9 15

From Knot Atlas

Jump to: navigation, search


9_14

9_16

Contents

Image:9 15.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9_15's page at Knotilus!

Visit 9 15's page at the original Knot Atlas!

[edit 9 15 Quick Notes]
[edit 9 15 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12
Gauss code -1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code 4 8 14 10 2 18 16 6 12
Conway Notation [2322]


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

Length is 10, width is 5,

Braid index is 5

Image:9 15_ML.gif Image:9 15_AP.gif
[{11, 4}, {5, 2}, {4, 10}, {1, 5}, {6, 11}, {3, 7}, {2, 6}, {8, 3}, {7, 9}, {10, 8}, {9, 1}]

[edit Notes on presentations of 9 15]


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["9 15"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
In[6]:= DTCode[K]
Out[6]= 4 8 14 10 2 18 16 6 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [2322]
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(5,{1,1,1,2,−1,−3,2,4,−3,4})
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]= { 5, 10, 5 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image: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: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:9 15_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{11, 4}, {5, 2}, {4, 10}, {1, 5}, {6, 11}, {3, 7}, {2, 6}, {8, 3}, {7, 9}, {10, 8}, {9, 1}]
In[14]:= Draw[ap]
Image:9 15_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index {4,5}
Nakanishi index 1
Maximal Thurston-Bennequin number [-1][-10]
Hyperbolic Volume 9.8855
A-Polynomial See Data:9 15/A-polynomial

[edit Notes for 9 15's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for 9 15's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −2t2 + 10t−15 + 10t−1−2t−2
Conway polynomial −2z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 39, 2 }
Jones polynomial −q8 + 2q7−4q6 + 6q5−6q4 + 7q3−6q2 + 4q−2 + q−1
HOMFLY-PT polynomial (db, data sources) −z4a−2−z4a−4−z2a−2 + 2z2a−6 + z2−a−2 + a−4 + a−6−a−8 + 1
Kauffman polynomial (db, data sources) z8a−4 + z8a−6 + 2z7a−3 + 4z7a−5 + 2z7a−7 + 2z6a−2 + z6a−4 + z6a−6 + 2z6a−8 + 2z5a−1−z5a−3−7z5a−5−3z5a−7 + z5a−9−4z4a−6−5z4a−8 + z4−3z3a−1 + 5z3a−5−z3a−7−3z3a−9−3z2a−2−2z2a−4 + 2z2a−6 + 3z2a−8−2z2 + za−1 + za−3−za−5 + za−7 + 2za−9 + a−2 + a−4−a−6−a−8 + 1
The A2 invariant q4 + 2q−2−2q−4 + 2q−12 + 2q−16−q−20 + q−22−q−24−q−26
The G2 invariant q18−q16 + 3q14−3q12 + 2q10−3q6 + 8q4−9q2 + 11−9q−2 + 5q−4 + 4q−6−13q−8 + 23q−10−26q−12 + 21q−14−13q−16−5q−18 + 20q−20−31q−22 + 33q−24−20q−26 + 2q−28 + 14q−30−23q−32 + 15q−34−2q−36−13q−38 + 24q−40−23q−42 + 9q−44 + 17q−46−34q−48 + 46q−50−42q−52 + 21q−54 + 6q−56−28q−58 + 43q−60−44q−62 + 36q−64−11q−66−11q−68 + 26q−70−30q−72 + 20q−74−q−76−15q−78 + 21q−80−15q−82 + 3q−84 + 18q−86−31q−88 + 33q−90−23q−92 + 18q−96−32q−98 + 33q−100−24q−102 + 10q−104 + 3q−106−15q−108 + 17q−110−15q−112 + 9q−114−3q−116−2q−118 + 3q−120−4q−122 + 3q−124−q−126 + q−128
Further Quantum Invariants
Further quantum knot invariants for 9_15.

A1 Invariants.

