10 113

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10_112

10_114

Contents

Image:10 113.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 113's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_113's page at Knotilus!

Visit 10 113's page at the original Knot Atlas!

[edit 10 113 Quick Notes]
[edit 10 113 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X12,7,13,8 X8,18,9,17 X6,19,7,20 X16,12,17,11 X18,13,19,14 X2,10,3,9
Gauss code 1, -10, 2, -1, 3, -7, 5, -6, 10, -2, 8, -5, 9, -3, 4, -8, 6, -9, 7, -4
Dowker-Thistlethwaite code 4 10 14 12 2 16 18 20 8 6
Conway Notation [8*21]


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

Length is 11, width is 4,

Braid index is 4

Image:10 113_ML.gif Image:10 113_AP.gif
[{3, 11}, {2, 5}, {1, 3}, {12, 7}, {10, 6}, {11, 8}, {7, 4}, {5, 9}, {8, 2}, {4, 10}, {9, 12}, {6, 1}]

[edit Notes on presentations of 10 113]


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["10 113"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X12,7,13,8 X8,18,9,17 X6,19,7,20 X16,12,17,11 X18,13,19,14 X2,10,3,9
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -1, 3, -7, 5, -6, 10, -2, 8, -5, 9, -3, 4, -8, 6, -9, 7, -4
In[6]:= DTCode[K]
Out[6]= 4 10 14 12 2 16 18 20 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [8*21]
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,{1,1,1,2,−3,2,−1,2,−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:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage: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:10 113_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 11}, {2, 5}, {1, 3}, {12, 7}, {10, 6}, {11, 8}, {7, 4}, {5, 9}, {8, 2}, {4, 10}, {9, 12}, {6, 1}]
In[14]:= Draw[ap]
Image:10 113_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 16.4735
A-Polynomial See Data:10 113/A-polynomial

[edit Notes for 10 113's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for 10 113's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 2t3−11t2 + 26t−33 + 26t−1−11t−2 + 2t−3
Conway polynomial 2z6 + z4 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 111, 2 }
Jones polynomial −q8 + 5q7−10q6 + 14q5−18q4 + 19q3−17q2 + 14q−8 + 4q−1−q−2
HOMFLY-PT polynomial (db, data sources) z6a−2 + z6a−4 + 2z4a−2 + z4a−4−z4a−6−z4 + 3z2a−2−2z2a−4−z2 + 3a−2−3a−4 + a−6
Kauffman polynomial (db, data sources) 3z9a−3 + 3z9a−5 + 7z8a−2 + 16z8a−4 + 9z8a−6 + 7z7a−1 + 12z7a−3 + 15z7a−5 + 10z7a−7−5z6a−2−23z6a−4−9z6a−6 + 5z6a−8 + 4z6 + az5−10z5a−1−30z5a−3−36z5a−5−16z5a−7 + z5a−9−6z4a−2 + z4a−4−4z4a−6−5z4a−8−6z4−az3 + 5z3a−1 + 17z3a−3 + 16z3a−5 + 5z3a−7 + 8z2a−2 + 8z2a−4 + 3z2a−6 + 3z2−za−1−za−3 + za−5 + za−7−3a−2−3a−4−a−6
The A2 invariant −q6 + 2q4−q2−1 + 5q−2−2q−4 + 4q−6−2q−10 + q−12−5q−14 + 3q−16−q−18−q−20 + 3q−22−q−24
The G2 invariant q32−3q30 + 7q28−13q26 + 15q24−15q22 + 4q20 + 20q18−50q16 + 86q14−108q12 + 100q10−52q8−47q6 + 182q4−301q2 + 359−303q−2 + 110q−4 + 168q−6−443q−8 + 605q−10−562q−12 + 315q−14 + 63q−16−415q−18 + 597q−20−516q−22 + 220q−24 + 165q−26−445q−28 + 484q−30−265q−32−118q−34 + 506q−36−702q−38 + 623q−40−277q−42−216q−44 + 661q−46−905q−48 + 841q−50−511q−52 + 19q−54 + 454q−56−748q−58 + 762q−60−503q−62 + 81q−64 + 313q−66−530q−68 + 465q−70−168q−72−207q−74 + 498q−76−548q−78 + 346q−80 + 29q−82−415q−84 + 644q−86−631q−88 + 398q−90−50q−92−270q−94 + 454q−96−460q−98 + 334q−100−139q−102−42q−104 + 147q−106−183q−108 + 147q−110−84q−112 + 31q−114 + 10q−116−25q−118 + 26q−120−20q−122 + 10q−124−4q−126 + q−128
Further Quantum Invariants
Further quantum knot invariants for 10_113.

