9 16

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9_15

9_17

Contents

Image:9 16.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9_16's page at Knotilus!

Visit 9 16's page at the original Knot Atlas!

[edit 9 16 Quick Notes]
[edit 9 16 Further Notes and Views]


[edit] Knot presentations

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


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

Length is 10, width is 3,

Braid index is 3

Image:9 16_ML.gif Image:9 16_AP.gif
[{3, 10}, {2, 6}, {1, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 5}, {6, 4}, {5, 2}, {4, 11}, {7, 1}]

[edit Notes on presentations of 9 16]


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

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3,3,2+]
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,{1,1,1,1,2,2,−1,2,2,2})
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, 10, 3 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.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 16_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 10}, {2, 6}, {1, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 5}, {6, 4}, {5, 2}, {4, 11}, {7, 1}]
In[14]:= Draw[ap]
Image:9 16_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index {4,7}
Nakanishi index 1
Maximal Thurston-Bennequin number [5][-16]
Hyperbolic Volume 9.88301
A-Polynomial See Data:9 16/A-polynomial

[edit Notes for 9 16's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 3
Topological 4 genus 3
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for 9 16's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 2t3−5t2 + 8t−9 + 8t−1−5t−2 + 2t−3
Conway polynomial 2z6 + 7z4 + 6z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 39, 6 }
Jones polynomial −q12 + 3q11−5q10 + 6q9−7q8 + 6q7−5q6 + 4q5−q4 + q3
HOMFLY-PT polynomial (db, data sources) z6a−6 + z6a−8 + 5z4a−6 + 3z4a−8−z4a−10 + 8z2a−6−2z2a−10 + 4a−6−3a−8
Kauffman polynomial (db, data sources) z8a−8 + z8a−10 + z7a−7 + 4z7a−9 + 3z7a−11 + z6a−6−z6a−8 + 3z6a−10 + 5z6a−12−2z5a−7−8z5a−9−z5a−11 + 5z5a−13−5z4a−6−4z4a−8−8z4a−10−6z4a−12 + 3z4a−14−2z3a−7−z3a−9−5z3a−11−5z3a−13 + z3a−15 + 8z2a−6 + 6z2a−8 + z2a−10 + 2z2a−12−z2a−14 + 4za−7 + 4za−9 + 2za−11 + 2za−13−4a−6−3a−8
The A2 invariant q−10 + 3q−14 + q−16 + 2q−18 + q−20−2q−22−3q−26 + q−34−q−36
The G2 invariant q−50 + 3q−54−2q−56 + 3q−58−q−60 + 8q−64−11q−66 + 16q−68−11q−70 + 6q−72 + 11q−74−21q−76 + 32q−78−26q−80 + 18q−82 + q−84−23q−86 + 33q−88−31q−90 + 19q−92−18q−96 + 22q−98−18q−100 + 11q−104−24q−106 + 22q−108−14q−110−8q−112 + 27q−114−40q−116 + 41q−118−27q−120 + 2q−122 + 22q−124−40q−126 + 46q−128−36q−130 + 16q−132 + 10q−134−26q−136 + 30q−138−20q−140 + 2q−142 + 13q−144−19q−146 + 13q−148−q−150−15q−152 + 25q−154−26q−156 + 19q−158−4q−160−14q−162 + 23q−164−27q−166 + 24q−168−14q−170 + 3q−172 + 6q−174−13q−176 + 15q−178−12q−180 + 9q−182−2q−184−q−186 + 2q−188−4q−190 + 3q−192−2q−194 + q−196
Further Quantum Invariants
Further quantum knot invariants for 9_16.

A1 Invariants.

