10 106

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10_105

10_107

Contents

Image:10 106.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 106's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_106's page at Knotilus!

Visit 10 106's page at the original Knot Atlas!

[edit 10 106 Quick Notes]
[edit 10 106 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X16,8,17,7 X10,3,11,4 X2,15,3,16 X14,5,15,6 X4,11,5,12 X18,10,19,9 X20,14,1,13 X8,18,9,17 X12,20,13,19
Gauss code 1, -4, 3, -6, 5, -1, 2, -9, 7, -3, 6, -10, 8, -5, 4, -2, 9, -7, 10, -8
Dowker-Thistlethwaite code 6 10 14 16 18 4 20 2 8 12
Conway Notation [30:2:20]


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

Length is 10, width is 3,

Braid index is 3

Image:10 106_ML.gif Image:10 106_AP.gif
[{9, 12}, {11, 4}, {12, 6}, {5, 3}, {4, 2}, {3, 7}, {6, 10}, {8, 11}, {7, 1}, {2, 9}, {1, 8}, {10, 5}]

[edit Notes on presentations of 10 106]


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["10 106"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X16,8,17,7 X10,3,11,4 X2,15,3,16 X14,5,15,6 X4,11,5,12 X18,10,19,9 X20,14,1,13 X8,18,9,17 X12,20,13,19
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -6, 5, -1, 2, -9, 7, -3, 6, -10, 8, -5, 4, -2, 9, -7, 10, -8
In[6]:= DTCode[K]
Out[6]= 6 10 14 16 18 4 20 2 8 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [30:2:20]
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,−2,1,−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:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.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:10 106_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{9, 12}, {11, 4}, {12, 6}, {5, 3}, {4, 2}, {3, 7}, {6, 10}, {8, 11}, {7, 1}, {2, 9}, {1, 8}, {10, 5}]
In[14]:= Draw[ap]
Image:10 106_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-9]
Hyperbolic Volume 13.933
A-Polynomial See Data:10 106/A-polynomial

[edit Notes for 10 106's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for 10 106's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 4t3−9t2 + 15t−17 + 15t−1−9t−2 + 4t−3−t−4
Conway polynomial −z8−4z6−5z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 75, 2 }
Jones polynomial q7−3q6 + 6q5−10q4 + 12q3−12q2 + 12q−9 + 6q−1−3q−2 + q−3
HOMFLY-PT polynomial (db, data sources) −z8a−2−6z6a−2 + z6a−4 + z6−13z4a−2 + 4z4a−4 + 4z4−11z2a−2 + 5z2a−4 + 5z2−2a−2 + a−4 + 2
Kauffman polynomial (db, data sources) 2z9a−1 + 2z9a−3 + 9z8a−2 + 5z8a−4 + 4z8 + 3az7 + z7a−1 + 4z7a−3 + 6z7a−5 + a2z6−23z6a−2−6z6a−4 + 5z6a−6−11z6−9az5−12z5a−1−13z5a−3−7z5a−5 + 3z5a−7−3a2z4 + 22z4a−2 + 4z4a−4−5z4a−6 + z4a−8 + 9z4 + 7az3 + 9z3a−1 + 8z3a−3 + 3z3a−5−3z3a−7 + 2a2z2−13z2a−2−3z2a−4 + 2z2a−6−z2a−8−5z2−az−2za−1−za−3 + za−5 + za−7 + 2a−2 + a−4 + 2
The A2 invariant q8−q6 + 2q4−q2 + 2q−2−2q−4 + 4q−6−2q−8 + q−10−q−12−2q−14 + 2q−16−q−18 + q−20
The G2 invariant q46−2q44 + 5q42−9q40 + 10q38−10q36 + 2q34 + 16q32−37q30 + 60q28−66q26 + 44q24 + 4q22−72q20 + 133q18−156q16 + 126q14−39q12−72q10 + 165q8−193q6 + 153q4−51q2−64 + 139q−2−149q−4 + 84q−6 + 21q−8−110q−10 + 151q−12−114q−14 + 21q−16 + 89q−18−181q−20 + 214q−22−176q−24 + 69q−26 + 67q−28−185q−30 + 250q−32−225q−34 + 127q−36 + 7q−38−128q−40 + 187q−42−171q−44 + 81q−46 + 34q−48−113q−50 + 129q−52−73q−54−27q−56 + 114q−58−152q−60 + 124q−62−52q−64−41q−66 + 115q−68−146q−70 + 135q−72−79q−74 + 16q−76 + 38q−78−74q−80 + 81q−82−71q−84 + 50q−86−19q−88−6q−90 + 24q−92−31q−94 + 28q−96−20q−98 + 11q−100−2q−102−4q−104 + 5q−106−6q−108 + 4q−110−2q−112 + q−114
Further Quantum Invariants
Further quantum knot invariants for 10_106.