A1 Invariants.

Weight Invariant
1 −q5 + 3q3−4q + 6q−1−3q−3 + 2q−5 + q−7−4q−9 + 4q−11−5q−13 + 4q−15−q−17
2 q16−3q14 + 10q10−15q8−6q6 + 39q4−26q2−34 + 67q−2−10q−4−61q−6 + 51q−8 + 19q−10−50q−12 + 6q−14 + 35q−16−10q−18−40q−20 + 32q−22 + 34q−24−65q−26 + 10q−28 + 61q−30−52q−32−17q−34 + 50q−36−17q−38−20q−40 + 16q−42 + q−44−4q−46 + q−48
3 −q33 + 3q31−6q27−q25 + 15q23 + 5q21−43q19−14q17 + 84q15 + 58q13−140q11−150q9 + 181q7 + 296q5−167q3−458q + 67q−1 + 608q−3 + 93q−5−663q−7−290q−9 + 618q−11 + 460q−13−492q−15−556q−17 + 304q−19 + 579q−21−105q−23−535q−25−86q−27 + 459q−29 + 252q−31−346q−33−413q−35 + 228q−37 + 541q−39−78q−41−643q−43−92q−45 + 678q−47 + 275q−49−626q−51−442q−53 + 498q−55 + 535q−57−304q−59−540q−61 + 107q−63 + 454q−65 + 41q−67−318q−69−104q−71 + 171q−73 + 108q−75−69q−77−75q−79 + 22q−81 + 31q−83−12q−87−q−89 + 4q−91−q−93

A2 Invariants.

Weight Invariant
1,0 −q6 + 2q4−q2−1 + 5q−2−2q−4 + 4q−6−2q−10 + q−12−5q−14 + 3q−16−q−18−q−20 + 3q−22−q−24
1,1 q20−6q18 + 20q16−50q14 + 109q12−218q10 + 392q8−662q6 + 1048q4−1548q2 + 2134−2722q−2 + 3230q−4−3482q−6 + 3384q−8−2820q−10 + 1769q−12−308q−14−1436q−16 + 3228q−18−4886q−20 + 6174q−22−6924q−24 + 7056q−26−6543q−28 + 5466q−30−3938q−32 + 2154q−34−340q−36−1288q−38 + 2552q−40−3344q−42 + 3642q−44−3508q−46 + 3056q−48−2430q−50 + 1773q−52−1192q−54 + 730q−56−400q−58 + 198q−60−88q−62 + 32q−64−8q−66 + q−68
2,0 q18−2q16−2q14 + 6q12 + 2q10−12q8−6q6 + 19q4 + 10q2−24−7q−2 + 35q−4 + 7q−6−29q−8 + 4q−10 + 24q−12−7q−14−22q−16 + 7q−18 + 6q−20−20q−22 + 8q−24 + 13q−26−15q−28−4q−30 + 28q−32 + 2q−34−30q−36 + 4q−38 + 27q−40−3q−42−29q−44 + 7q−46 + 19q−48−6q−50−11q−52 + 9q−56−q−58−3q−60 + q−62

A3 Invariants.