Weight Invariant
1 q−5 + 3q−9−q−11 + q−13−q−15−q−17 + q−19−2q−21 + 2q−23−q−25
2 q−10 + 4q−16 + q−18−4q−20 + 6q−22 + 3q−24−9q−26 + 3q−28 + 6q−30−9q−32−q−34 + 6q−36−4q−38−4q−40 + 3q−42 + 3q−44−4q−46−2q−48 + 9q−50−3q−52−8q−54 + 9q−56−q−58−6q−60 + 5q−62−2q−66 + q−68
3 q−15 + q−21 + 4q−23 + q−25−3q−27−2q−29 + 9q−31 + 8q−33−7q−35−16q−37 + 6q−39 + 20q−41 + 2q−43−26q−45−12q−47 + 25q−49 + 18q−51−21q−53−27q−55 + 14q−57 + 28q−59−7q−61−32q−63 + q−65 + 27q−67 + 8q−69−26q−71−9q−73 + 21q−75 + 16q−77−13q−79−19q−81 + 3q−83 + 21q−85 + 8q−87−22q−89−19q−91 + 18q−93 + 29q−95−14q−97−32q−99 + 8q−101 + 31q−103−3q−105−23q−107−q−109 + 16q−111−11q−115 + 2q−117 + 5q−119−3q−123 + 2q−127−q−129
4 q−20 + q−26 + q−28 + 4q−30−3q−34 + q−38 + 13q−40 + 6q−42−8q−44−15q−46−15q−48 + 22q−50 + 30q−52 + 11q−54−26q−56−60q−58−4q−60 + 44q−62 + 65q−64 + 16q−66−86q−68−68q−70−3q−72 + 93q−74 + 94q−76−44q−78−99q−80−84q−82 + 50q−84 + 135q−86 + 37q−88−67q−90−132q−92−17q−94 + 118q−96 + 88q−98−14q−100−130q−102−60q−104 + 80q−106 + 103q−108 + 18q−110−109q−112−75q−114 + 47q−116 + 101q−118 + 35q−120−77q−122−85q−124 + 86q−128 + 64q−130−17q−132−83q−134−72q−136 + 36q−138 + 91q−140 + 78q−142−46q−144−135q−146−46q−148 + 68q−150 + 147q−152 + 25q−154−129q−156−100q−158 + 6q−160 + 137q−162 + 67q−164−69q−166−76q−168−34q−170 + 71q−172 + 49q−174−24q−176−26q−178−25q−180 + 27q−182 + 15q−184−12q−186−q−188−8q−190 + 10q−192 + 3q−194−6q−196 + q−198−2q−200 + 3q−202−2q−206 + q−208
5 q−25 + q−31 + q−33 + q−35 + 3q−37−3q−41 + 4q−45 + 5q−47 + 10q−49 + 3q−51−11q−53−17q−55−9q−57 + 7q−59 + 33q−61 + 35q−63 + 4q−65−40q−67−67q−69−48q−71 + 27q−73 + 98q−75 + 102q−77 + 28q−79−97q−81−177q−83−115q−85 + 48q−87 + 201q−89 + 229q−91 + 69q−93−185q−95−320q−97−211q−99 + 71q−101 + 343q−103 + 366q−105 + 91q−107−288q−109−462q−111−280q−113 + 140q−115 + 481q−117 + 454q−119 + 46q−121−418q−123−555q−125−241q−127 + 281q−129 + 592q−131 + 400q−133−129q−135−548q−137−500q−139−21q−141 + 476q−143 + 542q−145 + 129q−147−374q−149−537q−151−202q−153 + 311q−155 + 495q−157 + 221q−159−238q−161−462q−163−233q−165 + 212q−167 + 424q−169 + 221q−171−174q−173−401q−175−242q−177 + 139q−179 + 380q−181 + 275q−183−64q−185−342q−187−333q−189−54q−191 + 274q−193 + 392q−195 + 209q−197−154q−199−423q−201−384q−203−23q−205 + 406q−207 + 536q−209 + 