A1 Invariants.

Weight Invariant
1 q7−2q5 + 3q3−3q + 3q−1 + 2q−7−4q−9 + 3q−11−2q−13 + q−15
2 q22−2q20−q18 + 8q16−5q14−12q12 + 18q10 + 4q8−26q6 + 15q4 + 17q2−27 + 3q−2 + 22q−4−13q−6−9q−8 + 15q−10 + 6q−12−17q−14−3q−16 + 24q−18−15q−20−18q−22 + 29q−24−4q−26−19q−28 + 15q−30 + q−32−8q−34 + 5q−36−2q−40 + q−42
3 q45−2q43−q41 + 4q39 + 5q37−8q35−17q33 + 10q31 + 37q29 + 3q27−59q25−37q23 + 71q21 + 83q19−55q17−130q15 + 12q13 + 160q11 + 45q9−159q7−102q5 + 131q3 + 147q−94q−1−164q−3 + 48q−5 + 169q−7−6q−9−156q−11−26q−13 + 138q−15 + 59q−17−109q−19−92q−21 + 67q−23 + 126q−25−21q−27−149q−29−45q−31 + 157q−33 + 103q−35−133q−37−151q−39 + 96q−41 + 167q−43−46q−45−148q−47−q−49 + 113q−51 + 20q−53−68q−55−21q−57 + 33q−59 + 13q−61−15q−63−5q−65 + 9q−67−q−69−4q−71 + q−73 + 2q−75−2q−79 + q−81
4 q76−2q74−q72 + 4q70 + q68 + 2q66−14q64−11q62 + 19q60 + 26q58 + 33q56−47q54−96q52−27q50 + 75q48 + 210q46 + 62q44−199q42−302q40−152q38 + 366q36 + 495q34 + 130q32−467q30−778q28−82q26 + 697q24 + 903q22 + 123q20−1048q18−978q16 + 37q14 + 1221q12 + 1121q10−406q8−1359q6−971q4 + 666q2 + 1556 + 524q−2−942q−4−1426q−6−108q−8 + 1268q−10 + 983q−12−352q−14−1274q−16−500q−18 + 798q−20 + 996q−22 + 7q−24−974q−26−667q−28 + 398q−30 + 979q−32 + 360q−34−664q−36−944q−38−179q−40 + 922q−42 + 953q−44−23q−46−1153q−48−1068q−50 + 408q−52 + 1389q−54 + 972q−56−734q−58−1666q−60−549q−62 + 1014q−64 + 1564q−66 + 197q−68−1288q−70−1082q−72 + 96q−74 + 1161q−76 + 706q−78−397q−80−743q−82−382q−84 + 385q−86 + 465q−88 + 61q−90−191q−92−255q−94 + 23q−96 + 120q−98 + 56q−100 + 12q−102−68q−104−6q−106 + 7q−108−q−110 + 18q−112−10q−114 + 4q−116−q−118−6q−120 + 5q−122−2q−124 + 2q−126−2q−130 + q−132
5 q115−2q113−q111 + 4q109 + q107−2q105−4q103−8q101−3q99 + 22q97 + 33q95 + 6q93−37q91−81q89−74q87 + 25q85 + 178q83 + 231q81 + 85q79−213q77−484q75−440q73 + 36q71 + 708q69 + 1018q67 + 561q65−545q63−1597q61−1657q59−330q57 + 1698q55 + 2876q53 + 2017q51−708q49−3548q47−4170q45−1535q43 + 2901q41 + 5852q39 + 4665q37−481q35−6115q33−7726q31−3350q29 + 4339q27 + 9500q25 + 7639q23−607q21−9216q19−11152q17−4163q15 + 6760q13 + 12846q11 + 8763q9−2714q7−12414q5−12140q3−1758q + 10187q−1 + 13621q−3 + 5673q−5−6931q−7−13374q−9−8268q−11 + 3640q−13 + 11804q−15 + 9362q−17−934q−19−9663q−21−9256q−23−771q−25 + 7594q−27 + 8418q−29 + 1607q−31−6002q−33−7460q−35−1929q−37 + 5004q−39 + 6855q−41 + 2218q−43−4368q−45−6831q−47−3018q−49 + 3666q−51 + 7297q−53 + 4619q−55−2340q−57−7850q−59−7018q−61 + 8q−63 + 7779q−65 + 9759q−67 + 3538q−69−6516q−71−12052q−73−7759q−75 + 3621q−77 + 12924q−79 + 11916q−81 + 612q−83−11842q−85−14738q−87−5337q−89 + 8656q−91 + 15445q−93 + 9364q−95−4217q−97−13764q−99−11559q−101−274q−103 + 10203q−105 + 11546q−107 + 3667q−109−6029q−111−9622q−113−5228q−115 + 2330q−117 + 6697q−119 + 5152q−121 + 103q−123−3846q−125−4005q−127−1180q−129 + 1713q−131 + 2546q−133 + 1313q−135−473q−137−1354q−139−989q−141−35q−143 + 586q−145 + 567q−147 + 176q−149−189q−151−286q−153−142q−155 + 45q−157 + 109q−159 + 75q−161 + 13q−163−34q−165−40q−167−14q−169 + 14q−171 + 10q−173 + 4q−175 + 5q−177−6q−179−6q−181 + 3q−183 + 2q−185−2q−187 + 2q−189−2q−193 + q−195
6 q162−2q160−q158 + 4q156 + q154−2q152−8q150 + 2q148 + 28q142 + 20q140−7q138−60q136−58q134−47q132−q130 + 151q128 + 228q126 + 184q124−76q122−314q120−558q118−560q116−23q114 + 716q112 + 1351q110 + 1216q108 + 463q106−1114q104−2678q102−2921q100−1456q98 + 1537q96 + 4345q94 + 5854q92 + 3881q90−1166q88−6867q86−10071q84−7998q82−978q80 + 9250q78 + 16021q76 + 14916q74 + 4752q72−10457q70−22829q68−24985q66−11829q64 + 10246q62 + 30703q60 + 36804q58 + 22735q56−7299q54−38783q52−51318q50−36410q48 + 2519q46 + 45216q44 + 67760q42 + 52791q40 + 3889q38−51963q36−84154q34−69232q32−11890q30 + 59181q28 + 100583q26 + 84373q24 + 17068q22−66349q20−114780q18−97115q16−17973q14 + 75300q12 + 126173q10 + 102446q8 + 14232q6−84737q4−134038q2−99524−3780q−2 + 94336q−4 + 133757q−6 + 88598q−8−11232q−10−102497q−12−125121q−14−68754q−16 + 28975q−18 + 103992q−20 + 108501q−22 + 43447q−24−45877q−26−98057q−28−83655q−30−15916q−32 + 56151q−34 + 84595q−36 + 54398q−38−9259q−40−58910q−42−63956q−44−24327q−46 + 27408q−48 + 54507q−50 + 39931q−52−2396q−54−38911q−56−44649q−58−16038q−60 + 23464q−62 + 45468q−64 + 33221q−66−4535q−68−40860q−70−50087q−72−23278q−74 + 21846q−76 + 56889q−78 + 55912q−80 + 16776q−82−38527q−84−74395q−86−64416q−88−11493q−90 + 56103q−92 + 94238q−94 + 75439q−96 + 4528q−98−76288q−100−115181q−102−84044q−104 + 4965q−106 + 97437q−108 + 134564q−110 + 86668q−112−18467q−114−116520q−116−145346q−118−82483q−120 + 33295q−122 + 130209q−124 + 144654q−126 + 70030q−128−46524q−130−132155q−132−133183q−134−53004q−136 + 55728q−138 + 122461q−140 + 111514q−142 + 34764q−144−56323q−146−104450q−148−85533q−150−18001q−152 + 50216q−154 + 80520q−156 + 59713q−158 + 7170q−160−40695q−162−56810q−164−37106q−166−968q−168 + 29071q−170 + 36496q−172 + 21387q−174−2257q−176−18999q−178−20775q−180−11148q−182 + 2568q−184 + 11330q−186 + 11137q−188 + 4840q−190−2223q−192−5788q−194−5470q−196−2000q−198 + 1586q−200 + 2961q−202 + 2247q−204 + 578q−206−724q−208−1423q−210−964q−212−43q−214 + 454q−216 + 527q−218 + 298q−220 + 61q−222−250q−224−254q−226−59q−228 + 34q−230 + 75q−232 + 64q−234 + 60q−236−36q−238−51q−240−7q−242−q−244 + 6q−246 + 2q−248 + 19q−250−5q−252−11q−254 + 3q−256 + 2q−260−2q−262 + 2q−264−2q−268 + q−270