Weight Invariant
1 q3−q + 2q−1−2q−3 + q−5 + q−7 + 2q−11−2q−13 + q−15−q−17
2 q10−q8−q6 + 3q4−3q2−2 + 9q−2−4q−4−6q−6 + 11q−8−q−10−7q−12 + 5q−14 + 2q−16−3q−18−3q−20 + 5q−22 + 2q−24−8q−26 + 5q−28 + 6q−30−10q−32 + q−34 + 7q−36−6q−38−2q−40 + 4q−42−q−44−q−46 + q−48
3 q21−q19−q17 + 2q13−q11−2q9 + 4q7 + 4q5−7q3−8q + 11q−1 + 16q−3−14q−5−25q−7 + 13q−9 + 33q−11−5q−13−38q−15 + 38q−19 + 8q−21−28q−23−14q−25 + 21q−27 + 16q−29−9q−31−19q−33−2q−35 + 17q−37 + 11q−39−19q−41−21q−43 + 19q−45 + 27q−47−14q−49−33q−51 + 10q−53 + 36q−55−37q−59−7q−61 + 28q−63 + 15q−65−21q−67−18q−69 + 12q−71 + 16q−73−3q−75−12q−77−q−79 + 7q−81 + 2q−83−3q−85−q−87 + q−89 + q−91−q−93
4 q36−q34−q32−q28 + 4q26−q24 + 2q20−5q18 + 3q16−7q14 + 2q12 + 15q10−q8−2q6−32q4−9q2 + 41 + 34q−2 + 13q−4−78q−6−62q−8 + 46q−10 + 92q−12 + 74q−14−95q−16−135q−18−7q−20 + 113q−22 + 151q−24−49q−26−159q−28−75q−30 + 68q−32 + 162q−34 + 20q−36−103q−38−97q−40−2q−42 + 107q−44 + 62q−46−25q−48−78q−50−50q−52 + 33q−54 + 76q−56 + 40q−58−56q−60−85q−62−25q−64 + 91q−66 + 95q−68−35q−70−115q−72−81q−74 + 96q−76 + 143q−78 + 5q−80−118q−82−136q−84 + 55q−86 + 152q−88 + 67q−90−67q−92−161q−94−20q−96 + 101q−98 + 99q−100 + 15q−102−113q−104−64q−106 + 17q−108 + 70q−110 + 61q−112−37q−114−47q−116−28q−118 + 16q−120 + 44q−122 + 4q−124−9q−126−21q−128−7q−130 + 14q−132 + 4q−134 + 3q−136−5q−138−4q−140 + 3q−142 + q−146−q−148−q−150 + q−152
5 q55−q53−q51−q47 + q45 + 4q43 + q41−2q39−5q35−5q33 + 3q31 + 6q29 + 7q27 + 6q25−5q23−20q21−19q19 + 2q17 + 30q15 + 45q13 + 21q11−37q9−88q7−68q5 + 34q3 + 137q + 146q−1 + 8q−3−182q−5−260q−7−97q−9 + 206q−11 + 382q−13 + 235q−15−169q−17−487q−19−410q−21 + 67q−23 + 553q−25 + 581q−27 + 84q−29−529q−31−704q−33−266q−35 + 436q−37 + 765q−39 + 412q−41−287q−43−721q−45−510q−47 + 116q−49 + 604q−51 + 547q−53 + 28q−55−451q−57−501q−59−150q−61 + 277q−63 + 434q−65 + 222q−67−136q−69−336q−71−267q−73 + 5q−75 + 263q−77 + 298q−79 + 91q−81−198q−83−334q−85−181q−87 + 156q−89 + 385q−91 + 269q−93−125q−95−443q−97−366q−99 + 79q−101 + 496q−103 + 476q−105−16q−107−530q−109−576q−111−88q−113 + 522q−115 + 667q−117 + 208q−119−441q−121−711q−123−352q−125 + 317q−127 + 695q−129 + 462q−131−137q−133−595q−135−548q−137−43q−139 + 449q−141 + 534q−143 + 195q−145−252q−147−461q−149−292q−151 + 79q−153 + 328q−155 + 302q−157 + 61q−159−178q−161−252q−163−137q−165 + 57q−167 + 168q−169 + 141q−171 + 26q−173−80q−175−110q−177−59q−179 + 21q−181 + 63q−183 + 54q−185 + 9q−187−25q−189−34q−191−19q−193 + 7q−195 + 18q−197 + 11q−199−4q−203−7q−205−3q−207 + 4q−209 + 2q−211−q−213−q−219 + q−221 + q−223−q−225

A2 Invariants.