229q−211−307q−213−634q−215−432q−217 + 153q−219 + 637q−221 + 589q−223 + 24q−225−555q−227−648q−229−184q−231 + 404q−233 + 617q−235 + 289q−237−251q−239−502q−241−307q−243 + 105q−245 + 359q−247 + 273q−249−21q−251−229q−253−193q−255−15q−257 + 122q−259 + 119q−261 + 24q−263−62q−265−68q−267−6q−269 + 28q−271 + 24q−273 + 6q−275−13q−277−11q−279 + q−281 + 9q−283 + 2q−285−5q−287−2q−289 + q−291 + q−295 + 2q−297−3q−299 + 2q−303−q−305
6 q−30 + q−36 + q−38 + q−40 + 3q−44−3q−48 + q−50 + 4q−52 + 7q−54 + 2q−56 + 7q−58−q−60−15q−62−14q−64−4q−66 + 14q−68 + 18q−70 + 39q−72 + 23q−74−21q−76−57q−78−70q−80−38q−82 + 2q−84 + 103q−86 + 142q−88 + 98q−90−16q−92−149q−94−226q−96−226q−98−12q−100 + 216q−102 + 378q−104 + 341q−106 + 108q−108−250q−110−594q−112−542q−114−230q−116 + 304q−118 + 733q−120 + 844q−122 + 446q−124−359q−126−939q−128−1136q−130−653q−132 + 242q−134 + 1195q−136 + 1504q−138 + 875q−140−194q−142−1363q−144−1799q−146−1273q−148 + 193q−150 + 1597q−152 + 2082q−154 + 1504q−156−118q−158−1745q−160−2514q−162−1636q−164 + 195q−166 + 1945q−168 + 2712q−170 + 1765q−172−273q−174−2326q−176−2802q−178−1647q−180 + 531q−182 + 2513q−184 + 2866q−186 + 1428q−188−1036q−190−2643q−192−2650q−194−948q−196 + 1406q−198 + 2752q−200 + 2262q−202 + 202q−204−1745q−206−2565q−208−1602q−210 + 424q−212 + 2035q−214 + 2152q−216 + 701q−218−1003q−220−1991q−222−1457q−224 + 73q−226 + 1462q−228 + 1680q−230 + 600q−232−764q−234−1576q−236−1118q−238 + 116q−240 + 1277q−242 + 1445q−244 + 522q−246−723q−248−1509q−250−1174q−252−96q−254 + 1125q−256 + 1590q−258 + 1040q−260−200q−262−1369q−264−1675q−266−1067q−268 + 310q−270 + 1560q−272 + 2010q−274 + 1189q−276−404q−278−1854q−280−2408q−282−1420q−284 + 534q−286 + 2418q−288 + 2785q−290 + 1487q−292−840q−294−2898q−296−3129q−298−1380q−300 + 1432q−302 + 3224q−304 + 3100q−306 + 995q−308−1852q−310−3415q−312−2780q−314−279q−316 + 2092q−318 + 3141q−320 + 2142q−322−230q−324−2201q−326−2606q−328−1226q−330 + 569q−332 + 1893q−334 + 1871q−336 + 601q−338−806q−340−1463q−342−1008q−344−171q−346 + 688q−348 + 960q−350 + 510q−352−133q−354−525q−356−424q−358−203q−360 + 144q−362 + 323q−364 + 193q−366−3q−368−131q−370−87q−372−73q−374 + 13q−376 + 80q−378 + 36q−380−9q−382−27q−384 + 4q−386−14q−388 + 16q−392−7q−396−7q−398 + 11q−400−3q−402−2q−404 + 5q−406−2q−408−q−410−2q−412 + 3q−414−2q−418 + q−420

