10 109

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10_108

10_110

Contents

Image:10 109.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 109's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_109's page at Knotilus!

Visit 10 109's page at the original Knot Atlas!

[edit 10 109 Quick Notes]
[edit 10 109 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,15,1,16 X12,19,13,20 X14,6,15,5 X2,10,3,9 X4,14,5,13
Gauss code 1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6
Dowker-Thistlethwaite code 6 10 14 16 2 18 4 20 8 12
Conway Notation [2.2.2.2]


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

Length is 10, width is 3,

Braid index is 3

Image:10 109_ML.gif Image:10 109_AP.gif
[{3, 13}, {2, 6}, {4, 7}, {6, 12}, {5, 3}, {1, 4}, {13, 11}, {12, 8}, {7, 9}, {8, 10}, {9, 5}, {11, 2}, {10, 1}]

[edit Notes on presentations of 10 109]


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["10 109"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X10,4,11,3 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,15,1,16 X12,19,13,20 X14,6,15,5 X2,10,3,9 X4,14,5,13
In[5]:= GaussCode[K]
Out[5]= 1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6
In[6]:= DTCode[K]
Out[6]= 6 10 14 16 2 18 4 20 8 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [2.2.2.2]
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,−1,2,−1,2,2,−1,−1,2,2})
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:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.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:10 109_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 13}, {2, 6}, {4, 7}, {6, 12}, {5, 3}, {1, 4}, {13, 11}, {12, 8}, {7, 9}, {8, 10}, {9, 5}, {11, 2}, {10, 1}]
In[14]:= Draw[ap]
Image:10 109_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 14.9002
A-Polynomial See Data:10 109/A-polynomial

[edit Notes for 10 109's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for 10 109's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−4t3 + 10t2−17t + 21−17t−1 + 10t−2−4t−3 + t−4
Conway polynomial z8 + 4z6 + 6z4 + 3z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 85, 0 }
Jones polynomial −q5 + 3q4−7q3 + 11q2−13q + 15−13q−1 + 11q−2−7q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−z6a−2 + 6z6−4a2z4−4z4a−2 + 14z4−6a2z2−6z2a−2 + 15z2−3a2−3a−2 + 7
Kauffman polynomial (db, data sources) 2az9 + 2z9a−1 + 5a2z8 + 5z8a−2 + 10z8 + 5a3z7 + 6az7 + 6z7a−1 + 5z7a−3 + 3a4z6−7a2z6−7z6a−2 + 3z6a−4−20z6 + a5z5−8a3z5−16az5−16z5a−1−8z5a−3 + z5a−5−5a4z4 + 6a2z4 + 6z4a−2−5z4a−4 + 22z4−2a5z3 + 4a3z3 + 13az3 + 13z3a−1 + 4z3a−3−2z3a−5 + 2a4z2−7a2z2−7z2a−2 + 2z2a−4−18z2 + a5z−a3z−5az−5za−1−za−3 + za−5 + 3a2 + 3a−2 + 7
The A2 invariant −q14 + q12−3q10 + q8−q4 + 5q2−1 + 5q−2−q−4 + q−8−3q−10 + q−12−q−14
The G2 invariant q80−2q78 + 5q76−8q74 + 9q72−8q70 + q68 + 14q66−32q64 + 51q62−62q60 + 51q58−20q56−39q54 + 113q52−171q50 + 187q48−138q46 + 19q44 + 124q42−249q40 + 297q38−239q36 + 89q34 + 90q32−231q30 + 264q28−177q26 + 13q24 + 150q22−232q20 + 187q18−37q16−151q14 + 301q12−336q10 + 248q8−53q6−170q4 + 353q2−417 + 353q−2−170q−4−53q−6 + 248q−8−336q−10 + 301q−12−151q−14−37q−16 + 187q−18−232q−20 + 150q−22 + 13q−24−177q−26 + 264q−28−231q−30 + 90q−32 + 89q−34−239q−36 + 297q−38−249q−40 + 124q−42 + 19q−44−138q−46 + 187q−48−171q−50 + 113q−52−39q−54−20q−56 + 51q−58−62q−60 + 51q−62−32q−64 + 14q−66 + q−68−8q−70 + 9q−72−8q−74 + 5q−76−2q−78 + q−80
Further Quantum Invariants
Further quantum knot invariants for 10_109.

