10 6

From Knot Atlas

Jump to: navigation, search


10_5

10_7

Contents

Image:10 6.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_6's page at Knotilus!

Visit 10 6's page at the original Knot Atlas!

[edit 10 6 Quick Notes]
[edit 10 6 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,16,6,17 X7,18,8,19 X9,20,10,1 X17,6,18,7 X19,8,20,9 X15,10,16,11
Gauss code -1, 4, -3, 1, -5, 8, -6, 9, -7, 10, -2, 3, -4, 2, -10, 5, -8, 6, -9, 7
Dowker-Thistlethwaite code 4 12 16 18 20 14 2 10 6 8
Conway Notation [532]


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

Length is 11, width is 4,

Braid index is 4

Image:10 6_ML.gif Image:10 6_AP.gif
[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 1}]

[edit Notes on presentations of 10 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["10 6"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,16,6,17 X7,18,8,19 X9,20,10,1 X17,6,18,7 X19,8,20,9 X15,10,16,11
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -5, 8, -6, 9, -7, 10, -2, 3, -4, 2, -10, 5, -8, 6, -9, 7
In[6]:= DTCode[K]
Out[6]= 4 12 16 18 20 14 2 10 6 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [532]
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,−1,−1,−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, 11, 4 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart3.gifImage:BraidPart3.gifImage: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:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage: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:10 6_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 1}]
In[14]:= Draw[ap]
Image:10 6_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-13][1]
Hyperbolic Volume 8.39094
A-Polynomial See Data:10 6/A-polynomial

[edit Notes for 10 6'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 10 6's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −2t3 + 6t2−7t + 7−7t−1 + 6t−2−2t−3
Conway polynomial −2z6−6z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 37, -4 }
Jones polynomial 1−q−1 + 3q−2−4q−3 + 5q−4−6q−5 + 6q−6−5q−7 + 3q−8−2q−9 + q−10
HOMFLY-PT polynomial (db, data sources) z4a8 + 3z2a8 + a8−z6a6−4z4a6−4z2a6−a6−z6a4−4z4a4−4z2a4−2a4 + z4a2 + 4z2a2 + 3a2
Kauffman polynomial (db, data sources) z4a12−2z2a12 + 2z5a11−4z3a11 + za11 + 2z6a10−3z4a10 + z2a10 + 2z7a9−4z5a9 + 4z3a9 + 2z8a8−7z6a8 + 12z4a8−5z2a8 + a8 + z9a7−3z7a7 + 5z5a7−2z3a7 + 3z8a6−12z6a6 + 18z4a6−10z2a6 + a6 + z9a5−4z7a5 + 8z5a5−10z3a5 + 3za5 + z8a4−2z6a4−3z4a4 + 5z2a4−2a4 + z7a3−3z5a3 + 2za3 + z6a2−5z4a2 + 7z2a2−3a2
The A2 invariant q30−q22 + q20−q18−q14−2q12 + q10 + 2q6 + q4 + q2 + 1
The G2 invariant q162−q160 + 2q158−3q156 + q154−3q150 + 5q148−6q146 + 6q144−4q142 + 4q138−7q136 + 9q134−8q132 + 6q130−4q128 + 7q124−9q122 + 13q120−10q118 + 6q116−7q112 + 9q110−9q108 + 5q106 + 5q104−9q102 + 7q100−q98−7q96 + 14q94−17q92 + 11q90−3q88−7q86 + 18q84−20q82 + 18q80−11q78−2q76 + 9q74−15q72 + 15q70−14q68 + 5q66 + 5q64−11q62 + 10q60−7q58−4q56 + 10q54−13q52 + 5q50−9q46 + 18q44−17q42 + 10q40−q38−8q36 + 14q34−13q32 + 11q30−4q28 + 2q26 + 4q24−5q22 + 7q20−4q18 + 4q16 + 2q10−q8 + 2q6 + q2
Further Quantum Invariants
Further quantum knot invariants for 10_6.

A1 Invariants.