A2 Invariants.

Weight Invariant
1,0 q−10 + 3q−14 + q−16 + 2q−18 + q−20−2q−22−3q−26 + q−34−q−36
1,1 q−20 + 6q−24−4q−26 + 20q−28−22q−30 + 50q−32−58q−34 + 83q−36−98q−38 + 102q−40−104q−42 + 75q−44−60q−46 + 4q−48 + 34q−50−85q−52 + 128q−54−162q−56 + 192q−58−189q−60 + 184q−62−156q−64 + 118q−66−73q−68 + 24q−70 + 16q−72−52q−74 + 73q−76−84q−78 + 88q−80−84q−82 + 74q−84−58q−86 + 46q−88−36q−90 + 22q−92−12q−94 + 8q−96−4q−98 + q−100
2,0 q−20 + 3q−26 + 4q−28 + 3q−32 + 6q−34 + 4q−36−q−38 + 2q−40 + 3q−42−4q−44−6q−46−q−48−5q−50−8q−52−2q−54−q−56−3q−58 + 7q−62 + 4q−64 + 3q−68 + 3q−70−4q−72−4q−74 + q−76 + q−84−q−88 + q−90

A3 Invariants.

Weight Invariant
0,1,0 q−20 + 3q−24 + 4q−26 + 2q−28 + 7q−30 + 6q−32−4q−34 + 4q−36−3q−38−12q−40−q−44−7q−46 + 3q−48 + 5q−50−q−52−q−54 + 3q−58−5q−60−2q−62 + 7q−64−5q−66−3q−68 + 8q−70−3q−72−3q−74 + 5q−76−q−78−2q−80 + q−82
1,0,0 q−15 + 3q−19 + q−21 + 4q−23 + q−25 + 2q−27−q−29−2q−31−2q−33−3q−35−q−39 + 2q−41−q−43 + q−45−q−47
1,0,1 q−30 + 6q−34 + 2q−36 + 11q−38 + 15q−40−q−42 + 38q−44−18q−46 + 27q−48 + 10q−50−48q−52 + 63q−54−97q−56 + 36q−58−17q−60−72q−62 + 84q−64−102q−66 + 61q−68−3q−70−27q−72 + 56q−74−21q−76 + 6q−78 + 21q−80−35q−84 + 64q−86−71q−88 + 26q−90 + 30q−92−84q−94 + 97q−96−61q−98 + 2q−100 + 53q−102−76q−104 + 61q−106−21q−108−24q−110 + 48q−112−42q−114 + 19q−116 + 9q−118−24q−120 + 19q−122−7q−124−2q−126 + 6q−128−4q−130 + q−132

A4 Invariants.

Weight Invariant
0,1,0,0 q−30 + 3q−34 + 4q−36 + 5q−38 + 7q−40 + 12q−42 + 6q−44 + 6q−46 + 6q−48−3q−50−10q−52−8q−54−8q−56−15q−58−8q−60 + 3q−62 + 3q−64−3q−66 + 8q−68 + 8q−70−5q−72−2q−74 + 3q−76−5q−78−6q−80 + 3q−82 + 2q−84−3q−86 + q−88 + 6q−90−3q−94 + 3q−96 + 2q−98−3q−100−q−102 + q−104
1,0,0,0 q−20 + 3q−24 + q−26 + 4q−28 + 3q−30 + 2q−32 + 2q−34−q−36−q−38−4q−40−2q−42−3q−44−q−48 + q−50 + q−52−q−54 + q−56−q−58

B2 Invariants.

Weight Invariant
0,1 q−20 + 3q−24−2q−26 + 6q−28−5q−30 + 8q−32−6q−34 + 8q−36−5q−38 + 2q−40−5q−44 + 7q−46−13q−48 + 13q−50−15q−52 + 13q−54−12q−56 + 9q−58−5q−60 + 2q−62 + 3q−64−5q−66 + 7q−68−8q−70 + 7q−72−7q−74 + 5q−76−3q−78 + 2q−80−q−82
1,0 q−30 + 3q−38 + 3q−40 + q−42−2q−44 + 2q−46 + 7q−48 + 6q−50−4q−52−5q−54 + q−56 + 8q−58 + q−60−10q−62−8q−64 + 2q−66 + 5q−68−3q−70−7q−72−2q−74 + 5q−76 + 2q−78−3q−80−2q−82 + 5q−84 + 4q−86−3q−88−6q−90 + 2q−92 + 6q−94−q−96−7q−98−2q−100 + 7q−102 + 4q−104−5q−106−7q−108 + 2q−110 + 8q−112 + 2q−114−5q−116−5q−118 + 2q−120 + 5q−122 + q−124−2q−126−2q−128 + q−132

D4 Invariants.

Weight Invariant
1,0,0,0 q−30 + 3q−34 + q−36 + 7q−38 + q−40 + 10q−42 + 10q−46−4q−48 + 4q−50−8q−52−8q−56−6q−58−2q−60−6q−62 + 5q−64−8q−66 + 11q−68−8q−70 + 12q−72−11q−74 + 10q−76−9q−78 + 8q−80−7q−82 + 2q−84−2q−86 + 3q−90−5q−92 + 4q−94−5q−96 + 8q−98−5q−100 + 4q−102−5q−104 + 5q−106−2q−108 + q−110−2q−112 + q−114

G2 Invariants.

Weight Invariant
1,0 q−50 + 3q−54−2q−56 + 3q−58−q−60 + 8q−64−11q−66 + 16q−68−11q−70 + 6q−72 + 11q−74−21q−76 + 32q−78−26q−80 + 18q−82 + q−84−23q−86 + 33q−88−31q−90 + 19q−92−18q−96 + 22q−98−18q−100 + 11q−104−24q−106 + 22q−108−14q−110−8q−112 + 27q−114−40q−116 + 41q−118−27q−120 + 2q−122 + 22q−124−40q−126 + 46q−128−36q−130 + 16q−132 + 10q−134−26q−136 + 30q−138−20q−140 + 2q−142 + 13q−144−19q−146 + 13q−148−q−150−15q−152 + 25q−154−26q−156 + 19q−158−4q−160−14q−162 + 23q−164−27q−166 + 24q−168−14q−170 + 3q−172 + 6q−174−13q−176 + 15q−178−12q−180 + 9q−182−2q−184−q−186 + 2q−188−4q−190 + 3q−192−2q−194 + q−196

.

