10 104

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10_103

10_105

Contents

Image:10 104.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 104's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_104's page at Knotilus!

Visit 10 104's page at the original Knot Atlas!

[edit 10 104 Quick Notes]
[edit 10 104 Further Notes and Views]


[edit] Knot presentations

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


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

Length is 10, width is 3,

Braid index is 3

Image:10 104_ML.gif Image:10 104_AP.gif
[{5, 11}, {7, 12}, {8, 6}, {4, 7}, {3, 5}, {9, 4}, {10, 8}, {11, 9}, {2, 10}, {1, 3}, {12, 2}, {6, 1}]

[edit Notes on presentations of 10 104]


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["10 104"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X16,4,17,3 X18,9,19,10 X14,7,15,8 X20,13,1,14 X8,17,9,18 X10,19,11,20 X12,6,13,5 X4,12,5,11 X2,16,3,15
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -9, 8, -1, 4, -6, 3, -7, 9, -8, 5, -4, 10, -2, 6, -3, 7, -5
In[6]:= DTCode[K]
Out[6]= 6 16 12 14 18 4 20 2 8 10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3:20:20]
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,−1,2,2,−1,2,−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:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.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:10 104_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{5, 11}, {7, 12}, {8, 6}, {4, 7}, {3, 5}, {9, 4}, {10, 8}, {11, 9}, {2, 10}, {1, 3}, {12, 2}, {6, 1}]
In[14]:= Draw[ap]
Image:10 104_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 14.1071
A-Polynomial See Data:10 104/A-polynomial

[edit Notes for 10 104'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 10 104's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−4t3 + 9t2−15t + 19−15t−1 + 9t−2−4t−3 + t−4
Conway polynomial z8 + 4z6 + 5z4 + z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 77, 0 }
Jones polynomial −q5 + 3q4−6q3 + 10q2−12q + 13−12q−1 + 10q−2−6q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−z6a−2 + 6z6−4a2z4−4z4a−2 + 13z4−5a2z2−5z2a−2 + 11z2−a2−a−2 + 3
Kauffman polynomial (db, data sources) 2az9 + 2z9a−1 + 4a2z8 + 5z8a−2 + 9z8 + 4a3z7 + 2az7 + 3z7a−1 + 5z7a−3 + 3a4z6−5a2z6−11z6a−2 + 3z6a−4−22z6 + a5z5−5a3z5−6az5−12z5a−1−11z5a−3 + z5a−5−6a4z4 + 3a2z4 + 12z4a−2−6z4a−4 + 27z4−2a5z3−a3z3 + 4az3 + 13z3a−1 + 8z3a−3−2z3a−5 + 3a4z2−4a2z2−6z2a−2 + 2z2a−4−15z2 + a5z + a3z−2az−4za−1−2za−3 + a2 + a−2 + 3
The A2 invariant −q14 + q12−2q10 + 2q8 + q6−q4 + 3q2−3 + 3q−2−q−4 + q−6 + 2q−8−2q−10 + q−12−q−14
The G2 invariant q80−2q78 + 5q76−8q74 + 8q72−6q70−2q68 + 16q66−29q64 + 41q62−43q60 + 30q58−6q56−36q54 + 82q52−118q50 + 121q48−87q46 + 9q44 + 85q42−164q40 + 201q38−162q36 + 66q34 + 57q32−157q30 + 181q28−122q26 + 15q24 + 99q22−156q20 + 131q18−27q16−104q14 + 206q12−236q10 + 168q8−34q6−123q4 + 244q2−286 + 243q−2−119q−4−35q−6 + 168q−8−234q−10 + 212q−12−111q−14−17q−16 + 126q−18−158q−20 + 111q−22−q−24−113q−26 + 180q−28−166q−30 + 68q−32 + 57q−34−166q−36 + 212q−38−177q−40 + 91q−42 + 12q−44−99q−46 + 137q−48−128q−50 + 84q−52−29q−54−16q−56 + 40q−58−47q−60 + 40q−62−25q−64 + 12q−66 + q−68−7q−70 + 7q−72−7q−74 + 4q−76−2q−78 + q−80
Further Quantum Invariants
Further quantum knot invariants for 10_104.

A1 Invariants.