A2 Invariants.

Weight Invariant
1,0 q8−q6 + 2q4−q2 + 2q−2−2q−4 + 4q−6−2q−8 + q−10−q−12−2q−14 + 2q−16−q−18 + q−20
1,1 q28−4q26 + 12q24−30q22 + 66q20−126q18 + 222q16−344q14 + 481q12−624q10 + 730q8−762q6 + 707q4−554q2 + 308 + 22q−2−377q−4 + 724q−6−1024q−8 + 1252q−10−1374q−12 + 1372q−14−1256q−16 + 1030q−18−728q−20 + 394q−22−60q−24−226q−26 + 442q−28−564q−30 + 600q−32−588q−34 + 532q−36−446q−38 + 358q−40−282q−42 + 213q−44−150q−46 + 102q−48−66q−50 + 38q−52−20q−54 + 10q−56−4q−58 + q−60
2,0 q24−q22−q20 + 4q18 + q16−6q14 + 7q10−8q6 + 2q4 + 10q2−6−8q−2 + 10q−4−6q−8 + 4q−10 + 7q−12−3q−14−3q−16 + 10q−18−q−20−12q−22 + 2q−24 + 9q−26−9q−28−4q−30 + 9q−32 + 2q−34−4q−36−3q−38 + 4q−40−3q−44 + 2q−46−q−50 + q−52

A3 Invariants.

Weight Invariant
0,1,0 q20−2q18 + q16 + 3q14−8q12 + 6q10 + 6q8−15q6 + 14q4 + 7q2−19 + 14q−2 + 8q−4−18q−6 + 6q−8 + 9q−10−7q−12−4q−14 + 2q−16 + 7q−18−11q−20−6q−22 + 21q−24−12q−26−9q−28 + 23q−30−9q−32−10q−34 + 14q−36−3q−38−7q−40 + 5q−42−2q−46 + q−48
1,0,0 q9−q7 + 3q5−2q3 + 3q−2q−1 + 2q−3−q−5 + q−7 + q−9−q−11 + q−13−3q−15 + 2q−17−3q−19 + 3q−21−q−23 + q−25
1,0,1 q34−4q32 + 10q30−16q28 + 13q26 + 12q24−60q22 + 117q20−130q18 + 69q16 + 82q14−280q12 + 414q10−408q8 + 212q6 + 130q4−473q2 + 684−630q−2 + 364q−4 + 17q−6−321q−8 + 420q−10−333q−12 + 152q−14−46q−16 + 88q−18−240q−20 + 363q−22−309q−24 + 47q−26 + 322q−28−600q−30 + 648q−32−420q−34 + 39q−36 + 320q−38−493q−40 + 411q−42−167q−44−105q−46 + 261q−48−244q−50 + 111q−52 + 38q−54−125q−56 + 125q−58−63q−60−5q−62 + 43q−64−45q−66 + 24q−68−2q−70−8q−72 + 8q−74−4q−76 + q−78