Weight Invariant
1 q21−q19 + q17−2q15 + q13−q9 + q7−q5 + 2q3 + q−1
2 q58−q56−q54 + 2q52−q50−q48 + 3q46−3q44−3q42 + 6q40−2q38−3q36 + 5q34 + q32−2q30−q28 + 2q26−q24−4q22 + 3q20 + q18−5q16 + 3q14 + 4q12−4q10 + 4q6−2q4−q2 + 2 + q−6
3 q111−q109−q107 + 2q103 + q101−2q99−2q97 + 2q93 + q91−2q87−2q85 + q83 + 7q81 + 2q79−8q77−8q75 + 9q73 + 10q71−7q69−11q67 + 2q65 + 11q63 + q61−7q59−4q57 + 3q55 + 6q53−q51−8q49 + 9q45 + 2q43−11q41−2q39 + 11q37 + 7q35−11q33−9q31 + 6q29 + 11q27−2q25−12q23−4q21 + 10q19 + 7q17−5q15−8q13 + 2q11 + 9q9 + q7−4q5−2q3 + 3q + q−1−q−3−q−5 + q−7 + q−15
4 q180−q178−q176 + 4q170−q168−2q166−3q164−4q162 + 8q160 + 4q158 + 2q156−5q154−13q152 + 4q150 + 7q148 + 13q146 + q144−21q142−10q140−q138 + 26q136 + 24q134−14q132−27q130−30q128 + 18q126 + 46q124 + 15q122−19q120−57q118−13q116 + 43q114 + 39q112 + 8q110−47q108−31q106 + 13q104 + 29q102 + 26q100−15q98−24q96−9q94 + 7q92 + 19q90 + 7q88−9q86−18q84−5q82 + 15q80 + 21q78−6q76−28q74−10q72 + 20q70 + 36q68−4q66−42q64−22q62 + 17q60 + 48q58 + 11q56−37q54−35q52−7q50 + 42q48 + 30q46−9q44−27q42−31q40 + 11q38 + 25q36 + 20q34 + 5q32−31q30−17q28−q26 + 19q24 + 27q22−6q20−15q18−19q16−q14 + 21q12 + 8q10 + q8−12q6−8q4 + 6q2 + 4 + 6q−2−2q−4−4q−6 + q−8−q−10 + 2q−12−q−16 + q−18−q−20 + q−28
5 q265−q263−q261 + 2q255 + 2q253−q251−4q249−2q247−q245 + 3q243 + 8q241 + 4q239−4q237−9q235−8q233−2q231 + 10q229 + 15q227 + 5q225−9q223−18q221−12q219 + 4q217 + 25q215 + 27q213−q211−31q209−41q207−16q205 + 34q203 + 66q201 + 40q199−33q197−91q195−80q193 + 14q191 + 112q189 + 127q187 + 29q185−116q183−176q181−82q179 + 98q177 + 200q175 + 144q173−48q171−209q169−186q167−4q165 + 176q163 + 198q161 + 56q159−119q157−185q155−88q153 + 65q151 + 140q149 + 95q147−10q145−90q143−88q141−19q139 + 45q137 + 65q135 + 33q133−13q131−49q129−43q127−4q125 + 38q123 + 49q121 + 15q119−42q117−61q115−19q113 + 55q111 + 86q109 + 28q107−68q105−110q103−47q101 + 77q99 + 142q97 + 70q95−77q93−158q91−103q89 + 51q87 + 172q85 + 133q83−20q81−155q79−157q77−29q75 + 121q73 + 163q71 + 73q69−68q67−146q65−101q63 + 11q61 + 98q59 + 108q57 + 39q55−44q53−85q51−63q49−13q47 + 37q45 + 65q43 + 49q41 + 7q39−38q37−56q35−43q33 + 47q29 + 56q27 + 28q25−16q23−46q21−44q19−7q17 + 31q15 + 38q13 + 22q11−8q9−28q7−23q5−4q3 + 13q + 18q−1 + 9q−3−5q−5−9q−7−7q−9−2q−11 + 6q−13 + 6q−15 + q−17−q−19−q−21−3q−23 + 2q−27 + q−33−q−35−q−37 + q−45

A2 Invariants.

Weight Invariant
1,0 q30−q22 + q20−q18−q14−2q12 + q10 + 2q6 + q4 + q2 + 1
1,1 q84−2q82 + 4q80−8q78 + 11q76−14q74 + 18q72−20q70 + 23q68−22q66 + 22q64−28q62 + 26q60−26q58 + 24q56−20q54 + 11q52 + 6q50−20q48 + 38q46−54q44 + 68q42−76q40 + 82q38−79q36 + 70q34−58q32 + 40q30−19q28−4q26 + 24q24−36q22 + 43q20−50q18 + 42q16−38q14 + 28q12−22q10 + 18q8−8q6 + 10q4−2q2 + 4 + q−4
2,0 q76−q70 + q66−2q60−q58−3q52 + 4q48 + 2q46 + q42 + 4q40−q38−3q36 + q34−q30−3q24−q22 + 2q20−q18−3q16 + 3q12−q10−q8 + 2q6 + 3q4 + q2 + 1 + q−2 + q−4

A3 Invariants.