Weight Invariant
1 −q11 + 2q9−3q7 + 4q5−2q3 + q + q−1−2q−3 + 4q−5−3q−7 + 2q−9−q−11
2 q32−2q30−q28 + 6q26−7q24−5q22 + 20q20−10q18−20q16 + 30q14−30q10 + 20q8 + 12q6−21q4 + 16−20q−4 + 12q−6 + 20q−8−30q−10 + q−12 + 30q−14−21q−16−10q−18 + 21q−20−5q−22−9q−24 + 6q−26−2q−30 + q−32
3 −q63 + 2q61 + q59−2q57−3q55 + 4q53 + 5q51−12q49−9q47 + 23q45 + 25q43−37q41−58q39 + 43q37 + 106q35−25q33−152q31−25q29 + 187q27 + 83q25−184q23−144q21 + 150q19 + 186q17−100q15−196q13 + 42q11 + 182q9 + 15q7−148q5−63q3 + 110q + 105q−1−68q−3−143q−5 + 21q−7 + 176q−9 + 30q−11−195q−13−84q−15 + 195q−17 + 135q−19−162q−21−176q−23 + 106q−25 + 190q−27−44q−29−164q−31−16q−33 + 124q−35 + 46q−37−72q−39−50q−41 + 29q−43 + 36q−45−8q−47−20q−49 + 2q−51 + 8q−53−3q−57 + 2q−61−q−63
4 q104−2q102−q100 + 2q98−q96 + 6q94−4q92−q90 + 6q88−13q86 + 3q84−17q82 + 14q80 + 59q78−2q76−37q74−139q72−39q70 + 201q68 + 216q66 + 68q64−404q62−455q60 + 99q58 + 619q56 + 712q54−286q52−1082q50−708q48 + 488q46 + 1553q44 + 648q42−1031q40−1655q38−494q36 + 1580q34 + 1633q32−98q30−1731q28−1431q26 + 730q24 + 1728q22 + 789q20−995q18−1542q16−151q14 + 1115q12 + 1070q10−204q8−1127q6−653q4 + 471q2 + 1058 + 374q−2−725q−4−1067q−6−73q−8 + 1107q−10 + 991q−12−316q−14−1535q−16−788q−18 + 970q−20 + 1658q−22 + 449q−24−1609q−26−1581q−28 + 226q−30 + 1774q−32 + 1368q−34−824q−36−1739q−38−798q−40 + 938q−42 + 1594q−44 + 274q−46−928q−48−1106q−50−124q−52 + 874q−54 + 645q−56 + 2q−58−577q−60−442q−62 + 123q−64 + 301q−66 + 244q−68−75q−70−200q−72−65q−74 + 21q−76 + 100q−78 + 24q−80−34q−82−16q−84−14q−86 + 18q−88 + 6q−90−6q−92 + q−94−3q−96 + 3q−98−2q−102 + q−104
5 −q155 + 2q153 + q151−2q149 + q147−2q145−6q143 + 7q139 + 4q137 + 12q135 + 10q133−17q131−36q129−32q127−3q125 + 56q123 + 116q121 + 87q119−66q117−240q115−284q113−73q111 + 348q109 + 680q107 + 494q105−287q103−1156q101−1335q99−316q97 + 1447q95 + 2588q93 + 1730q91−1032q89−3792q87−4020q85−680q83 + 4223q81 + 6723q79 + 3881q77−3075q75−8851q73−8085q71−149q69 + 9311q67 + 12280q65 + 5034q63−7480q61−15006q59−10508q57 + 3365q55 + 15430q53 + 15145q51 + 1960q49−13320q47−17683q45−7211q43 + 9314q41 + 17762q39 + 11109q37−4611q35−15671q33−12973q31 + 295q29 + 12259q27 + 12965q25 + 2839q23−8578q21−11604q19−4646q17 + 5353q15 + 9734q13 + 5462q11−2959q9−8065q7−5830q5 + 1317q3 + 6996q + 6351q−1−16q−3−6591q−5−7451q−7−1423q−9 + 6534q−11 + 9215q−13 + 3487q−15−6274q−17−11394q−19−6417q−21 + 5223q−23 + 13382q−25 + 10045q−27−2878q−29−14348q−31−13827q−33−870q−35 + 13628q−37 + 16794q−39 + 5499q−41−10758q−43−17996q−45−10100q−47 + 6084q−49 + 16818q−51 + 13399q−53−605q−55−13207q−57−14458q−59−4393q−61 + 8092q−63 + 13040q−65 + 