10 57

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10_56

10_58

Contents

Image:10 57.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 57's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_57's page at Knotilus!

Visit 10 57's page at the original Knot Atlas!

[edit 10 57 Quick Notes]
[edit 10 57 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X3849 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X7283
Gauss code -1, 10, -2, 1, -4, 5, -10, 2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code 4 8 12 2 14 18 6 20 10 16
Conway Notation [221,21,2]


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

Length is 11, width is 4,

Braid index is 4

Image:10 57_ML.gif Image:10 57_AP.gif
[{12, 2}, {1, 10}, {9, 11}, {10, 12}, {11, 14}, {3, 13}, {2, 9}, {8, 4}, {7, 3}, {5, 8}, {14, 7}, {4, 6}, {13, 5}, {6, 1}]

[edit Notes on presentations of 10 57]


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 57"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3849 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X7283
In[5]:= GaussCode[K]
Out[5]= -1, 10, -2, 1, -4, 5, -10, 2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7
In[6]:= DTCode[K]
Out[6]= 4 8 12 2 14 18 6 20 10 16

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [221,21,2]
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(4,{1,1,1,2,−1,2,−3,2,2,−3,−3})
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]= { 4, 11, 4 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.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 57_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{12, 2}, {1, 10}, {9, 11}, {10, 12}, {11, 14}, {3, 13}, {2, 9}, {8, 4}, {7, 3}, {5, 8}, {14, 7}, {4, 6}, {13, 5}, {6, 1}]
In[14]:= Draw[ap]
Image:10 57_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 13.5886
A-Polynomial See Data:10 57/A-polynomial

[edit Notes for 10 57's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for 10 57's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 2t3−8t2 + 18t−23 + 18t−1−8t−2 + 2t−3
Conway polynomial 2z6 + 4z4 + 4z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 79, 2 }
Jones polynomial −q8 + 3q7−7q6 + 10q5−12q4 + 14q3−12q2 + 10q−6 + 3q−1−q−2
HOMFLY-PT polynomial (db, data sources) z6a−2 + z6a−4 + 3z4a−2 + 3z4a−4−z4a−6−z4 + 4z2a−2 + 4z2a−4−2z2a−6−2z2 + 2a−2 + 2a−4−2a−6−1
Kauffman polynomial (db, data sources) z9a−3 + z9a−5 + 3z8a−2 + 7z8a−4 + 4z8a−6 + 4z7a−1 + 9z7a−3 + 10z7a−5 + 5z7a−7 + 2z6a−2−7z6a−4−3z6a−6 + 3z6a−8 + 3z6 + az5−5z5a−1−19z5a−3−23z5a−5−9z5a−7 + z5a−9−11z4a−2−z4a−4−z4a−6−5z4a−8−6z4−2az3 + 12z3a−3 + 18z3a−5 + 6z3a−7−2z3a−9 + 8z2a−2−2z2a−6 + 2z2a−8 + 4z2 + az + za−1−2za−3−6za−5−3za−7 + za−9−2a−2 + 2a−4 + 2a−6−1
The A2 invariant −q6 + q4−q2−1 + 3q−2−2q−4 + 3q−6 + q−8 + q−10 + 3q−12−2q−14 + 2q−16−2q−18−2q−20 + q−22−q−24
The G2 invariant q32−2q30 + 5q28−8q26 + 8q24−7q22−2q20 + 16q18−32q16 + 46q14−51q12 + 35q10−4q8−46q6 + 99q4−131q2 + 132−89q−2 + 5q−4 + 95q−6−177q−8 + 210q−10−173q−12 + 77q−14 + 42q−16−142q−18 + 181q−20−139q−22 + 45q−24 + 70q−26−141q−28 + 131q−30−45q−32−84q−34 + 200q−36−242q−38 + 197q−40−59q−42−109q−44 + 259q−46−323q−48 + 286q−50−155q−52−17q−54 + 169q−56−248q−58 + 240q−60−143q−62 + 13q−64 + 100q−66−156q−68 + 120q−70−25q−72−91q−74 + 166q−76−168q−78 + 91q−80 + 31q−82−152q−84 + 220q−86−214q−88 + 134q−90−21q−92−92q−94 + 155q−96−160q−98 + 122q−100−55q−102−6q−104 + 46q−106−64q−108 + 54q−110−34q−112 + 15q−114 + q−116−8q−118 + 9q−120−8q−122 + 5q−124−2q−126 + q−128
Further Quantum Invariants
Further quantum knot invariants for 10_57.

A1 Invariants.