Weight Invariant
1,0 q4 + 2q−2−2q−4 + 2q−12 + 2q−16−q−20 + q−22−q−24−q−26
1,1 q12−2q10 + 6q8−10q6 + 17q4−26q2 + 36−52q−2 + 68q−4−82q−6 + 98q−8−102q−10 + 106q−12−82q−14 + 52q−16−8q−18−47q−20 + 102q−22−156q−24 + 194q−26−217q−28 + 220q−30−204q−32 + 174q−34−126q−36 + 72q−38−12q−40−38q−42 + 78q−44−108q−46 + 118q−48−116q−50 + 98q−52−80q−54 + 58q−56−38q−58 + 23q−60−12q−62 + 6q−64−2q−66 + q−68
2,0 q12−q8 + 2q4−4 + 7q−4−q−6−6q−8 + 3q−10 + 7q−12 + q−14−4q−16 + 3q−18 + 3q−20−3q−22−q−24−3q−28−q−30 + 5q−32−q−34−2q−36 + 3q−38 + 6q−40−2q−42−6q−44 + 2q−46 + 3q−48−3q−50−5q−52 + 3q−56−2q−60 + q−64 + q−66

A3 Invariants.

Weight Invariant
0,1,0 q8−q6 + q4 + 3q2−3 + 6q−4−6q−6−2q−8 + 9q−10−5q−12−2q−14 + 6q−16−2q−18−2q−20 + q−22 + 4q−24−2q−28 + 6q−30 + 3q−32−8q−34 + 4q−36 + 2q−38−9q−40 + 2q−42 + q−44−4q−46 + 2q−48 + q−50−q−52 + q−54
1,0,0 q5 + q + 2q−3−2q−5−q−9 + q−15 + 2q−17 + 2q−21 + q−25−q−27 + q−29−q−31−q−33−q−35

B2 Invariants.

Weight Invariant
0,1 q8−q6 + 3q4−3q2 + 5−6q−2 + 8q−4−8q−6 + 6q−8−5q−10 + q−12 + 2q−14−6q−16 + 10q−18−12q−20 + 15q−22−14q−24 + 14q−26−10q−28 + 8q−30−3q−32 + 4q−36−6q−38 + 7q−40−8q−42 + 7q−44−6q−46 + 4q−48−3q−50 + q−52−q−54
1,0 q14−q10−q8 + 2q6 + 3q4−4−3q−2 + 3q−4 + 7q−6−8q−10−4q−12 + 6q−14 + 8q−16−2q−18−7q−20 + 7q−24 + 2q−26−6q−28−3q−30 + 4q−32 + 4q−34−3q−36−4q−38 + 2q−40 + 6q−42−4q−46 + 7q−50 + 3q−52−6q−54−7q−56 + 4q−58 + 8q−60−q−62−9q−64−5q−66 + 5q−68 + 5q−70−2q−72−5q−74−q−76 + 3q−78 + 2q−80−q−82−q−84 + q−88

G2 Invariants.

Weight Invariant
1,0 q18−q16 + 3q14−3q12 + 2q10−3q6 + 8q4−9q2 + 11−9q−2 + 5q−4 + 4q−6−13q−8 + 23q−10−26q−12 + 21q−14−13q−16−5q−18 + 20q−20−31q−22 + 33q−24−20q−26 + 2q−28 + 14q−30−23q−32 + 15q−34−2q−36−13q−38 + 24q−40−23q−42 + 9q−44 + 17q−46−34q−48 + 46q−50−42q−52 + 21q−54 + 6q−56−28q−58 + 43q−60−44q−62 + 36q−64−11q−66−11q−68 + 26q−70−30q−72 + 20q−74−q−76−15q−78 + 21q−80−15q−82 + 3q−84 + 18q−86−31q−88 + 33q−90−23q−92 + 18q−96−32q−98 + 33q−100−24q−102 + 10q−104 + 3q−106−15q−108 + 17q−110−15q−112 + 9q−114−3q−116−2q−118 + 3q−120−4q−122 + 3q−124−q−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["9 15"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −2t2 + 10t−15 + 10t−1−2t−2
In[5]:= Conway[K][z]
Out[5]= −2z4 + 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]= { 39, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 2q7−4q6 + 6q5−6q4 + 7q3−6q2 + 4q−2 + q−1
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z4a−2−z4a−4−z2a−2 + 2z2a−6 + z2−a−2 + a−4 + a−6−a−8 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z8a−4 + z8a−6 + 2z7a−3 + 4z7a−5 + 2z7a−7 + 2z6a−2 + z6a−4 + z6a−6 + 2z6a−8 + 2z5a−1−z5a−3−7z5a−5−3z5a−7 + z5a−9−4z4a−6−5z4a−8 + z4−3z3a−1 + 5z3a−5−z3a−7−3z3a−9−3z2a−2−2z2a−4 + 2z2a−6 + 3z2a−8−2z2 + za−1 + za−3−za−5 + za−7 + 2za−9 + a−2 + a−4−a−6−a−8 + 1