A1 Invariants.

Weight Invariant
1 −q11 + 2q9−4q7 + 4q5−2q3 + 2q + 2q−1−2q−3 + 4q−5−4q−7 + 2q−9−q−11
2 q32−2q30 + 7q26−10q24−6q22 + 26q20−14q18−27q16 + 38q14−38q10 + 26q8 + 16q6−26q4 + 21−26q−4 + 16q−6 + 26q−8−38q−10 + 38q−14−27q−16−14q−18 + 26q−20−6q−22−10q−24 + 7q−26−2q−30 + q−32
3 −q63 + 2q61−3q57−q55 + 9q53 + 4q51−23q49−13q47 + 44q45 + 41q43−62q41−100q39 + 65q37 + 172q35−28q33−238q31−52q29 + 280q27 + 139q25−267q23−226q21 + 215q19 + 277q17−135q15−291q13 + 55q11 + 264q9 + 30q7−219q5−98q3 + 162q + 162q−1−98q−3−219q−5 + 30q−7 + 264q−9 + 55q−11−291q−13−135q−15 + 277q−17 + 215q−19−226q−21−267q−23 + 139q−25 + 280q−27−52q−29−238q−31−28q−33 + 172q−35 + 65q−37−100q−39−62q−41 + 41q−43 + 44q−45−13q−47−23q−49 + 4q−51 + 9q−53−q−55−3q−57 + 2q−61−q−63
4 q104−2q102 + 3q98−3q96 + 2q94−7q92 + 4q90 + 20q88−11q86−14q84−47q82 + 15q80 + 121q78 + 49q76−62q74−276q72−140q70 + 316q68 + 462q66 + 219q64−653q62−905q60−12q58 + 1048q56 + 1376q54−258q52−1835q50−1447q48 + 613q46 + 2634q44 + 1386q42−1520q40−2875q38−1093q36 + 2454q34 + 2872q32 + 93q30−2801q28−2520q26 + 986q24 + 2865q22 + 1476q20−1528q18−2602q16−408q14 + 1822q12 + 1882q10−243q8−1912q6−1218q4 + 745q2 + 1861 + 745q−2−1218q−4−1912q−6−243q−8 + 1882q−10 + 1822q−12−408q−14−2602q−16−1528q−18 + 1476q−20 + 2865q−22 + 986q−24−2520q−26−2801q−28 + 93q−30 + 2872q−32 + 2454q−34−1093q−36−2875q−38−1520q−40 + 1386q−42 + 2634q−44 + 613q−46−1447q−48−1835q−50−258q−52 + 1376q−54 + 1048q−56−12q−58−905q−60−653q−62 + 219q−64 + 462q−66 + 316q−68−140q−70−276q−72−62q−74 + 49q−76 + 121q−78 + 15q−80−47q−82−14q−84−11q−86 + 20q−88 + 4q−90−7q−92 + 2q−94−3q−96 + 3q−98−2q−102 + q−104
5 −q155 + 2q153−3q149 + 3q147 + 2q145−4q143−q141−q139−7q137 + 11q135 + 28q133 + 4q131−31q129−63q127−57q125 + 36q123 + 184q121 + 222q119 + 11q117−352q115−592q113−343q111 + 438q109 + 1253q107 + 1227q105−88q103−1972q101−2832q99−1334q97 + 2094q95 + 4991q93 + 4286q91−701q89−6719q87−8568q85−3193q83 + 6606q81 + 13159q79 + 9577q77−3359q75−16099q73−17271q71−3553q69 + 15601q67 + 24222q65 + 12985q63−10867q61−27958q59−22765q57 + 2383q55 + 27293q53 + 30347q51 + 7663q49−22195q47−33817q45−16957q43 + 14195q41 + 32822q39 + 23342q37−5386q35−28197q33−26026q31−2235q29 + 21581q27 + 25347q25 + 7640q23−14764q21−22543q19−10608q17 + 8924q15 + 18941q13 + 11978q11−4543q9−15841q7−12669q5 + 1379q3 + 13741q + 13741q−1 + 1379q−3−12669q−5−15841q−7−4543q−9 + 11978q−11 + 18941q−13 + 8924q−15−10608q−17−22543q−19−14764q−21 + 7640q−23 + 25347q−25 + 21581q−27−2235q−29−26026q−31−28197q−33−5386q−35 + 23342q−37 + 32822q−39 + 14195q−41−16957q−43−33817q−45−22195q−47 + 7663q−49 + 30347q−51 + 27293q−53 + 2383q−55−22765q−57−27958q−59−10867q−61 + 12985q−63 + 24222q−65 + 15601q−67−3553q−69−17271q−71−16099q−73−3359q−75 + 9577q−77 + 13159q−79 + 6606q−81−3193q−83−8568q−85−6719q−87−701q−89 + 4286q−91 + 4991q−93 + 2094q−95−1334q−97−2832q−99−1972q−101−88q−103 + 1227q−105 + 1253q−107 + 438q−109−343q−111−592q−113−352q−115 + 11q−117 + 222q−119 + 184q−121 + 36q−123−57q−125−63q−127−31q−129 + 4q−131 + 28q−133 + 11q−135−7q−137−q−139−q−141−4q−143 + 2q−145 + 3q−147−3q−149 + 2q−153−q−155
6 q216−2q214 + 3q210−3q208−2q206 + 12q202−2q200−12q198 + 7q196−14q194−15q192 + 4q190 + 62q188 + 45q186−22q184−28q182−130q180−166q178−61q176 + 268q174 + 447q172 + 364q170 + 140q168−596q166−1263q164−1334q162−123q160 + 1590q158 + 2958q156 + 3267q154 + 805q152−3423q150−7276q148−7082q146−2166q144 + 6351q142 + 14700q140 + 15389q138 + 5690q136−11964q134−26799q132−29393q130−12890q128 + 18819q126 + 46840q124 + 52118q122 + 23431q120−27179q118−74733q116−85092q114−40712q112 + 39679q110 + 112342q108 + 126233q106 + 63203q104−55078q102−158976q100−176923q98−84719q96 + 76363q94 + 210285q92 + 230455q90 + 102841q88−104731q86−266784q84−275385q82−109284q80 + 138428q78 + 319539q76 + 306278q74 + 99016q72−180371q70−356738q68−313153q66−71885q64 + 221829q62 + 374479q60 + 292206q58 + 26369q56−251667q54−365377q52−246762q50 + 25519q48 + 266682q46 + 329262q44 + 180904q42−70774q40−260866q38−272938q36−110226q34 + 105095q32 + 235464q30 + 203724q28 + 47380q26−123571q24−197604q22−135835q20 + 5058q18 + 128690q16 + 154331q14 + 76682q12−46056q10−127127q8−115073q6−24477q4 + 80664q2 + 125109 + 80664q−2−24477q−4−115073q−6−127127q−8−46056q−10 + 76682q−12 + 154331q−14 + 128690q−16 + 5058q−18−135835q−20−197604q−22−123571q−24 + 47380q−26 + 203724q−28 + 235464q−30 + 105095q−32−110226q−34−272938q−36−260866q−38−70774q−40 + 180904q−42 + 329262q−44 + 266682q−46 + 25519q−48−246762q−50−365377q−52−251667q−54 + 26369q−56 + 292206q−58 + 374479q−60 + 221829q−62−71885q−64−313153q−66−356738q−68−180371q−70 + 99016q−72 + 306278q−74 + 319539q−76 + 138428q−78−109284q−80−275385q−82−266784q−84−104731q−86 + 102841q−88 + 230455q−90 + 210285q−92 + 76363q−94−84719q−96−176923q−98−158976q−100−55078q−102 + 63203q−104 + 126233q−106 + 112342q−108 + 39679q−110−40712q−112−85092q−114−74733q−116−27179q−118 + 23431q−120 + 52118q−122 + 46840q−124 + 18819q−126−12890q−128−29393q−130−26799q−132−11964q−134 + 5690q−136 + 15389q−138 + 14700q−140 + 6351q−142−2166q−144−7082q−146−7276q−148−3423q−150 + 805q−152 + 3267q−154 + 2958q−156 + 1590q−158−123q−160−1334q−162−1263q−164−596q−166 + 140q−168 + 364q−170 + 447q−172 + 268q−174−61q−176−166q−178−130q−180−28q−182−22q−184 + 45q−186 + 62q−188 + 4q−190−15q−192−14q−194 + 7q−196−12q−198−2q−200 + 12q−202−2q−206−3q−208 + 3q−210−2q−214 + q−216