7534q−67−2798q−69−9589q−71−8378q−73−1344q−75 + 5448q−77 + 7124q−79 + 3503q−81−1776q−83−4729q−85−3816q−87−556q−89 + 2305q−91 + 2921q−93 + 1441q−95−578q−97−1659q−99−1356q−101−269q−103 + 680q−105 + 875q−107 + 431q−109−133q−111−412q−113−323q−115−59q−117 + 155q−119 + 169q−121 + 60q−123−35q−125−62q−127−40q−129 + q−131 + 28q−133 + 13q−135−5q−137−3q−139−2q−141−3q−143 + 2q−145 + 3q−147−3q−149 + 2q−153−q−155
6 q216−2q214−q212 + 2q210−q208 + 2q206 + 2q204 + 10q202−6q200−17q198−3q196−13q194−q192 + 18q190 + 64q188 + 32q186−30q184−46q182−110q180−107q178−16q176 + 221q174 + 288q172 + 191q170 + 22q168−402q166−719q164−663q162 + 120q160 + 965q158 + 1509q156 + 1455q154 + 40q152−2039q150−3613q148−2902q146−184q144 + 3784q142 + 7091q140 + 6335q138 + 887q136−7324q134−12778q132−11978q130−2741q128 + 11913q126 + 22565q124 + 21343q122 + 5211q120−17867q118−36180q116−35286q114−10285q112 + 26717q110 + 54694q108 + 52634q106 + 17382q104−37270q102−77786q100−74642q98−23577q96 + 50689q94 + 102814q92 + 98204q90 + 28659q88−67117q86−130261q84−117497q82−28938q80 + 84770q78 + 155408q76 + 130572q74 + 22279q72−105255q70−171854q68−132481q66−9272q64 + 123887q62 + 178331q60 + 121446q58−11339q56−134837q54−171214q52−99656q50 + 33358q48 + 137568q46 + 150892q44 + 68755q42−50367q40−129519q38−121518q36−36909q34 + 61371q32 + 112138q30 + 86762q28 + 10286q26−64462q24−89710q22−54145q20 + 10755q18 + 61572q16 + 65831q14 + 27076q12−26133q10−56386q8−45327q6−3919q4 + 38752q2 + 51908 + 28086q−2−17504q−4−51727q−6−50221q−8−10989q−10 + 40669q−12 + 67436q−14 + 48458q−16−9260q−18−67262q−20−85269q−22−43241q−24 + 35261q−26 + 98116q−28 + 100004q−30 + 31346q−32−66062q−34−129418q−36−108427q−38−11799q−40 + 100315q−42 + 153682q−44 + 107507q−46−12011q−48−131484q−50−168070q−52−97266q−54 + 37826q−56 + 151378q−58 + 169818q−60 + 81856q−62−58799q−64−158959q−66−159612q−68−62701q−70 + 68653q−72 + 152982q−74 + 141879q−76 + 45428q−78−69353q−80−135727q−82−117868q−84−34108q−86 + 61500q−88 + 112599q−90 + 93187q−92 + 25736q−94−48170q−96−85693q−98−71766q−100−20451q−102 + 34132q−104 + 60737q−106 + 52052q−108 + 17061q−110−20425q−112−40974q−114−35931q−116−13594q−118 + 10413q−120 + 25035q−122 + 23690q−124 + 10983q−126−4816q−128−14087q−130−14363q−132−7978q−134 + 1259q−136 + 7438q−138 + 8505q−140 + 4808q−142 + 131q−144−3439q−146−4580q−148−2905q−150−329q−152 + 1697q−154 + 2045q−156 + 1509q−158 + 338q−160−756q−162−989q−164−644q−166−59q−168 + 225q−170 + 398q−172 + 291q−174 + 5q−176−134q−178−137q−180−47q−182−25q−184 + 44q−186 + 61q−188 + 9q−190−14q−192−16q−194 + 5q−196−10q−198−q−200 + 11q−202−2q−206−3q−208 + 3q−210−2q−214 + q−216