Weight Invariant
0,1,0 q68−q66 + q62−3q60 + 3q56−3q54−q52 + 6q50−2q48−2q46 + 4q44−q42−q40 + q38 + 2q36−2q32 + 3q30−5q26−7q20−q18 + q16−2q14 + 4q12 + 3q10 + 2q8 + 3q6 + 2q4 + 1
1,0,0 q39 + q35−q33 + q31−q29 + q27−q25−q21−2q19−q17−2q15 + q13 + 3q9 + q7 + 2q5 + q3 + q

A4 Invariants.

Weight Invariant
0,1,0,0 q86−q82 + q80 + q78−3q76−2q74 + 2q72−3q68 + 4q64−2q60 + 3q58 + q56−2q54 + 2q52 + 3q50−q48 + q46 + 5q44 + q42−q40 + q38 + 3q36−4q34−6q32−4q30−6q28−7q26−5q24−2q22 + 4q18 + 4q16 + 5q14 + 5q12 + 5q10 + 3q8 + 2q6 + q4 + q2
1,0,0,0 q48 + q44 + q38−q36 + q34−q32−q28−q26−2q24−2q22−q20−2q18 + q16 + 3q12 + 2q10 + 2q8 + 2q6 + q4 + q2

B2 Invariants.

Weight Invariant
0,1 q68−q66 + 2q64−3q62 + 3q60−4q58 + 5q56−5q54 + 5q52−4q50 + 2q48−2q44 + 5q42−7q40 + 9q38−10q36 + 10q34−10q32 + 7q30−6q28 + 3q26−2q24−2q22 + 3q20−5q18 + 5q16−4q14 + 6q12−3q10 + 4q8−q6 + 2q4 + 1
1,0 q110−q106−q104 + q102 + 2q100−q98−3q96−2q94 + 2q92 + 4q90−4q86−3q84 + 3q82 + 6q80−5q76−2q74 + 4q72 + 3q70−3q68−3q66 + 2q64 + 4q62−3q58 + 3q54 + q52−3q50−q48 + 2q46 + 2q44−3q42−5q40 + 5q36 + q34−6q32−6q30 + 2q28 + 5q26−4q22−2q20 + 4q18 + 3q16 + q14−q12 + q10 + 2q8 + 2q6 + q−2

D4 Invariants.

Weight Invariant
1,0,0,0 q94−q92 + q90−2q88 + 2q86−3q84 + 2q82−3q80 + 4q78−4q76 + 3q74−3q72 + 5q70−2q68 + q66 + 4q60−4q58 + 5q56−6q54 + 8q52−7q50 + 8q48−8q46 + 7q44−5q42 + 5q40−5q38 + q36−3q34−3q32−2q30−6q28 + q26−6q24 + 3q22−3q20 + 7q18 + 6q14 + q12 + 5q10 + q8 + 2q6 + q2

G2 Invariants.

Weight Invariant
1,0 q162−q160 + 2q158−3q156 + q154−3q150 + 5q148−6q146 + 6q144−4q142 + 4q138−7q136 + 9q134−8q132 + 6q130−4q128 + 7q124−9q122 + 13q120−10q118 + 6q116−7q112 + 9q110−9q108 + 5q106 + 5q104−9q102 + 7q100−q98−7q96 + 14q94−17q92 + 11q90−3q88−7q86 + 18q84−20q82 + 18q80−11q78−2q76 + 9q74−15q72 + 15q70−14q68 + 5q66 + 5q64−11q62 + 10q60−7q58−4q56 + 10q54−13q52 + 5q50−9q46 + 18q44−17q42 + 10q40−q38−8q36 + 14q34−13q32 + 11q30−4q28 + 2q26 + 4q24−5q22 + 7q20−4q18 + 4q16 + 2q10−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["10 6"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −2t3 + 6t2−7t + 7−7t−1 + 6t−2−2t−3
In[5]:= Conway[K][z]
Out[5]= −2z6−6z4−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]= { 37, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= 1−q−1 + 3q−2−4q−3 + 5q−4−6q−5 + 6q−6−5q−7 + 3q−8−2q−9 + q−10
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z4a8 + 3z2a8 + a8−z6a6−4z4a6−4z2a6−a6−z6a4−4z4a4−4z2a4−2a4 + z4a2 + 4z2a2 + 3a2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z4a12−2z2a12 + 2z5a11−4z3a11 + za11 + 2z6a10−3z4a10 + z2a10 + 2z7a9−4z5a9 + 4z3a9 + 2z8a8−7z6a8 + 12z4a8−5z2a8 + a8 + z9a7−3z7a7 + 5z5a7−2z3a7 + 3z8a6−12z6a6 + 18z4a6−10z2a6 + a6 + z9a5−4z7a5 + 8z5a5−10z3a5 + 3za5 + z8a4−2z6a4−3z4a4 + 5z2a4−2a4 + z7a3−3z5a3 + 2za3 + z6a2−5z4a2 + 7z2a2−3a2