A2 Invariants.

Weight Invariant
1,0 −q14 + q12−3q10 + q8−q4 + 5q2−1 + 5q−2−q−4 + q−8−3q−10 + q−12−q−14
1,1 q44−4q42 + 12q40−28q38 + 60q36−116q34 + 204q32−334q30 + 514q28−738q26 + 976q24−1190q22 + 1334q20−1352q18 + 1186q16−840q14 + 312q12 + 332q10−1030q8 + 1690q6−2220q4 + 2582q2−2694 + 2582q−2−2220q−4 + 1690q−6−1030q−8 + 332q−10 + 312q−12−840q−14 + 1186q−16−1352q−18 + 1334q−20−1190q−22 + 976q−24−738q−26 + 514q−28−334q−30 + 204q−32−116q−34 + 60q−36−28q−38 + 12q−40−4q−42 + q−44
2,0 q38−q36 + 4q32−3q30−5q28 + 7q26 + 3q24−9q22−7q20 + 9q18 + 5q16−19q14 + 2q12 + 14q10−8q8−6q6 + 12q4 + 7q2−6 + 7q−2 + 12q−4−6q−6−8q−8 + 14q−10 + 2q−12−19q−14 + 5q−16 + 9q−18−7q−20−9q−22 + 3q−24 + 7q−26−5q−28−3q−30 + 4q−32−q−36 + q−38

A3 Invariants.

Weight Invariant
0,1,0 q34−2q32 + q30 + 5q28−9q26 + 2q24 + 15q22−22q20 + 2q18 + 23q16−32q14−q12 + 22q10−22q8−3q6 + 19q4 + 4q2 + 4q−2 + 19q−4−3q−6−22q−8 + 22q−10−q−12−32q−14 + 23q−16 + 2q−18−22q−20 + 15q−22 + 2q−24−9q−26 + 5q−28 + q−30−2q−32 + q−34
1,0,0 −q17 + q15−4q13 + 2q11−4q9 + 2q7−q5 + 4q3 + 3q + 3q−1 + 4q−3−q−5 + 2q−7−4q−9 + 2q−11−4q−13 + q−15−q−17
1,0,1 q56−4q54 + 10q52−14q50 + 7q48 + 23q46−72q44 + 110q42−85q40−44q38 + 252q36−425q34 + 405q32−93q30−430q28 + 923q26−1076q24 + 723q22 + 48q20−926q18 + 1449q16−1414q14 + 762q12 + 89q10−756q8 + 914q6−568q4 + 142q2 + 113 + 142q−2−568q−4 + 914q−6−756q−8 + 89q−10 + 762q−12−1414q−14 + 1449q−16−926q−18 + 48q−20 + 723q−22−1076q−24 + 923q−26−430q−28−93q−30 + 405q−32−425q−34 + 252q−36−44q−38−85q−40 + 110q−42−72q−44 + 23q−46 + 7q−48−14q−50 + 10q−52−4q−54 + q−56

A4 Invariants.