A4 Invariants.

Weight Invariant
0,1,0,0 q22−q20 + 3q16−4q14−q12 + 6q10−4q8−4q6 + 13q4 + 2q2−9 + 8q−2 + 14q−4−8q−6−13q−8 + 15q−10 + 4q−12−22q−14 + 2q−16 + 19q−18−19q−20−8q−22 + 18q−24−5q−26−15q−28 + 11q−30 + 13q−32−11q−34−2q−36 + 16q−38 + q−40−15q−42 + 5q−44 + 7q−46−8q−48−2q−50 + 4q−52−q−54−q−56 + q−58
1,0,0,0 q10−q8 + 3q6−q4 + 2q2 + 1−q−2 + 2q−4−2q−6 + 2q−8−2q−10 + 2q−12−2q−14 + q−16−2q−18 + q−22−2q−24 + 3q−26−q−28 + q−30

B2 Invariants.

Weight Invariant
0,1 q20−2q18 + 5q16−9q14 + 14q12−20q10 + 24q8−25q6 + 26q4−21q2 + 15−4q−2−8q−4 + 22q−6−34q−8 + 43q−10−49q−12 + 50q−14−46q−16 + 37q−18−25q−20 + 14q−22−q−24−10q−26 + 19q−28−25q−30 + 25q−32−24q−34 + 20q−36−15q−38 + 11q−40−7q−42 + 4q−44−2q−46 + q−48
1,0 q34−2q30−2q28 + 3q26 + 6q24−2q22−11q20−4q18 + 15q16 + 14q14−12q12−22q10 + 2q8 + 28q6 + 12q4−23q2−22 + 12q−2 + 26q−4 + q−6−23q−8−8q−10 + 17q−12 + 12q−14−12q−16−12q−18 + 10q−20 + 15q−22−7q−24−19q−26 + 2q−28 + 20q−30 + 2q−32−22q−34−11q−36 + 20q−38 + 19q−40−13q−42−25q−44 + 3q−46 + 27q−48 + 10q−50−17q−52−18q−54 + 5q−56 + 17q−58 + 5q−60−9q−62−9q−64 + q−66 + 6q−68 + 2q−70−2q−72−2q−74 + q−78

D4 Invariants.

Weight Invariant
1,0,0,0 q26−2q24 + 3q22−5q20 + 8q18−12q16 + 14q14−17q12 + 21q10−20q8 + 21q6−17q4 + 20q2−10 + 6q−2 + 2q−4−6q−6 + 14q−8−25q−10 + 28q−12−33q−14 + 36q−16−41q−18 + 36q−20−35q−22 + 33q−24−27q−26 + 17q−28−12q−30 + 8q−32 + 4q−34−9q−36 + 12q−38−16q−40 + 23q−42−19q−44 + 16q−46−18q−48 + 17q−50−11q−52 + 8q−54−9q−56 + 6q−58−3q−60 + 2q−62−2q−64 + q−66

G2 Invariants.