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

Same Alexander/Conway Polynomial: {}

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["10 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]= { −2t3 + 6t2−7t + 7−7t−1 + 6t−2−2t−3, 1−q−1 + 3q−2−4q−3 + 5q−4−6q−5 + 6q−6−5q−7 + 3q−8−2q−9 + q−10 }
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]= {}
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, 4)
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 32 8 -\frac{110}{3} \frac{158}{3} −128 -\frac{1120}{3} -\frac{544}{3} −256 -\frac{32}{3} 512 \frac{440}{3} -\frac{632}{3} \frac{71009}{30} -\frac{1298}{15} \frac{68818}{45} \frac{5983}{18} \frac{929}{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 = -4 is the signature of 10 6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-1012χ
1          11
-1           0
-3        31 2
-5       21  -1
-7      32   1
-9     32    -1
-11    33     0
-13   23      1
-15  13       -2
-17 12        1
-19 1         -1
-211          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −5 i = −3
r = −8 {\mathbb Z}
r = −7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 0 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
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 + 3q−1−4q−2−q−3 + 9q−4−8q−5−5q−6 + 17q−7−9q−8−13q−9 + 23q−10−7q−11−20q−12 + 26q−13−4q−14−23q−15 + 25q−16−q−17−19q−18 + 17q−19−11q−21 + 8q−22−5q−24 + 4q−25−2q−27 + q−28
3 q6−q5 + 2q2−3q + 2q−1 + 4q−2−8q−3−2q−4 + 7q−5 + 12q−6−15q−7−12q−8 + 10q−9 + 24q−10−12q−11−26q−12 + 2q−13 + 34q−14 + q−15−31q−16−13q−17 + 32q−18 + 19q−19−27q−20−26q−21 + 23q−22 + 32q−23−20q−24−35q−25 + 15q−26 + 39q−27−13q−28−38q−29 + 8q−30 + 36q−31−5q−32−28q−33−q−34 + 23q−35−q−36−11q−37−2q−38 + 6q−39−q−40−q−41 + 3q−42−4q−44−q−45 + 5q−46 + q−47−3q−48−3q−49 + 3q−50 + q−51−2q−53 + q−54
4 q12−q11−q8 + 3q7−3q6 + q5 + 2q4−4q3 + 5q2−8q + 3 + 10q−1−6q−2 + 7q−3−22q−4−q−5 + 23q−6 + q−7 + 20q−8−44q−9−19q−10 + 27q−11 + 10q−12 + 53q−13−52q−14−39q−15 + 11q−16−4q−17 + 89q−18−37q−19−34q−20−3q−21−46q−22 + 93q−23−19q−24 + 5q−25 + 9q−26−95q−27 + 65q−28−21q−29 + 53q−30 + 46q−31−126q−32 + 26q−33−41q−34 + 91q−35 + 86q−36−142q−37−4q−38−59q−39 + 113q−40 + 113q−41−148q−42−24q−43−72q−44 + 122q−45 + 129q−46−136q−47−36q−48−88q−49 + 107q−50 + 138q−51−95q−52−33q−53−104q−54 + 63q−55 + 122q−56−40q−57−2q−58−100q−59 + 7q−60 + 78q−61−2q−62 + 32q−63−69q−64−21q−65 + 30q−66 + q−67 + 45q−68−31q−69−19q−70 + 3q−71−8q−72 + 34q−73−9q−74−7q−75−3q−76−11q−77 + 17q−78−q−79−q−81−7q−82 + 5q−83 + q−85−2q−87 + q−88

[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.

10_5

10_7

Retrieved from "http://katlas.math.toronto.edu/wiki/10_6"
Personal tools