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

Same Alexander/Conway Polynomial: {10_165, K11n63, K11n101,}

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["9 15"];
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 + 10t−15 + 10t−1−2t−2, −q8 + 2q7−4q6 + 6q5−6q4 + 7q3−6q2 + 4q−2 + q−1 }
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_165, K11n63, K11n101,}
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, 5)
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 40 32 \frac{508}{3} \frac{92}{3} 320 \frac{2416}{3} \frac{352}{3} 168 \frac{256}{3} 800 \frac{4064}{3} \frac{736}{3} \frac{59071}{15} -\frac{1628}{5} \frac{92164}{45} \frac{881}{9} \frac{4351}{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 = 2 is the signature of 9 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-101234567χ
17         1-1
15        1 1
13       31 -2
11      31  2
9     33   0
7    43    1
5   23     1
3  24      -2
1 13       2
-1 1        -1
-31         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −2 {\mathbb Z}
r = −1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r = 1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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 q23−2q22 + 6q20−8q19−4q18 + 19q17−14q16−15q15 + 35q14−15q13−28q12 + 45q11−12q10−36q9 + 45q8−7q7−33q6 + 33q5−q4−21q3 + 16q2 + q−8 + 5q−1−2q−3 + q−4
3 −q45 + 2q44−2q42−3q41 + 7q40 + 5q39−10q38−14q37 + 16q36 + 24q35−14q34−44q33 + 13q32 + 60q31−q30−79q29−17q28 + 97q27 + 35q26−105q25−60q24 + 116q23 + 76q22−113q21−100q20 + 118q19 + 106q18−107q17−119q16 + 101q15 + 116q14−82q13−114q12 + 66q11 + 102q10−46q9−84q8 + 28q7 + 64q6−13q5−46q4 + 8q3 + 26q2−2q−16 + 3q−1 + 7q−2−q−3−5q−4 + 3q−5 + q−6−2q−8 + q−9
4 q74−2q73 + 2q71−q70 + 4q69−9q68−q67 + 10q66 + 14q64−30q63−15q62 + 22q61 + 13q60 + 54q59−58q58−59q57 + 3q56 + 23q55 + 152q54−49q53−112q52−78q51−26q50 + 280q49 + 35q48−110q47−199q46−167q45 + 374q44 + 169q43−25q42−296q41−358q40 + 392q39 + 292q38 + 113q37−343q36−535q35 + 358q34 + 372q33 + 243q32−347q31−651q30 + 298q29 + 401q28 + 339q27−311q26−694q25 + 215q24 + 373q23 + 392q22−224q21−649q20 + 106q19 + 278q18 + 386q17−101q16−507q15 + 12q14 + 135q13 + 302q12 + 9q11−307q10−26q9 + 15q8 + 174q7 + 49q6−138q5−8q4−31q3 + 66q2 + 33q−47 + 13q−1−24q−2 + 16q−3 + 10q−4−17q−5 + 14q−6−8q−7 + 3q−8 + q−9−7q−10 + 6q−11−q−12 + q−13−2q−15 + q−16