Weight Invariant
0,1,0 q14−3q12 + q10 + 7q8−14q6 + 3q4 + 24q2−31 + 2q−2 + 44q−4−44q−6 + q−8 + 48q−10−34q−12−7q−14 + 27q−16−10q−18−17q−20−5q−22 + 20q−24−6q−26−31q−28 + 41q−30 + 9q−32−48q−34 + 39q−36 + 9q−38−42q−40 + 26q−42 + 5q−44−19q−46 + 11q−48 + 2q−50−4q−52 + q−54
1,0,0 −q7 + 2q5−2q3 + 2q−2q−1 + 5q−3−2q−5 + 5q−7 + q−9 + q−11−q−13−3q−15−5q−19 + 3q−21−2q−23 + 3q−25−2q−27 + 3q−29−q−31

A4 Invariants.

Weight Invariant
0,1,0,0 q16−2q14−q12 + 5q10−2q8−9q6 + 5q4 + 14q2−10−16q−2 + 22q−4 + 22q−6−31q−8−8q−10 + 47q−12 + 2q−14−40q−16 + 18q−18 + 33q−20−32q−22−27q−24 + 30q−26−4q−28−45q−30 + 20q−32 + 34q−34−35q−36−8q−38 + 51q−40−2q−42−40q−44 + 13q−46 + 30q−48−22q−50−24q−52 + 21q−54 + 12q−56−17q−58−2q−60 + 11q−62−2q−64−3q−66 + q−68
1,0,0,0 −q8 + 2q6−2q4 + q2 + 1−2q−2 + 5q−4−2q−6 + 5q−8 + 2q−10 + 2q−12 + q−14−q−16−2q−18−4q−20−5q−24 + 3q−26−2q−28 + 2q−30 + 2q−32−2q−34 + 3q−36−q−38

B2 Invariants.

Weight Invariant
0,1 −q14 + 3q12−7q10 + 13q8−22q6 + 33q4−44q2 + 55−58q−2 + 58q−4−46q−6 + 29q−8−2q−10−26q−12 + 57q−14−83q−16 + 104q−18−115q−20 + 113q−22−102q−24 + 78q−26−51q−28 + 19q−30 + 9q−32−34q−34 + 51q−36−59q−38 + 60q−40−54q−42 + 45q−44−31q−46 + 19q−48−10q−50 + 4q−52−q−54
1,0 q24−3q20−3q18 + 4q16 + 10q14−18q10−14q8 + 18q6 + 34q4−q2−47−25q−2 + 42q−4 + 54q−6−17q−8−64q−10−11q−12 + 60q−14 + 35q−16−39q−18−44q−20 + 22q−22 + 45q−24−8q−26−45q−28−4q−30 + 39q−32 + 9q−34−39q−36−22q−38 + 35q−40 + 32q−42−30q−44−45q−46 + 21q−48 + 60q−50 + 5q−52−61q−54−32q−56 + 48q−58 + 54q−60−21q−62−57q−64−10q−66 + 40q−68 + 26q−70−19q−72−25q−74 + q−76 + 15q−78 + 6q−80−4q−82−4q−84 + q−88

D4 Invariants.

Weight Invariant
1,0,0,0 q18−3q16 + 4q14−6q12 + 11q10−18q8 + 21q6−25q4 + 37q2−43 + 43q−2−43q−4 + 50q−6−40q−8 + 29q−10−17q−12 + 12q−14 + 18q−16−32q−18 + 43q−20−59q−22 + 77q−24−88q−26 + 79q−28−93q−30 + 85q−32−74q−34 + 60q−36−52q−38 + 35q−40−6q−42−q−44 + 14q−46−30q−48 + 44q−50−45q−52 + 45q−54−50q−56 + 44q−58−35q−60 + 30q−62−25q−64 + 16q−66−8q−68 + 6q−70−4q−72 + q−74

G2 Invariants.