Weight Invariant
1,0 q46−2q44 + 5q42−9q40 + 10q38−10q36 + 2q34 + 16q32−37q30 + 60q28−66q26 + 44q24 + 4q22−72q20 + 133q18−156q16 + 126q14−39q12−72q10 + 165q8−193q6 + 153q4−51q2−64 + 139q−2−149q−4 + 84q−6 + 21q−8−110q−10 + 151q−12−114q−14 + 21q−16 + 89q−18−181q−20 + 214q−22−176q−24 + 69q−26 + 67q−28−185q−30 + 250q−32−225q−34 + 127q−36 + 7q−38−128q−40 + 187q−42−171q−44 + 81q−46 + 34q−48−113q−50 + 129q−52−73q−54−27q−56 + 114q−58−152q−60 + 124q−62−52q−64−41q−66 + 115q−68−146q−70 + 135q−72−79q−74 + 16q−76 + 38q−78−74q−80 + 81q−82−71q−84 + 50q−86−19q−88−6q−90 + 24q−92−31q−94 + 28q−96−20q−98 + 11q−100−2q−102−4q−104 + 5q−106−6q−108 + 4q−110−2q−112 + q−114

.

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["10 106"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 4t3−9t2 + 15t−17 + 15t−1−9t−2 + 4t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−4z6−5z4−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]= { 75, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q7−3q6 + 6q5−10q4 + 12q3−12q2 + 12q−9 + 6q−1−3q−2 + q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−6z6a−2 + z6a−4 + z6−13z4a−2 + 4z4a−4 + 4z4−11z2a−2 + 5z2a−4 + 5z2−2a−2 + a−4 + 2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z9a−1 + 2z9a−3 + 9z8a−2 + 5z8a−4 + 4z8 + 3az7 + z7a−1 + 4z7a−3 + 6z7a−5 + a2z6−23z6a−2−6z6a−4 + 5z6a−6−11z6−9az5−12z5a−1−13z5a−3−7z5a−5 + 3z5a−7−3a2z4 + 22z4a−2 + 4z4a−4−5z4a−6 + z4a−8 + 9z4 + 7az3 + 9z3a−1 + 8z3a−3 + 3z3a−5−3z3a−7 + 2a2z2−13z2a−2−3z2a−4 + 2z2a−6−z2a−8−5z2−az−2za−1−za−3 + za−5 + za−7 + 2a−2 + a−4 + 2

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_59,}

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["10 106"];
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 + 4t3−9t2 + 15t−17 + 15t−1−9t−2 + 4t−3−t−4, q7−3q6 + 6q5−10q4 + 12q3−12q2 + 12q−9 + 6q−1−3q−2 + q−3 }
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_59,}

[edit] Vassiliev invariants

V2 and V3: (-1, -1)

