44−17q−45−28q−46−25q−47−9q−48 + 14q−49 + 26q−50 + 22q−51 + 10q−52−8q−53−20q−54−21q−55−13q−56 + 3q−57 + 16q−58 + 19q−59 + 13q−60 + 3q−61−10q−62−15q−63−14q−64−8q−65 + 4q−66 + 11q−67 + 14q−68 + 11q−69−5q−71−10q−72−13q−73−7q−74 + 7q−76 + 13q−77 + 6q−78 + 6q−79−10q−81−9q−82−8q−83−3q−84 + 5q−85 + 5q−86 + 10q−87 + 9q−88−2q−89−3q−90−6q−91−9q−92−3q−93−3q−94 + 5q−95 + 9q−96 + 2q−97 + 4q−98 + q−99−5q−100−3q−101−6q−102−2q−103 + 4q−104−q−105 + 3q−106 + 3q−107 + 2q−109−2q−110−2q−111 + 2q−112−3q−113−q−114−2q−116 + 3q−117 + q−118 + 3q−120−q−123−4q−124−q−127 + 2q−128 + q−129 + 2q−130 + q−131−2q−132−q−133−q−135 + q−138 + q−139−q−140

[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_19

8_21

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