Weight Invariant
1,0 q32−3q30 + 7q28−13q26 + 15q24−15q22 + 4q20 + 20q18−50q16 + 86q14−108q12 + 100q10−52q8−47q6 + 182q4−301q2 + 359−303q−2 + 110q−4 + 168q−6−443q−8 + 605q−10−562q−12 + 315q−14 + 63q−16−415q−18 + 597q−20−516q−22 + 220q−24 + 165q−26−445q−28 + 484q−30−265q−32−118q−34 + 506q−36−702q−38 + 623q−40−277q−42−216q−44 + 661q−46−905q−48 + 841q−50−511q−52 + 19q−54 + 454q−56−748q−58 + 762q−60−503q−62 + 81q−64 + 313q−66−530q−68 + 465q−70−168q−72−207q−74 + 498q−76−548q−78 + 346q−80 + 29q−82−415q−84 + 644q−86−631q−88 + 398q−90−50q−92−270q−94 + 454q−96−460q−98 + 334q−100−139q−102−42q−104 + 147q−106−183q−108 + 147q−110−84q−112 + 31q−114 + 10q−116−25q−118 + 26q−120−20q−122 + 10q−124−4q−126 + q−128

.

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["10 113"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 2t3−11t2 + 26t−33 + 26t−1−11t−2 + 2t−3
In[5]:= Conway[K][z]
Out[5]= 2z6 + z4 + 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]= { 111, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 5q7−10q6 + 14q5−18q4 + 19q3−17q2 + 14q−8 + 4q−1−q−2
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−2 + z6a−4 + 2z4a−2 + z4a−4−z4a−6−z4 + 3z2a−2−2z2a−4−z2 + 3a−2−3a−4 + a−6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3z9a−3 + 3z9a−5 + 7z8a−2 + 16z8a−4 + 9z8a−6 + 7z7a−1 + 12z7a−3 + 15z7a−5 + 10z7a−7−5z6a−2−23z6a−4−9z6a−6 + 5z6a−8 + 4z6 + az5−10z5a−1−30z5a−3−36z5a−5−16z5a−7 + z5a−9−6z4a−2 + z4a−4−4z4a−6−5z4a−8−6z4−az3 + 5z3a−1 + 17z3a−3 + 16z3a−5 + 5z3a−7 + 8z2a−2 + 8z2a−4 + 3z2a−6 + 3z2−za−1−za−3 + za−5 + za−7−3a−2−3a−4−a−6