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 16"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 2t3−5t2 + 8t−9 + 8t−1−5t−2 + 2t−3
In[5]:= Conway[K][z]
Out[5]= 2z6 + 7z4 + 6z2 + 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, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q12 + 3q11−5q10 + 6q9−7q8 + 6q7−5q6 + 4q5−q4 + q3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−6 + z6a−8 + 5z4a−6 + 3z4a−8−z4a−10 + 8z2a−6−2z2a−10 + 4a−6−3a−8
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z8a−8 + z8a−10 + z7a−7 + 4z7a−9 + 3z7a−11 + z6a−6−z6a−8 + 3z6a−10 + 5z6a−12−2z5a−7−8z5a−9−z5a−11 + 5z5a−13−5z4a−6−4z4a−8−8z4a−10−6z4a−12 + 3z4a−14−2z3a−7−z3a−9−5z3a−11−5z3a−13 + z3a−15 + 8z2a−6 + 6z2a−8 + z2a−10 + 2z2a−12−z2a−14 + 4za−7 + 4za−9 + 2za−11 + 2za−13−4a−6−3a−8

[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["9 16"];
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]= { 2t3−5t2 + 8t−9 + 8t−1−5t−2 + 2t−3, −q12 + 3q11−5q10 + 6q9−7q8 + 6q7−5q6 + 4q5−q4 + q3 }
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: (6, 14)