Weight Invariant
0,1,0,0 q40−q38 + 5q34−3q32−4q30 + 13q28−3q26−15q24 + 10q22 + 9q20−25q18−14q16 + 13q14−7q12−28q10 + 9q8 + 30q6−14q4 + 6q2 + 46 + 6q−2−14q−4 + 30q−6 + 9q−8−28q−10−7q−12 + 13q−14−14q−16−25q−18 + 9q−20 + 10q−22−15q−24−3q−26 + 13q−28−4q−30−3q−32 + 5q−34−q−38 + q−40
1,0,0,0 −q20 + q18−4q16 + q14−3q12−2q10 + q8−q6 + 5q4 + 2q2 + 7 + 2q−2 + 5q−4−q−6 + q−8−2q−10−3q−12 + q−14−4q−16 + q−18−q−20

B2 Invariants.

Weight Invariant
0,1 −q34 + 2q32−5q30 + 9q28−15q26 + 22q24−29q22 + 34q20−36q18 + 31q16−24q14 + 11q12 + 4q10−22q8 + 41q6−53q4 + 66q2−66 + 66q−2−53q−4 + 41q−6−22q−8 + 4q−10 + 11q−12−24q−14 + 31q−16−36q−18 + 34q−20−29q−22 + 22q−24−15q−26 + 9q−28−5q−30 + 2q−32−q−34
1,0 q56−2q52−2q50 + 3q48 + 7q46−12q42−9q40 + 12q38 + 22q36−3q34−31q32−15q30 + 28q28 + 29q26−16q24−39q22−4q20 + 34q18 + 15q16−26q14−22q12 + 17q10 + 24q8−6q6−22q4 + 6q2 + 27 + 6q−2−22q−4−6q−6 + 24q−8 + 17q−10−22q−12−26q−14 + 15q−16 + 34q−18−4q−20−39q−22−16q−24 + 29q−26 + 28q−28−15q−30−31q−32−3q−34 + 22q−36 + 12q−38−9q−40−12q−42 + 7q−46 + 3q−48−2q−50−2q−52 + q−56

D4 Invariants.

Weight Invariant
1,0,0,0 q46−2q44 + 3q42−4q40 + 8q38−12q36 + 14q34−17q32 + 24q30−28q28 + 26q26−27q24 + 26q22−24q20 + 8q18−11q16−3q14 + 11q12−28q10 + 31q8−34q6 + 56q4−43q2 + 58−43q−2 + 56q−4−34q−6 + 31q−8−28q−10 + 11q−12−3q−14−11q−16 + 8q−18−24q−20 + 26q−22−27q−24 + 26q−26−28q−28 + 24q−30−17q−32 + 14q−34−12q−36 + 8q−38−4q−40 + 3q−42−2q−44 + q−46

G2 Invariants.

Weight Invariant
1,0 q80−2q78 + 5q76−8q74 + 9q72−8q70 + q68 + 14q66−32q64 + 51q62−62q60 + 51q58−20q56−39q54 + 113q52−171q50 + 187q48−138q46 + 19q44 + 124q42−249q40 + 297q38−239q36 + 89q34 + 90q32−231q30 + 264q28−177q26 + 13q24 + 150q22−232q20 + 187q18−37q16−151q14 + 301q12−336q10 + 248q8−53q6−170q4 + 353q2−417 + 353q−2−170q−4−53q−6 + 248q−8−336q−10 + 301q−12−151q−14−37q−16 + 187q−18−232q−20 + 150q−22 + 13q−24−177q−26 + 264q−28−231q−30 + 90q−32 + 89q−34−239q−36 + 297q−38−249q−40 + 124q−42 + 19q−44−138q−46 + 187q−48−171q−50 + 113q−52−39q−54−20q−56 + 51q−58−62q−60 + 51q−62−32q−64 + 14q−66 + q−68−8q−70 + 9q−72−8q−74 + 5q−76−2q−78 + q−80

.