5 −q110 + 2q109−2q107 + q106−2q104 + 5q103 + 2q102−9q101−3q100 + 3q99 + 2q98 + 16q97 + 9q96−20q95−29q94−12q93 + 11q92 + 50q91 + 54q90−11q89−71q88−92q87−40q86 + 80q85 + 160q84 + 104q83−44q82−203q81−234q80−35q79 + 234q78 + 343q77 + 197q76−177q75−483q74−406q73 + 65q72 + 552q71 + 644q70 + 162q69−568q68−898q67−440q66 + 505q65 + 1102q64 + 761q63−335q62−1276q61−1109q60 + 146q59 + 1372q58 + 1410q57 + 124q56−1421q55−1719q54−342q53 + 1418q52 + 1924q51 + 616q50−1401q49−2136q48−787q47 + 1341q46 + 2242q45 + 1010q44−1285q43−2365q42−1124q41 + 1188q40 + 2378q39 + 1299q38−1078q37−2400q36−1382q35 + 916q34 + 2309q33 + 1499q32−720q31−2188q30−1539q29 + 489q28 + 1958q27 + 1549q26−241q25−1669q24−1482q23 + q22 + 1332q21 + 1338q20 + 193q19−970q18−1129q17−328q16 + 630q15 + 900q14 + 368q13−357q12−632q11−356q10 + 144q9 + 423q8 + 291q7−39q6−228q5−209q4−32q3 + 120q2 + 128q + 39−38q−1−71q−2−41q−3 + 17q−4 + 26q−5 + 19q−6 + 13q−7−13q−8−17q−9 + 2q−10−2q−11−2q−12 + 12q−13 + 2q−14−6q−15 + 3q−16−3q−17−5q−18 + 4q−19 + 2q−20−q−21 + q−22−2q−24 + q−25
6 q153−2q152 + 2q150−q149−2q147 + 6q146−6q145−3q144 + 11q143−q142−2q141−13q140 + 12q139−14q138−8q137 + 36q136 + 14q135 + 4q134−42q133 + 9q132−56q131−40q130 + 78q129 + 73q128 + 74q127−47q126 + 18q125−169q124−189q123 + 32q122 + 127q121 + 245q120 + 106q119 + 225q118−239q117−465q116−296q115−103q114 + 287q113 + 379q112 + 870q111 + 139q110−498q109−797q108−869q107−338q106 + 251q105 + 1722q104 + 1220q103 + 356q102−797q101−1808q100−1854q99−973q98 + 1961q97 + 2517q96 + 2241q95 + 419q94−2012q93−3651q92−3328q91 + 884q90 + 3095q89 + 4451q88 + 2782q87−882q86−4793q85−6017q84−1336q83 + 2457q82 + 6087q81 + 5471q80 + 1273q79−4876q78−8170q77−3863q76 + 959q75 + 6803q74 + 7686q73 + 3621q72−4237q71−9459q70−5964q69−679q68 + 6860q67 + 9134q66 + 5537q65−3414q64−10060q63−7406q62−2014q61 + 6618q60 + 9940q59 + 6889q58−2632q57−10192q56−8327q55−3070q54 + 6131q53 + 10254q52 + 7867q51−1723q50−9805q49−8855q48−4092q47 + 5144q46 + 9955q45 + 8556q44−408q43−8580q42−8795q41−5116q40 + 3405q39 + 8681q38 + 8653q37 + 1237q36−6307q35−7713q34−5715q33 + 1143q32 + 6283q31 + 7664q30 + 2591q29−3415q28−5504q27−5290q26−813q25 + 3352q24 + 5548q23 + 2923q22−933q21−2863q20−3801q19−1647q18 + 976q17 + 3078q16 + 2182q15 + 326q14−831q13−2006q12−1377q11−174q10 + 1227q9 + 1096q8 + 480q7 + 99q6−724q5−722q4−356q3 + 333q2 + 346q + 231 + 252q−1−159q−2−254q−3−206q−4 + 67q−5 + 50q−6 + 41q−7 + 159q−8−11q−9−62q−10−79q−11 + 18q−12−10q−13−17q−14 + 69q−15 + 7q−16−8q−17−25q−18 + 10q−19−10q−20−18q−21 + 24q−22 + 3q−23 + 3q−24−7q−25 + 5q−26−3q−27−9q−28 + 6q−29 + 2q−31−q−32 + q−33−2q−35 + q−36

[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.

9_14

9_16

Retrieved from "http://katlas.math.toronto.edu/wiki/9_15"
Personal tools