[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 10 106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-10123456χ
15          11
13         2 -2
11        41 3
9       62  -4
7      64   2
5     66    0
3    66     0
1   47      3
-1  25       -3
-3 14        3
-5 2         -2
-71          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −4 {\mathbb Z}
r = −3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r = 1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 6 {\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 q20−3q19 + 2q18 + 6q17−16q16 + 11q15 + 20q14−50q13 + 26q12 + 53q11−97q10 + 29q9 + 92q8−124q7 + 15q6 + 115q5−115q4−9q3 + 111q2−80q−28 + 81q−1−36q−2−30q−3 + 40q−4−6q−5−16q−6 + 10q−7 + q−8−3q−9 + q−10
3 q39−3q38 + 2q37 + 2q36−8q34 + 5q33 + 12q32−14q31−18q30 + 33q29 + 32q28−68q27−65q26 + 121q25 + 125q24−182q23−212q22 + 223q21 + 338q20−253q19−459q18 + 241q17 + 574q16−199q15−661q14 + 137q13 + 702q12−52q11−720q10−22q9 + 685q8 + 116q7−641q6−186q5 + 555q4 + 266q3−466q2−307q + 343 + 336q−1−225q−2−323q−3 + 110q−4 + 279q−5−21q−6−208q−7−38q−8 + 137q−9 + 54q−10−70q−11−50q−12 + 29q−13 + 32q−14−8q−15−16q−16 + 2q−17 + 5q−18 + q−19−3q−20 + q−21
4 q64−3q63 + 2q62 + 2q61−4q60 + 8q59−14q58 + 7q57 + 7q56−18q55 + 36q54−33q53 + 15q52−6q51−80q50 + 116q49 + 11q48 + 79q47−103q46−358q45 + 180q44 + 263q43 + 483q42−183q41−1125q40−181q39 + 609q38 + 1586q37 + 272q36−2190q35−1359q34 + 403q33 + 3071q32 + 1639q31−2740q30−2922q29−714q28 + 4003q27 + 3345q26−2323q25−3903q24−2190q23 + 3918q22 + 4475q21−1347q20−3934q19−3291q18 + 3153q17 + 4755q16−323q15−3315q14−3872q13 + 2088q12 + 4448q11 + 658q10−2326q9−4070q8 + 790q7 + 3674q6 + 1580q5−991q4−3785q3−586q2 + 2356q + 2064 + 475q−1−2753q−2−1476q−3 + 719q−4 + 1676q−5 + 1428q−6−1226q−7−1376q−8−465q−9 + 661q−10 + 1358q−11−55q−12−596q−13−671q−14−118q−15 + 662q−16 + 256q−17 + q−18−306q−19−247q−20 + 144q−21 + 106q−22 + 104q−23−45q−24−99q−25 + 9q−26 + 4q−27 + 35q−28 + 4q−29−19q−30 + 2q−31−3q−32 + 5q−33 + q−34−3q−35 + q−36
5 q95−3q94 + 2q93 + 2q92−4q91 + 4q90 + 2q89−12q88 + 2q87 + 13q86−5q85 + 10q84 + 6q83−40q82−24q81 + 19q80 + 42q79 + 72q78 + 40q77−104q76−211q75−125q74 + 139q73 + 437q72 + 431q71−85q70−832q69−1079q68−226q67 + 1318q66 + 2217q65 + 1148q64−1665q63−3972q62−3051q61 + 1477q60 + 6166q59 + 6197q58−120q57−8339q56−10609q55−2917q54 + 9759q53 + 15893q52 + 7759q51−9682q50−21086q49−14202q48 + 7554q47 + 25440q46 + 21340q45−3601q44−27875q43−28195q42−1847q41 + 28336q40 + 33794q39 + 7703q38−26867q37−37498q36−13227q35 + 24043q34 + 39330q33 + 17757q32−20646q31−39478q30−20998q29 + 17017q28 + 38498q27 + 23267q26−13687q25−36800q24−24629q23 + 10333q22 + 34685q21 + 25730q20−7101q19−32163q18−26480q17 + 3400q16 + 29154q15 + 27188q14 + 508q13−25352q12−27304q11−4965q10 + 20669q9 + 26781q8 + 9237q7−15056q6−24862q5−13129q4 + 8761q3 + 21675q2 + 15680q−2452−16914q−1−16563q−2−3184q−3 + 11293q−4 + 15406q−5 + 7248q−6−5437q−7−12480q−8−9270q−9 + 370q−10 + 8417q−11 + 9184q−12 + 3172q−13−4234q−14−7409q−15−4791q−16 + 728q−17 + 4808q−18 + 4783q−19 + 1400q−20−2263q−21−3604q−22−2223q−23 + 372q−24 + 2148q−25 + 2022q−26 + 577q−27−879q−28−1364q−29−806q−30 + 120q−31 + 695q−32 + 637q−33 + 173q−34−258q−35−349q−36−190q−37 + 23q−38 + 161q−39 + 129q−40 + 13q−41−51q−42−44q−43−30q−44 + 8q−45 + 30q−46 + 6q−47−7q−48−q−49−3q−50−3q−51 + 5q−52 + q−53−3q−54 + q−55