[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 = 6 is the signature of 9 16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 0123456789χ
25         1-1
23        2 2
21       31 -2
19      32  1
17     43   -1
15    23    -1
13   34     1
11  12      -1
9  3       3
711        0
51         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 5 i = 7
r = 0 {\mathbb Z} {\mathbb Z}
r = 1 {\mathbb Z}
r = 2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 9 {\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 q33−3q32 + 2q31 + 6q30−14q29 + 7q28 + 16q27−31q26 + 12q25 + 28q24−42q23 + 10q22 + 35q21−42q20 + 3q19 + 35q18−32q17−4q16 + 27q15−17q14−7q13 + 15q12−5q11−4q10 + 5q9−q7 + q6
3 −q63 + 3q62−2q61−3q60 + 2q59 + 8q58−5q57−16q56 + 13q55 + 24q54−22q53−38q52 + 33q51 + 58q50−45q49−78q48 + 51q47 + 101q46−56q45−115q44 + 48q43 + 131q42−43q41−133q40 + 26q39 + 137q38−14q37−128q36−4q35 + 120q34 + 20q33−109q32−30q31 + 87q30 + 45q29−74q28−44q27 + 46q26 + 51q25−35q24−37q23 + 9q22 + 37q21−7q20−19q19−5q18 + 15q17 + 2q16−4q15−4q14 + 4q13 + q12−q10 + q9
4 q102−3q101 + 2q100 + 3q99−5q98 + 4q97−10q96 + 11q95 + 10q94−23q93 + 11q92−21q91 + 38q90 + 22q89−75q88 + 10q87−19q86 + 111q85 + 44q84−180q83−32q82−12q81 + 247q80 + 114q79−311q78−138q77−41q76 + 401q75 + 236q74−390q73−252q72−130q71 + 490q70 + 360q69−377q68−307q67−238q66 + 479q65 + 426q64−296q63−285q62−324q61 + 394q60 + 434q59−184q58−219q57−378q56 + 272q55 + 400q54−57q53−134q52−401q51 + 132q50 + 330q49 + 59q48−32q47−371q46−3q45 + 215q44 + 124q43 + 72q42−273q41−88q40 + 81q39 + 109q38 + 127q37−135q36−89q35−15q34 + 44q33 + 109q32−33q31−40q30−36q29−4q28 + 53q27 + q26−3q25−17q24−12q23 + 16q22 + q21 + 4q20−3q19−5q18 + 4q17 + q15−q13 + q12
5 −q150 + 3q149−2q148−3q147 + 5q146−q145−2q144 + 4q143−5q142−6q141 + 12q140 + 6q139−10q138−8q137−7q136 + 13q135 + 30q134 + 10q133−44q132−70q131−q130 + 99q129 + 125q128 + 13q127−181q126−248q125−37q124 + 307q123 + 419q122 + 99q121−435q120−660q119−232q118 + 558q117 + 959q116 + 427q115−648q114−1248q113−696q112 + 651q111 + 1538q110 + 992q109−600q108−1732q107−1281q106 + 449q105 + 1865q104 + 1528q103−293q102−1862q101−1710q100 + 88q99 + 1826q98 + 1797q97 + 70q96−1679q95−1828q94−240q93 + 1547q92 + 1788q91 + 348q90−1340q89−1723q88−481q87 + 1166q86 + 1629q85 + 575q84−945q83−1520q82−693q81 + 721q80 + 1400q79 + 799q78−486q77−1246q76−877q75 + 208q74 + 1065q73 + 962q72 + 17q71−833q70−943q69−289q68 + 586q67 + 914q66 + 436q65−304q64−751q63−600q62 + 64q61 + 600q60 + 573q59 + 160q58−343q57−573q56−277q55 + 180q54 + 391q53 + 334q52 + 36q51−298q50−300q49−92q48 + 109q47 + 225q46 + 171q45−44q44−140q43−120q42−44q41 + 62q40 + 109q39 + 36q38−15q37−46q36−48q35−9q34 + 34q33 + 17q32 + 12q31−2q30−17q29−11q28 + 8q27 + q26 + 4q25 + 4q24−3q23−4q22 + 3q21 + q18−q16 + q15
6 q207−3q206 + 2q205 + 3q204−5q203 + q202−q201 + 8q200−10q199 + q198 + 17q197−23q196 + q195 + 6q194 + 24q193−26q192−13q191 + 35q190−54q189 + 19q188 + 51q187 + 68q186−93q185−99q184 + 21q183−98q182 + 147q181 + 247q180 + 198q179−272q178−446q177−200q176−199q175 + 539q174 + 890q173 + 648q172−544q171−1305q170−1037q169−654q168 + 1196q167 + 2297q166 + 1918q165−522q164−2629q163−2832q162−2034q161 + 1601q160 + 4268q159 + 4290q158 + 477q157−3678q156−5203q155−4535q154 + 966q153 + 5831q152 + 7137q151 + 2582q150−3554q149−6995q148−7347q147−783q146 + 6062q145 + 9195q144 + 4909q143−2256q142−7362q141−9231q140−2737q139 + 5074q138 + 9749q137 + 6359q136−663q135−6541q134−9681q133−3987q132 + 3697q131 + 9141q130 + 6665q129 + 506q128−5301q127−9131q126−4452q125 + 2476q124 + 8063q123 + 6330q122 + 1292q121−4056q120−8208q119−4620q118 + 1315q117 + 6829q116 + 5872q115 + 2104q114−2692q113−7128q112−4838q111−74q110 + 5299q109 + 5338q108 + 3092q107−988q106−5677q105−4955q104−1685q103 + 3273q102 + 4375q101 + 3912q100 + 959q99−3617q98−4465q97−3031q96 + 919q95 + 2673q94 + 3919q93 + 2566q92−1153q91−3027q90−3384q89−1063q88 + 495q87 + 2764q86 + 3042q85 + 900q84−989q83−2437q82−1830q81−1255q80 + 933q79 + 2164q78 + 1669q77 + 638q76−815q75−1252q74−1740q73−471q72 + 698q71 + 1143q70 + 1074q69 + 354q68−183q67−1111q66−780q65−255q64 + 248q63 + 591q62 + 551q61 + 397q60−306q59−382q58−366q57−181q56 + 57q55 + 239q54 + 345q53 + 38q52−24q51−133q50−144q49−105q48 + 11q47 + 131q46 + 38q45 + 53q44−30q42−61q41−28q40 + 30q39−2q38 + 21q37 + 13q36 + 6q35−17q34−12q33 + 9q32−6q31 + 3q30 + 3q29 + 5q28−3q27−4q26 + 4q25−q24 + q21−q19 + q18

[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_15

9_17

Retrieved from "http://katlas.org/wiki/9_16"
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