A2 Invariants.

Weight Invariant
1,0 −q14 + q12−2q10 + 2q8 + q6−q4 + 3q2−3 + 3q−2−q−4 + q−6 + 2q−8−2q−10 + q−12−q−14
1,1 q44−4q42 + 12q40−28q38 + 54q36−96q34 + 154q32−236q30 + 341q28−470q26 + 610q24−728q22 + 816q20−832q18 + 740q16−528q14 + 208q12 + 180q10−618q8 + 1040q6−1381q4 + 1618q2−1706 + 1654q−2−1453q−4 + 1130q−6−728q−8 + 288q−10 + 134q−12−476q−14 + 726q−16−856q−18 + 868q−20−788q−22 + 654q−24−510q−26 + 365q−28−238q−30 + 148q−32−90q−34 + 48q−36−22q−38 + 10q−40−4q−42 + q−44
2,0 q38−q36−q34 + 2q32−3q30−4q28 + 5q26 + 3q24−5q22−2q20 + 10q18 + 6q16−14q14 + 2q12 + 10q10−8q8−7q6 + 7q4 + 3q2−8 + 4q−2 + 8q−4−4q−6−6q−8 + 12q−10 + 2q−12−13q−14 + 7q−16 + 9q−18−4q−20−7q−22 + 3q−24 + 4q−26−5q−28−3q−30 + 3q−32−q−36 + q−38

A3 Invariants.

Weight Invariant
0,1,0 q34−2q32 + q30 + 4q28−8q26 + 2q24 + 11q22−17q20 + 2q18 + 19q16−24q14 + q12 + 20q10−16q8−2q6 + 13q4 + q2−4 + 13q−4−3q−6−16q−8 + 19q−10 + q−12−24q−14 + 19q−16 + 2q−18−17q−20 + 12q−22 + 2q−24−7q−26 + 4q−28−2q−32 + q−34
1,0,0 −q17 + q15−3q13 + 3q11−2q9 + 3q7−q5 + 2q3 + 2q−3−q−5 + 3q−7−2q−9 + 3q−11−3q−13 + q−15−q−17
1,0,1 q56−4q54 + 10q52−14q50 + 7q48 + 18q46−57q44 + 82q42−65q40−20q38 + 149q36−257q34 + 246q32−61q30−242q28 + 540q26−622q24 + 402q22 + 63q20−586q18 + 887q16−836q14 + 431q12 + 105q10−503q8 + 600q6−388q4 + 105q2 + 46 + 67q−2−326q−4 + 506q−6−412q−8 + 21q−10 + 497q−12−860q−14 + 894q−16−552q−18 + 18q−20 + 456q−22−679q−24 + 576q−26−275q−28−51q−30 + 247q−32−269q−34 + 170q−36−38q−38−49q−40 + 72q−42−54q−44 + 20q−46 + 4q−48−10q−50 + 8q−52−4q−54 + q−56

A4 Invariants.

Weight Invariant
0,1,0,0 q40−q38 + 4q34−3q32−4q30 + 9q28−3q26−13q24 + 8q22 + 9q20−16q18−7q16 + 15q14−19q10 + 8q8 + 22q6−14q4−q2 + 29−2q−2−17q−4 + 19q−6 + 5q−8−21q−10−3q−12 + 14q−14−6q−16−15q−18 + 10q−20 + 10q−22−10q−24−2q−26 + 9q−28−4q−30−3q−32 + 3q−34−q−36−q−38 + q−40
1,0,0,0 −q20 + q18−3q16 + 2q14−q12 + 2q8−q6 + 3q4−q2 + 3−q−2 + 3q−4−q−6 + 2q−8−q−12 + 2q−14−3q−16 + q−18−q−20

B2 Invariants.

Weight Invariant
0,1 −q34 + 2q32−5q30 + 8q28−12q26 + 18q24−23q22 + 27q20−28q18 + 25q16−18q14 + 9q12 + 4q10−18q8 + 32q6−43q4 + 51q2−54 + 52q−2−43q−4 + 33q−6−18q−8 + 5q−10 + 9q−12−18q−14 + 25q−16−28q−18 + 27q−20−24q−22 + 18q−24−13q−26 + 8q−28−4q−30 + 2q−32−q−34
1,0 q56−2q52−2q50 + 3q48 + 6q46−q44−10q42−7q40 + 10q38 + 17q36−3q34−24q32−11q30 + 22q28 + 24q26−12q24−30q22−3q20 + 28q18 + 13q16−20q14−17q12 + 13q10 + 19q8−6q6−18q4 + 3q2 + 20 + 3q−2−18q−4−6q−6 + 18q−8 + 13q−10−17q−12−20q−14 + 12q−16 + 28q−18−2q−20−30q−22−13q−24 + 24q−26 + 23q−28−11q−30−25q−32−3q−34 + 18q−36 + 10q−38−7q−40−10q−42 + 6q−46 + 2q−48−2q−50−2q−52 + q−56

D4 Invariants.