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

Same Alexander/Conway Polynomial: {K11a107, K11a347,}

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["10 113"];
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]= { 2t3−11t2 + 26t−33 + 26t−1−11t−2 + 2t−3, −q8 + 5q7−10q6 + 14q5−18q4 + 19q3−17q2 + 14q−8 + 4q−1−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]= {K11a107, K11a347,}
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 −32 −8 0 -\frac{176}{3} -\frac{32}{3} −8 0 32 0 0 0 -\frac{328}{3} \frac{488}{3} -\frac{112}{3} 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 = 2 is the signature of 10 113. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-101234567χ
17          1-1
15         4 4
13        61 -5
11       84  4
9      106   -4
7     98    1
5    810     2
3   69      -3
1  39       6
-1 15        -4
-3 3         3
-51          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −3 {\mathbb Z}
r = −2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r = 1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 7 {\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 q23−5q22 + 5q21 + 16q20−41q19 + 8q18 + 83q17−108q16−27q15 + 196q14−159q13−102q12 + 295q11−161q10−174q9 + 325q8−116q7−203q6 + 269q5−47q4−171q3 + 157q2 + 4q−94 + 56q−1 + 12q−2−29q−3 + 11q−4 + 3q−5−4q−6 + q−7
3 −q45 + 5q44−5q43−11q42 + 11q41 + 36q40−14q39−108q38 + 17q37 + 213q36 + 49q35−383q34−197q33 + 572q32 + 462q31−730q30−844q29 + 808q28 + 1301q27−767q26−1784q25 + 624q24 + 2202q23−364q22−2554q21 + 73q20 + 2767q19 + 255q18−2867q17−568q16 + 2834q15 + 853q14−2660q13−1113q12 + 2385q11 + 1283q10−1976q9−1388q8 + 1525q7 + 1347q6−1024q5−1230q4 + 617q3 + 974q2−268q−715 + 76q−1 + 449q−2 + 23q−3−252q−4−39q−5 + 118q−6 + 33q−7−54q−8−13q−9 + 20q−10 + 4q−11−6q−12−3q−13 + 4q−14−q−15
4 q74−5q73 + 5q72 + 11q71−16q70−6q69−30q68 + 54q67 + 103q66−82q65−112q64−251q63 + 212q62 + 645q61 + 48q60−469q59−1402q58−26q57 + 2075q56 + 1563q55−206q54−4248q53−2542q52 + 3229q51 + 5529q50 + 3239q49−7254q48−8631q47 + 897q46 + 10069q45 + 11259q44−6767q43−15986q42−6381q41 + 11329q40 + 21118q39−1369q38−20470q37−15829q36 + 7975q35 + 28515q34 + 6390q33−20505q32−23517q31 + 2084q30 + 31572q29 + 13360q28−17291q27−27794q26−4111q25 + 30657q24 + 18367q23−11987q22−28602q21−9929q20 + 26018q19 + 21092q18−4872q17−25517q16−14666q15 + 17731q14 + 20332q13 + 2697q12−18207q11−16185q10 + 7772q9 + 15153q8 + 7485q7−8838q6−12951q5 + 302q4 + 7617q3 + 7305q2−1746q−6996−2153q−1 + 1895q−2 + 4049q−3 + 829q−4−2326q−5−1348q−6−201q−7 + 1302q−8 + 662q−9−441q−10−330q−11−268q−12 + 248q−13 + 179q−14−69q−15−17q−16−70q−17 + 37q−18 + 26q−19−19q−20 + 5q−21−9q−22 + 6q−23 + 3q−24−4q−25 + q−26
5 −q110 + 5q109−5q108−11q107 + 16q106 + 11q105−10q103−49q102−53q101 + 77q100 + 193q99 + 105q98−147q97−470q96−463q95 + 146q94 + 1142q93 + 1425q92 + 120q91−2076q90−3380q89−1743q88 + 2924q87 + 7090q86 + 5594q85−2529q84−11815q83−13351q82−1683q81 + 16722q80 + 25488q79 + 11986q78−18233q77−40936q76−31019q75 + 12436q74 + 56673q73 + 58646q72 + 4708q71−67378q70−92461q69−35372q68 + 67642q67 + 127438q66 + 78290q65−53411q64−156999q63−129444q62 + 23740q61 + 175703q60 + 182272q59 + 18982q58−180144q57−230663q56−69435q55 + 170524q54 + 269064q53 + 122008q52−149290q51−295846q50−170921q49 + 120958q48 + 310323q47 + 213088q46−89127q45−315143q44−247013q43 + 57310q42 + 312043q41 + 273152q40−26342q39−303206q38−292904q37−3679q36 + 289392q35 + 307134q34 + 33780q33−269756q32−316379q31−65076q30 + 243562q29 + 319348q28 + 97142q27−208866q26−314188q25−128881q24 + 166215q23 + 298125q22 + 156560q21−116345q20−269958q19−176317q18 + 64083q17 + 229332q16 + 183631q15−13836q14−179900q13−176795q12−26835q11 + 126211q10 + 155697q9 + 54716q8−75683q7−124947q6−66442q5 + 33970q4 + 89488q3 + 64870q2−5068q−56439−53407q−1−10729q−2 + 29891q−3 + 38155q−4 + 15804q−5−12198q−6−23416q−7−14367q−8 + 2521q−9 + 12382q−10 + 10079q−11 + 1302q−12−5328q−13−5903q−14−2040q−15 + 1877q−16 + 2885q−17 + 1431q−18−405q−19−1165q−20−799q−21−6q−22 + 435q−23 + 328q−24 + 26q−25−111q−26−98q−27−46q−28 + 40q−29 + 47q−30−15q−31−9q−32 + 6q−33−6q−34 + 9q−36−6q−37−3q−38 + 4q−39−q−40

[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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10_112

10_114

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