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["10 109"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−4t3 + 10t2−17t + 21−17t−1 + 10t−2−4t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 4z6 + 6z4 + 3z2 + 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]= { 85, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 3q4−7q3 + 11q2−13q + 15−13q−1 + 11q−2−7q−3 + 3q−4−q−5
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−a2z6−z6a−2 + 6z6−4a2z4−4z4a−2 + 14z4−6a2z2−6z2a−2 + 15z2−3a2−3a−2 + 7
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2az9 + 2z9a−1 + 5a2z8 + 5z8a−2 + 10z8 + 5a3z7 + 6az7 + 6z7a−1 + 5z7a−3 + 3a4z6−7a2z6−7z6a−2 + 3z6a−4−20z6 + a5z5−8a3z5−16az5−16z5a−1−8z5a−3 + z5a−5−5a4z4 + 6a2z4 + 6z4a−2−5z4a−4 + 22z4−2a5z3 + 4a3z3 + 13az3 + 13z3a−1 + 4z3a−3−2z3a−5 + 2a4z2−7a2z2−7z2a−2 + 2z2a−4−18z2 + a5z−a3z−5az−5za−1−za−3 + za−5 + 3a2 + 3a−2 + 7

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_81,}

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["10 109"];
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]= { t4−4t3 + 10t2−17t + 21−17t−1 + 10t−2−4t−3 + t−4, −q5 + 3q4−7q3 + 11q2−13q + 15−13q−1 + 11q−2−7q−3 + 3q−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]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {10_81,}

[edit] Vassiliev invariants

V2 and V3: (3, 0)