Weight Invariant
1,0,0,0 q46−2q44 + 3q42−4q40 + 7q38−10q36 + 11q34−14q32 + 19q30−22q28 + 20q26−21q24 + 21q22−18q20 + 8q18−6q16 + 11q12−21q10 + 25q8−28q6 + 42q4−38q2 + 42−38q−2 + 42q−4−29q−6 + 25q−8−22q−10 + 11q−12−q−14−6q−16 + 7q−18−18q−20 + 21q−22−21q−24 + 21q−26−22q−28 + 20q−30−14q−32 + 12q−34−10q−36 + 7q−38−4q−40 + 2q−42−2q−44 + q−46

G2 Invariants.

Weight Invariant
1,0 q80−2q78 + 5q76−8q74 + 8q72−6q70−2q68 + 16q66−29q64 + 41q62−43q60 + 30q58−6q56−36q54 + 82q52−118q50 + 121q48−87q46 + 9q44 + 85q42−164q40 + 201q38−162q36 + 66q34 + 57q32−157q30 + 181q28−122q26 + 15q24 + 99q22−156q20 + 131q18−27q16−104q14 + 206q12−236q10 + 168q8−34q6−123q4 + 244q2−286 + 243q−2−119q−4−35q−6 + 168q−8−234q−10 + 212q−12−111q−14−17q−16 + 126q−18−158q−20 + 111q−22−q−24−113q−26 + 180q−28−166q−30 + 68q−32 + 57q−34−166q−36 + 212q−38−177q−40 + 91q−42 + 12q−44−99q−46 + 137q−48−128q−50 + 84q−52−29q−54−16q−56 + 40q−58−47q−60 + 40q−62−25q−64 + 12q−66 + q−68−7q−70 + 7q−72−7q−74 + 4q−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["10 104"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−4t3 + 9t2−15t + 19−15t−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]= { 77, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 3q4−6q3 + 10q2−12q + 13−12q−1 + 10q−2−6q−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 + 13z4−5a2z2−5z2a−2 + 11z2−a2−a−2 + 3
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2az9 + 2z9a−1 + 4a2z8 + 5z8a−2 + 9z8 + 4a3z7 + 2az7 + 3z7a−1 + 5z7a−3 + 3a4z6−5a2z6−11z6a−2 + 3z6a−4−22z6 + a5z5−5a3z5−6az5−12z5a−1−11z5a−3 + z5a−5−6a4z4 + 3a2z4 + 12z4a−2−6z4a−4 + 27z4−2a5z3−a3z3 + 4az3 + 13z3a−1 + 8z3a−3−2z3a−5 + 3a4z2−4a2z2−6z2a−2 + 2z2a−4−15z2 + a5z + a3z−2az−4za−1−2za−3 + a2 + a−2 + 3

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_71,}

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["10 104"];
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 + 19−15t−1 + 9t−2−4t−3 + t−4, −q5 + 3q4−6q3 + 10q2−12q + 13−12q−1 + 10q−2−6q−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_71,}