6 q132−3q131 + 2q130 + 2q129−4q128 + 4q127−2q126 + 4q125−17q124 + 8q123 + 26q122−21q121 + 8q120−9q119−2q118−61q117 + 23q116 + 122q115−17q114 + 19q113−50q112−95q111−256q110 + 27q109 + 433q108 + 220q107 + 248q106−123q105−592q104−1177q103−432q102 + 1132q101 + 1522q100 + 1917q99 + 591q98−1967q97−4763q96−3902q95 + 814q94 + 5087q93 + 8980q92 + 6888q91−1774q90−13525q89−17618q88−8813q87 + 6863q86 + 25722q85 + 30532q84 + 13335q83−20950q82−47657q81−44951q80−12841q79 + 41837q78 + 78397q77 + 65878q76−143q75−77961q74−113168q73−80373q72 + 22920q71 + 126524q70 + 156965q69 + 76607q68−67014q67−179901q66−189105q65−57259q64 + 127552q63 + 242596q62 + 193161q61 + 7610q60−194346q59−286019q58−173037q57 + 64689q56 + 271422q55 + 291214q54 + 112745q53−146450q52−323146q51−265509q50−24320q49 + 240285q48 + 330107q47 + 193561q46−75539q45−304347q44−303644q43−91916q42 + 187362q41 + 320128q40 + 229429q39−20236q38−265211q37−302667q36−126959q35 + 142238q34 + 293319q33 + 238656q32 + 16089q31−227455q30−290420q29−148963q28 + 102736q27 + 264708q26 + 244394q25 + 52604q24−185128q23−275844q22−176062q21 + 51001q20 + 225079q19 + 249440q18 + 102255q17−121471q16−245647q15−204506q14−21066q13 + 157340q12 + 235038q11 + 154435q10−32147q9−180593q8−209015q7−96083q6 + 59611q5 + 178671q4 + 177059q3 + 59118q2−80763q−163856−135504q−1−38505q−2 + 82927q−3 + 142545q−4 + 108419q−5 + 18206q−6−75642q−7−111777q−8−89378q−9−10346q−10 + 63403q−11 + 90754q−12 + 66637q−13 + 7775q−14−44472q−15−73168q−16−51748q−17−7668q−18 + 33412q−19 + 51715q−20 + 39966q−21 + 11380q−22−24266q−23−36779q−24−30212q−25−9285q−26 + 12786q−27 + 25058q−28 + 23915q−29 + 7218q−30−6745q−31−16143q−32−15386q−33−7671q−34 + 2983q−35 + 10759q−36 + 9374q−37 + 5627q−38−934q−39−5222q−40−6566q−41−3788q−42 + 531q−43 + 2354q−44 + 3554q−45 + 2270q−46 + 479q−47−1519q−48−1815q−49−978q−50−454q−51 + 561q−52 + 805q−53 + 722q−54 + 45q−55−238q−56−225q−57−319q−58−74q−59 + 66q−60 + 185q−61 + 47q−62 + 6q−63 + 13q−64−59q−65−30q−66−11q−67 + 33q−68 + q−69−5q−70 + 11q−71−6q−72−3q−73−3q−74 + 5q−75 + q−76−3q−77 + q−78
7 q175−3q174 + 2q173 + 2q172−4q171 + 4q170−2q169−q167−11q166 + 21q165 + 10q164−23q163 + 3q162−12q161 + q160−q159−33q158 + 92q157 + 62q156−62q155−45q154−107q153−25q152 + 7q151−23q150 + 323q149 + 293q148−57q147−245q146−606q145−414q144−55q143 + 252q142 + 1291q141 + 1389q140 + 472q139−824q138−2773q137−3006q136−1609q135 + 963q134 + 5268q133 + 7195q132 + 5297q131−389q130−9640q129−15347q128−13838q127−3792q126 + 14530q125 + 29634q124 + 32602q123 + 17333q122−16781q121−51260q120−67228q119−49181q118 + 7271q117 + 76035q116 + 122556q115 + 112686q114 + 29786q113−92961q112−197895q111−219105q110−115150q109 + 79590q108 + 280008q107 + 372645q106 + 270836q105−5088q104−342023q103−562357q102−507743q101−161749q100</