[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 10 109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-1012345χ
11          1-1
9         2 2
7        51 -4
5       62  4
3      75   -2
1     86    2
-1    68     2
-3   57      -2
-5  26       4
-7 15        -4
-9 2         2
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r = 1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 5 {\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 q15−3q14 + 2q13 + 8q12−20q11 + 6q10 + 40q9−60q8−7q7 + 105q6−98q5−45q4 + 169q3−108q2−87q + 195−87q−1−108q−2 + 169q−3−45q−4−98q−5 + 105q−6−7q−7−60q−8 + 40q−9 + 6q−10−20q−11 + 8q−12 + 2q−13−3q−14 + q−15
3 −q30 + 3q29−2q28−3q27 + q26 + 13q25−7q24−30q23 + 11q22 + 70q21−10q20−133q19−27q18 + 235q17 + 97q16−333q15−237q14 + 421q13 + 429q12−474q11−643q10 + 462q9 + 870q8−412q7−1055q6 + 306q5 + 1216q4−203q3−1289q2 + 57q + 1337 + 57q−1−1289q−2−203q−3 + 1216q−4 + 306q−5−1055q−6−412q−7 + 870q−8 + 462q−9−643q−10−474q−11 + 429q−12 + 421q−13−237q−14−333q−15 + 97q−16 + 235q−17−27q−18−133q−19−10q−20 + 70q−21 + 11q−22−30q−23−7q−24 + 13q−25 + q−26−3q−27−2q−28 + 3q−29−q−30
4 q50−3q49 + 2q48 + 3q47−6q46 + 6q45−12q44 + 13q43 + 19q42−37q41 + 3q40−45q39 + 75q38 + 125q37−109q36−108q35−259q34 + 211q33 + 581q32 + 37q31−351q30−1131q29−41q28 + 1474q27 + 1097q26−23q25−2765q24−1618q23 + 1862q22 + 3157q21 + 1998q20−4013q19−4524q18 + 507q17 + 4939q16 + 5545q15−3595q14−7303q13−2387q12 + 5220q11 + 9051q10−1716q9−8692q8−5391q7 + 4146q6 + 11245q5 + 514q4−8632q3−7516q2 + 2477q + 11939 + 2477q−1−7516q−2−8632q−3 + 514q−4 + 11245q−5 + 4146q−6−5391q−7−8692q−8−1716q−9 + 9051q−10 + 5220q−11−2387q−12−7303q−13−3595q−14 + 5545q−15 + 4939q−16 + 507q−17−4524q−18−4013q−19 + 1998q−20 + 3157q−21 + 1862q−22−1618q−23−2765q−24−23q−25 + 1097q−26 + 1474q−27−41q−28−1131q−29−351q−30 + 37q−31 + 581q−32 + 211q−33−259q−34−108q−35−109q−36 + 125q−37 + 75q−38−45q−39 + 3q−40−37q−41 + 19q−42 + 13q−43−12q−44 + 6q−45−6q−46 + 3q−47 + 2q−48−3q−49 + q−50
5 −q75 + 3q74−2q73−3q72 + 6q71−q70−7q69 + 6q68−2q67−9q66 + 24q65 + 16q64−31q63−29q62−34q61−3q60 + 117q59 + 164q58 + 7q57−240q56−397q55−243q54 + 366q53 + 945q52 + 822q51−266q50−1712q49−2127q48−494q47 + 2443q46 + 4250q45 + 2631q44−2417q43−7114q42−6512q41 + 594q40 + 9625q39 + 12430q38 + 4136q37−10696q36−19448q35−12146q34 + 8453q33 + 26148q32 + 23290q31−2075q30−30685q29−35998q28−8638q27 + 31341q26 + 48438q25 + 22835q24−27631q23−58682q22−38496q21 + 19719q20 + 65298q19 + 53987q18−9004q17−68162q16−67224q15−3092q14 + 67469q13 + 77778q12 + 14812q11−64396q10−84931q9−25496q8 + 59690q7 + 89713q6 + 34344q5−54379q4−91894q3−42017q2 + 48392q + 92885 + 48392q−1−42017q−2−91894q−3−54379q−4 + 34344q−5 + 89713q−6 + 59690q−7−25496q−8−84931q−9−64396q−10 + 14812q−11 + 77778q−12 + 67469q−13−3092q−14−67224q−15−68162q−16−9004q−17 + 53987q−18 + 65298q−19 + 19719q−20−38496q−21−58682q−22−27631q−23 + 22835q−24 + 48438q−25 + 31341q−26−8638q−27−35998q−28−30685q−29−2075q−30 + 23290q−31 + 26148q−32 + 8453q−33−12146q−34−19448q−35−10696q−36 + 4136q−37 + 12430q−38 + 9625q−39 + 594q−40−6512q−41−7114q−42−2417q−43 + 2631q−44 + 4250q−45 + 2443q−46−494q−47−2127q−48−1712q−49−266q−50 + 822q−51 + 945q−52 + 366q−53−243q−54−397q−55−240q−56 + 7q−57 + 164q−58 + 117q−59−3q−60−34q−61−29q−62−31q−63 + 16q−64 + 24q−65−9q−66−2q−67 + 6q−68−7q−69−q−70 + 6q−71−3q−72−2q−73 + 3q−74−q−75