[edit] Vassiliev invariants

V2 and V3: (1, 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 10 104. 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        41 -3
5       62  4
3      64   -2
1     76    1
-1    67     1
-3   46      -2
-5  26       4
-7 14        -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r = 1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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 + 7q12−18q11 + 6q10 + 33q9−49q8−5q7 + 84q6−78q5−36q4 + 134q3−86q2−68q + 154−70q−1−84q−2 + 133q−3−37q−4−76q−5 + 83q−6−7q−7−46q−8 + 33q−9 + 3q−10−16q−11 + 8q−12 + q−13−3q−14 + q−15
3 −q30 + 3q29−2q28−3q27 + 2q26 + 11q25−8q24−25q23 + 14q22 + 55q21−15q20−104q19−8q18 + 173q17 + 63q16−244q15−156q14 + 293q13 + 297q12−328q11−438q10 + 307q9 + 594q8−268q7−717q6 + 196q5 + 819q4−122q3−872q2 + 32q + 894 + 51q−1−867q−2−141q−3 + 809q−4 + 214q−5−700q−6−281q−7 + 571q−8 + 310q−9−414q−10−317q−11 + 277q−12 + 270q−13−147q−14−213q−15 + 65q−16 + 143q−17−20q−18−82q−19 + 2q−20 + 42q−21 + q−22−20q−23 + 10q−25−2q−26−3q−27−q−28 + 3q−29−q−30
4 q50−3q49 + 2q48 + 3q47−6q46 + 5q45−10q44 + 14q43 + 15q42−38q41 + 3q40−28q39 + 72q38 + 91q37−117q36−83q35−163q34 + 197q33 + 410q32−60q31−261q30−728q29 + 62q28 + 989q27 + 583q26−32q25−1726q24−920q23 + 1167q22 + 1785q21 + 1288q20−2382q19−2656q18 + 226q17 + 2700q16 + 3480q15−1976q14−4204q13−1581q12 + 2672q11 + 5538q10−767q9−4892q8−3339q7 + 1925q6 + 6757q5 + 540q4−4776q3−4519q2 + 931q + 7099 + 1639q−1−4092q−2−5106q−3−193q−4 + 6625q−5 + 2562q−6−2818q−7−5061q−8−1459q−9 + 5234q−10 + 3109q−11−1034q−12−4122q−13−2457q−14 + 3073q−15 + 2809q−16 + 599q−17−2391q−18−2510q−19 + 999q−20 + 1648q−21 + 1223q−22−712q−23−1605q−24−66q−25 + 452q−26 + 849q−27 + 84q−28−607q−29−159q−30−68q−31 + 295q−32 + 135q−33−135q−34−11q−35−83q−36 + 55q−37 + 35q−38−33q−39 + 24q−40−22q−41 + 10q−42 + 4q−43−13q−44 + 8q−45−3q−46 + 3q−47 + q−48−3q−49 + q−50
5 −q75 + 3q74−2q73−3q72 + 6q71−q70−6q69 + 4q68−3q67−5q66 + 24q65 + 14q64−33q63−37q62−25q61 + 22q60 + 119q59 + 123q58−47q57−251q56−289q55−67q54 + 398q53 + 687q52 + 397q51−446q50−1238q49−1154q48 + 95q47 + 1768q46 + 2416q45 + 1034q44−1854q43−4015q42−3165q41 + 855q40 + 5369q39 + 6313q38 + 1767q37−5691q36−9957q35−6179q34 + 4158q33 + 13104q32 + 12099q31−185q30−14905q29−18664q28−5907q27 + 14355q26 + 24701q25 + 13819q24−11486q23−29398q22−22091q21 + 6459q20 + 31939q19 + 30076q18−191q17−32564q16−36589q15−6498q14 + 31418q13 + 41546q12 + 12732q11−29227q10−44748q9−18138q8 + 26441q7 + 46666q6 + 22493q5−23499q4−47429q3−26095q2 + 20413q + 47526 + 29068q−1−17132q−2−46784q−3−31774q−4 + 13266q−5 + 45291q−6 + 34174q−7−8711q−8−42512q−9−36155q−10 + 3267q−11 + 38333q−12 + 37200q−13 + 2706q−14−32386q−15−36861q−16−8697q−17 + 25065q−18 + 34502q−19 + 13766q−20−16666q−21−30208q−22−17145q−23 + 8540q−24 + 24030q−25 + 18129q−26−1386q−27−17023q−28−16860q−29−3525q−30 + 10157q−31 + 13631q−32 + 6140q−33−4509q−34−9614q−35−6494q−36 + 697q−37 + 5695q−38 + 5374q−39 + 1267q−40−2658q−41−3652q−42−1803q−43 + 792q−44 + 2034q−45 + 1495q−46 + 102q−47−890q−48−945q−49−349q−50 + 271q−51 + 476q−52 + 281q−53−21q−54−164q−55−163q−56−61q−57 + 55q−58 + 70q−59 + 23q−60 + 10q−61−10q−62−32q−63−5q−64 + 11q−65−6q−66 + 6q−67 + 9q−68−5q−69−3q−70 + 3q−71−3q−72−q−73 + 3q−74−q−75