Weight Invariant
1 −q5 + 2q3−3q + 4q−1−2q−3 + 2q−5 + 2q−7−2q−9 + 3q−11−4q−13 + 2q−15−q−17
2 q16−2q14−q12 + 7q10−7q8−7q6 + 20q4−10q2−20 + 32q−2−2q−4−30q−6 + 25q−8 + 11q−10−22q−12 + 3q−14 + 16q−16−q−18−21q−20 + 12q−22 + 19q−24−32q−26 + 2q−28 + 30q−30−25q−32−8q−34 + 24q−36−9q−38−9q−40 + 8q−42−2q−46 + q−48
3 −q33 + 2q31 + q29−3q27−4q25 + 7q23 + 10q21−14q19−20q17 + 20q15 + 39q13−26q11−69q9 + 25q7 + 108q5−11q3−147q−25q−1 + 183q−3 + 68q−5−190q−7−121q−9 + 178q−11 + 165q−13−136q−15−185q−17 + 84q−19 + 182q−21−18q−23−162q−25−39q−27 + 126q−29 + 92q−31−80q−33−142q−35 + 34q−37 + 173q−39 + 18q−41−201q−43−65q−45 + 198q−47 + 117q−49−182q−51−153q−53 + 141q−55 + 174q−57−90q−59−167q−61 + 34q−63 + 142q−65 + 7q−67−104q−69−25q−71 + 59q−73 + 32q−75−28q−77−23q−79 + 10q−81 + 11q−83−2q−85−4q−87 + 2q−91−q−93
4 q56−2q54−q52 + 3q50 + 4q46−10q44−5q42 + 15q40 + 5q38 + 12q36−41q34−29q32 + 49q30 + 48q28 + 46q26−127q24−136q22 + 78q20 + 194q18 + 219q16−235q14−440q12−70q10 + 403q8 + 688q6−124q4−862q2−607 + 353q−2 + 1321q−4 + 458q−6−974q−8−1342q−10−221q−12 + 1572q−14 + 1244q−16−471q−18−1664q−20−993q−22 + 1125q−24 + 1593q−26 + 308q−28−1282q−30−1366q−32 + 300q−34 + 1297q−36 + 849q−38−538q−40−1225q−42−438q−44 + 696q−46 + 1065q−48 + 196q−50−852q−52−1015q−54 + 67q−56 + 1141q−58 + 862q−60−411q−62−1466q−64−590q−66 + 1028q−68 + 1439q−70 + 206q−72−1598q−74−1235q−76 + 526q−78 + 1647q−80 + 928q−82−1152q−84−1518q−86−253q−88 + 1219q−90 + 1318q−92−326q−94−1156q−96−757q−98 + 413q−100 + 1059q−102 + 270q−104−448q−106−663q−108−134q−110 + 471q−112 + 313q−114 + 14q−116−278q−118−196q−120 + 85q−122 + 117q−124 + 86q−126−47q−128−73q−130−2q−132 + 10q−134 + 28q−136−12q−140−2q−144 + 4q−146−2q−150 + q−152
5 −q85 + 2q83 + q81−3q79−q73 + 5q71 + 4q69−11q67−8q65 + 6q63 + 13q61 + 19q59−43q55−56q53 + 6q51 + 97q49 + 121q47 + 23q45−171q43−284q41−126q39 + 276q37 + 563q35 + 367q33−310q31−976q29−898q27 + 169q25 + 1494q23 + 1771q21 + 348q19−1918q17−3020q15−1465q13 + 1989q11 + 4488q9 + 3296q7−1380q5−5823q3−5720q−226q−1 + 6544q−3 + 8462q−5 + 2799q−7−6276q−9−10816q−11−6085q−13 + 4694q−15 + 12319q−17 + 9499q−19−2066q−21−12438q−23−12278q−25−1267q−27 + 11155q−29 + 13928q−31 + 4554q−33−8711q−35−14147q−37−7210q−39 + 5650q−41 + 13040q−43 + 8886q−45−2517q−47−11013q−49−9553q−51−189q−53 + 8510q−55 + 9383q−57 + 2399q−59−6004q−61−8807q−63−4102q−65 + 3733q−67 + 8085q−69 + 5531q−71−1679q−73−7485q−75−6949q−77−240q−79 + 7027q−81 + 8404q−83 + 2269q−85−6459q−87−10013q−89−4531q−91 + 5644q−93 + 11399q−95 + 7055q−97−4159q−99−12354q−101−9674q−103 + 2021q−105 + 12377q−107 + 11974q−109 + 827q−111−11296q−113−13511q−115−3913q−117 + 8970q−119 + 13880q−121 + 6769q−123−5791q−125−12831q−127−8755q−129 + 2218q−131 + 10530q−133 + 9539q−135 + 957q−137−7424q−139−8926q−141−3237q−143 + 4153q−145 + 7271q−147 + 4321q−149−1433q−151−5062q−153−4222q−155−426q−157 + 2882q−159 + 3401q−161 + 1304q−163−1247q−165−2240q−167−1396q−169 + 205q−171 + 1218q−173 + 1095q−175 + 230q−177−522q−179−660q−181−304q−183 + 139q−185 + 320q−187 + 227q−189 + 6q−191−132q−193−115q−195−28q−197 + 35q−199 + 44q−201 + 28q−203−9q−205−21q−207−5q−209 + 2q−211 + q−213 + 4q−215 + 2q−217−4q−219 + 2q−223−q−225

A2 Invariants.