6 q105−3q104 + 2q103 + 3q102−6q101 + q100 + 2q99 + 13q98−17q97−8q96 + 22q95−27q94 + 21q92 + 71q91−34q90−75q89 + 16q88−129q87−36q86 + 126q85 + 400q84 + 145q83−158q82−208q81−865q80−703q79 + 55q78 + 1611q77 + 1858q76 + 1210q75 + 101q74−3327q73−4931q72−3798q71 + 1805q70 + 6774q69 + 9727q68 + 8450q67−2638q66−14630q65−21452q64−13030q63 + 4180q62 + 26230q61 + 40159q60 + 25383q59−9352q58−50139q57−63640q56−43374q55 + 15871q54 + 84539q53 + 105774q52 + 63311q51−36248q50−126670q49−161655q48−88027q47 + 66592q46 + 197978q45 + 224393q44 + 97674q43−106500q42−289269q41−295599q40−95461q39 + 189377q38 + 390494q37 + 345386q36 + 74611q35−302530q34−502861q33−374848q32 + 13010q31 + 434079q30 + 586654q29 + 368325q28−149880q27−585134q26−640685q25−265026q24 + 320369q23 + 705269q22 + 640606q21 + 91283q20−522554q19−789043q18−516704q17 + 130277q16 + 693197q15 + 803318q14 + 306604q13−392185q12−820783q11−673048q10−40674q9 + 619164q8 + 865087q7 + 447497q6−268553q5−795142q4−750697q3−163412q2 + 538093q + 877141 + 538093q−1−163412q−2−750697q−3−795142q−4−268553q−5 + 447497q−6 + 865087q−7 + 619164q−8−40674q−9−673048q−10−820783q−11−392185q−12 + 306604q−13 + 803318q−14 + 693197q−15 + 130277q−16−516704q−17−789043q−18−522554q−19 + 91283q−20 + 640606q−21 + 705269q−22 + 320369q−23−265026q−24−640685q−25−585134q−26−149880q−27 + 368325q−28 + 586654q−29 + 434079q−30 + 13010q−31−374848q−32−502861q−33−302530q−34 + 74611q−35 + 345386q−36 + 390494q−37 + 189377q−38−95461q−39−295599q−40−289269q−41−106500q−42 + 97674q−43 + 224393q−44 + 197978q−45 + 66592q−46−88027q−47−161655q−48−126670q−49−36248q−50 + 63311q−51 + 105774q−52 + 84539q−53 + 15871q−54−43374q−55−63640q−56−50139q−57−9352q−58 + 25383q−59 + 40159q−60 + 26230q−61 + 4180q−62−13030q−63−21452q−64−14630q−65−2638q−66 + 8450q−67 + 9727q−68 + 6774q−69 + 1805q−70−3798q−71−4931q−72−3327q−73 + 101q−74 + 1210q−75 + 1858q−76 + 1611q−77 + 55q−78−703q−79−865q−80−208q−81−158q−82 + 145q−83 + 400q−84 + 126q−85−36q−86−129q−87 + 16q−88−75q−89−34q−90 + 71q−91 + 21q−92−27q−94 + 22q−95−8q−96−17q−97 + 13q−98 + 2q−99 + q−100−6q−101 + 3q−102 + 2q−103−3q−104 + q−105
7 −q140 + 3q139−2q138−3q137 + 6q136−q135−2q134−8q133−2q132 + 27q131−5q130−19q129 + 11q128−6q127−3q126−39q125−21q124 + 123q123 + 54q122−28q121−19q120−113q119−83q118−202q117−127q116 + 423q115 + 523q114 + 433q113 + 187q112−528q111−932q110−1542q109−1445q108 + 481q107 + 2262q106 + 3828q105 + 3942q104 + 1189q103−2557q102−7683q101−10796q100−7503q99 + 356q98 + 12216q97 + 22258q96 + 22340q95 + 11840q94−11241q93−37619q92−50791q91−43098q90−6817q89 + 46720q88 + 90727q87 + 103530q86 + 62047q85−27592q84−127434q83−194314q82−174586q81−54007q80 + 125828q79 + 294953q78 + 350927q77 + 235030q76−29753q75−353701q74−569223q73−535658q72−219825q71 + 291598q70 + 761077q69 + 932844q68 + 663707q67−16844q66−821391q65−1351760q64−1288831q63−532987q62 + 629230q61 + 1660551q60 + 2013990q59 + 1366120q58−89043q57−1713017q56−2697937q55−2404293q54−825147q53 + 1383588q52 + 3167221q51 + 3501745q50 + 2054700q49−619529q48−3275526q47−4475059q46−3455196q45−538986q44 + 2940592q43 + 5154249