6 q105−3q104 + 2q103 + 3q102−6q101 + q100 + 2q99 + 12q98−15q97−7q96 + 18q95−27q94 + 3q93 + 25q92 + 64q91−32q90−76q89−4q88−117q87 + 6q86 + 164q85 + 350q84 + 75q83−249q82−288q81−702q80−339q79 + 397q78 + 1444q77 + 1246q76 + 287q75−636q74−2728q73−2915q72−1278q71 + 2585q70 + 4816q69 + 4964q68 + 3061q67−3795q66−9094q65−10515q64−3800q63 + 5092q62 + 14235q61 + 18860q60 + 8912q59−7749q58−25137q57−27807q56−17245q55 + 9192q54 + 39409q53 + 46398q52 + 27242q51−16452q50−54412q49−71828q48−42123q47 + 25482q46 + 83921q45 + 101148q44 + 50999q43−35000q42−122927q41−137731q40−58278q39 + 66062q38 + 167522q37 + 165780q36 + 61451q35−111824q34−222060q33−189632q32−30849q31 + 168175q30 + 265940q29 + 202106q28−23862q27−240500q26−303184q25−165941q24 + 97371q23 + 302526q22 + 321579q21 + 95656q20−194325q19−356551q18−278055q17 + 743q16 + 281535q15 + 384935q14 + 193064q13−125667q12−358439q11−340910q10−77759q9 + 239507q8 + 402942q7 + 251066q6−67949q5−339461q4−367677q3−129417q2 + 200275q + 401436 + 285289q−1−22359q−2−315639q−3−380833q−4−172088q−5 + 158867q−6 + 390377q−7 + 315542q−8 + 30850q−9−276884q−10−385092q−11−222905q−12 + 93967q−13 + 354812q−14 + 340790q−15 + 105598q−16−200408q−17−359716q−18−273672q−19−4313q−20 + 270203q−21 + 332789q−22 + 184750q−23−81286q−24−277579q−25−286996q−26−108523q−27 + 137189q−28 + 261231q−29 + 221127q−30 + 42212q−31−144794q−32−230111q−33−162967q−34 + 4030q−35 + 138022q−36 + 181754q−37 + 108811q−38−17260q−39−121818q−40−138131q−41−66608q−42 + 24314q−43 + 93195q−44 + 96047q−45 + 45884q−46−26042q−47−68586q−48−61998q−49−27811q−50 + 18929q−51 + 44982q−52 + 42740q−53 + 14474q−54−13934q−55−26746q−56−25751q−57−9048q−58 + 7980q−59 + 17739q−60 + 13580q−61 + 4379q−62−3668q−63−9619q−64−7826q−65−2672q−66 + 3085q−67 + 4343q−68 + 3579q−69 + 1786q−70−1408q−71−2378q−72−1916q−73−222q−74 + 375q−75 + 861q−76 + 1075q−77 + 166q−78−299q−79−501q−80−168q−81−169q−82 + 16q−83 + 292q−84 + 110q−85 + 18q−86−77q−87 + q−88−72q−89−51q−90 + 55q−91 + 19q−92 + 15q−93−13q−94 + 17q−95−10q−96−19q−97 + 9q−98 + 3q−100−3q−101 + 3q−102 + q−103−3q−104 + q−105
7 −q140 + 3q139−2q138−3q137 + 6q136−q135−2q134−8q133−q132 + 25q131−6q130−15q129 + 11q128−9q127−8q126−34q125−8q124 + 113q123 + 50q122−21q121−31q120−137q119−103q118−149q117−23q116 + 451q115 + 491q114 + 318q113−49q112−740q111−969q110−1121q109−610q108 + 1155q107 + 2364q106 + 2882q105 + 1930q104−968q103−3601q102−6008q101−6031q100−1635q99 + 4304q98 + 10556q97 + 13220q96 + 8636q95−463q94−13353q93−23673q92−23275q91−12545q90 + 9142q89 + 32659q88 + 44420q87 + 39575q86 + 12071q85−29499q84−65205q83−81143q82−59083q81−1221q80 + 68789q79 + 126227q78 + 133437q77 + 75698q76−29758q75−150480q74−222562q73−199339q72−76206q71 + 117808q70 + 292893q69 + 357405q68 + 262171q67 + 9059q66−296859q65−511236q64−514971q63−251172q62 + 184864q61 + 601362q60 + 791184q59 + 601389q58 + 76731q57−566306q56−1025922q55−1016784q54−486598q53 + 362231q52 + 1146031q51 + 1427166q50 + 1007808q49 + 21565q48−1098365q47−1756628q46−1570252q45−552151q44 + 861955q43 + 1939263q42 + 2094478q41 + 1168781q40−457096q39−1946542q38−2511320q37−1790364q36−62409q35 + 1782232q34 + 2777646q33