Weight Invariant
1,0 −q6 + q4−q2−1 + 3q−2−2q−4 + 3q−6 + q−8 + q−10 + 3q−12−2q−14 + 2q−16−2q−18−2q−20 + q−22−q−24
1,1 q20−4q18 + 12q16−28q14 + 58q12−108q10 + 182q8−288q6 + 423q4−582q2 + 744−896q−2 + 992q−4−1002q−6 + 900q−8−664q−10 + 312q−12 + 162q−14−668q−16 + 1184q−18−1616q−20 + 1938q−22−2086q−24 + 2048q−26−1838q−28 + 1462q−30−998q−32 + 470q−34 + 36q−36−480q−38 + 814q−40−1004q−42 + 1065q−44−1012q−46 + 876q−48−696q−50 + 506q−52−344q−54 + 214q−56−120q−58 + 62q−60−28q−62 + 12q−64−4q−66 + q−68
2,0 q18−q16−2q14 + 3q12 + 3q10−5q8−5q6 + 7q4 + 5q2−13−6q−2 + 16q−4 + 2q−6−15q−8 + 3q−10 + 14q−12−q−14−6q−16 + 11q−18 + 8q−20−8q−22 + 6q−24 + 7q−26−11q−28−7q−30 + 12q−32−q−34−15q−36 + 12q−40−3q−42−14q−44 + 4q−46 + 8q−48−3q−50−5q−52 + q−54 + 4q−56−q−60 + q−62

A3 Invariants.

Weight Invariant
0,1,0 q14−2q12 + q10 + 4q8−9q6 + 2q4 + 11q2−19 + 2q−2 + 21q−4−24q−6 + q−8 + 25q−10−16q−12−2q−14 + 18q−16 + q−18−3q−20 + q−22 + 15q−24−6q−26−20q−28 + 17q−30−2q−32−28q−34 + 19q−36 + 5q−38−20q−40 + 14q−42 + 3q−44−9q−46 + 5q−48 + q−50−2q−52 + q−54
1,0,0 −q7 + q5−2q3 + q−2q−1 + 3q−3−2q−5 + 3q−7 + q−9 + 2q−11 + 2q−13 + q−15 + 3q−17−2q−19 + 2q−21−3q−23−3q−27 + q−29−q−31

A4 Invariants.

Weight Invariant
0,1,0,0 q16−q14 + 3q10−q8−5q6 + 2q4 + 4q2−8−9q−2 + 9q−4 + 8q−6−17q−8−3q−10 + 21q−12−2q−14−19q−16 + 12q−18 + 18q−20−7q−22 + q−24 + 27q−26 + 9q−28−11q−30 + 14q−32 + 12q−34−25q−36−13q−38 + 12q−40−13q−42−25q−44 + 4q−46 + 13q−48−9q−50−8q−52 + 12q−54 + 7q−56−7q−58 + 5q−62−q−64−q−66 + q−68
1,0,0,0 −q8 + q6−2q4−2q−2 + 3q−4−2q−6 + 3q−8 + q−10 + 2q−12 + 2q−14 + 2q−16 + 2q−18 + q−20 + 3q−22−2q−24 + 2q−26−3q−28−q−30−q−32−3q−34 + q−36−q−38

B2 Invariants.

Weight Invariant
0,1 −q14 + 2q12−5q10 + 8q8−13q6 + 18q4−23q2 + 27−28q−2 + 27q−4−20q−6 + 11q−8 + 3q−10−16q−12 + 32q−14−42q−16 + 53q−18−55q−20 + 55q−22−47q−24 + 36q−26−22q−28 + 7q−30 + 6q−32−18q−34 + 25q−36−29q−38 + 28q−40−26q−42 + 21q−44−15q−46 + 9q−48−5q−50 + 2q−52−q−54
1,0 q24−2q20−2q18 + 3q16 + 6q14−q12−11q10−7q8 + 11q6 + 17q4−4q2−25−11q−2 + 22q−4 + 25q−6−11q−8−31q−10−4q−12 + 29q−14 + 16q−16−18q−18−19q−20 + 13q−22 + 22q−24−3q−26−19q−28 + 3q−30 + 21q−32 + 5q−34−18q−36−7q−38 + 18q−40 + 13q−42−18q−44−22q−46 + 10q−48 + 26q−50−2q−52−32q−54−16q−56 + 23q−58 + 26q−60−10q−62−27q−64−5q−66 + 20q−68 + 13q−70−8q−72−12q−74 + 7q−78 + 3q−80−2q−82−2q−84 + q−88

D4 Invariants.

Weight Invariant
1,0,0,0 q18−2q16 + 3q14−4q12 + 7q10−11q8 + 11q6−15q4 + 18q2−23 + 19q−2−21q−4 + 23q−6−18q−8 + 11q−10−5q−12 + 4q−14 + 12q−16−18q−18 + 25q−20−27q−22 + 44q−24−38q−26 + 44q−28−39q−30 + 45q−32−32q−34 + 26q−36−27q−38 + 10q−40−6q−42−7q−44 + 3q−46−18q−48 + 21q−50−21q−52 + 22q−54−23q−56 + 22q−58−16q−60 + 14q−62−12q−64 + 8q−66−4q−68 + 3q−70−2q−72 + q−74

G2 Invariants.

Weight Invariant
1,0 q32−2q30 + 5q28−8q26 + 8q24−7q22−2q20 + 16q18−32q16 + 46q14−51q12 + 35q10−4q8−46q6 + 99q4−131q2 + 132−89q−2 + 5q−4 + 95q−6−177q−8 + 210q−10−173q−12 + 77q−14 + 42q−16−142q−18 + 181q−20−139q−22 + 45q−24 + 70q−26−141q−28 + 131q−30−45q−32−84q−34 + 200q−36−242q−38 + 197q−40−59q−42−109q−44 + 259q−46−323q−48 + 286q−50−155q−52−17q−54 + 169q−56−248q−58 + 240q−60−143q−62 + 13q−64 + 100q−66−156q−68 + 120q−70−25q−72−91q−74 + 166q−76−168q−78 + 91q−80 + 31q−82−152q−84 + 220q−86−214q−88 + 134q−90−21q−92−92q−94 + 155q−96−160q−98 + 122q−100−55q−102−6q−104 + 46q−106−64q−108 + 54q−110−34q−112 + 15q−114 + q−116−8q−118 + 9q−120−8q−122 + 5q−124−2q−126 + q−128

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 57"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 2t3−8t2 + 18t−23 + 18t−1−8t−2 + 2t−3
In[5]:= Conway[K][z]
Out[5]= 2z6 + 4z4 + 4z2 + 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]= { 79, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 3q7−7q6 + 10q5−12q4 + 14q3−12q2 + 10q−6 + 3q−1−q−2
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−2 + z6a−4 + 3z4a−2 + 3z4a−4−z4a−6−z4 + 4z2a−2 + 4z2a−4−2z2a−6−2z2 + 2a−2 + 2a−4−2a−6−1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z9a−3 + z9a−5 + 3z8a−2 + 7z8a−4 + 4z8a−6 + 4z7a−1 + 9z7a−3 + 10z7a−5 + 5z7a−7 + 2z6a−2−7z6a−4−3z6a−6 + 3z6a−8 + 3z6 + az5−5z5a−1−19z5a−3−23z5a−5−9z5a−7 + z5a−9−11z4a−2−z4a−4−z4a−6−5z4a−8−6z4−2az3 + 12z3a−3 + 18z3a−5 + 6z3a−7−2z3a−9 + 8z2a−2−2z2a−6 + 2z2a−8 + 4z2 + az + za−1−2za−3−6za−5−3za−7 + za−9−2a−2 + 2a−4 + 2a−6−1

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

Same Alexander/Conway Polynomial: {K11n40, K11n46,}

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["10 57"];
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]= { 2t3−8t2 + 18t−23 + 18t−1−8t−2 + 2t−3, −q8 + 3q7−7q6 + 10q5−12q4 + 14q3−12q2 + 10q−6 + 3q−1−q−2 }
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]= {K11n40, K11n46,}
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: (4, 6)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
16 48 128 \frac{728}{3} \frac{88}{3} 768 1248 192 176 \frac{2048}{3} 1152 \frac{11648}{3} \frac{1408}{3} \frac{101102}{15} \frac{1592}{15} \frac{114248}{45} \frac{850}{9} \frac{4622}{15}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 2 is the signature of 10 57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-101234567χ
17          1-1
15         2 2
13        51 -4
11       52  3
9      75   -2
7     75    2
5    57     2
3   57      -2
1  26       4
-1 14        -3
-3 2         2
-51          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −3 {\mathbb Z}
r = −2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r = 1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 7 {\mathbb Z}_2 {\mathbb Z}

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the n + 1-dimensional representation of sl(2))

 n  Jn
2 q23−3q22 + 2q21 + 9q20−20q19 + 2q18 + 42q17−52q16−15q15 + 97q14−80q13−49q12 + 148q11−87q10−82q9 + 168q8−70q7−95q6 + 143q5−37q4−81q3 + 88q2−9q−47 + 36q−1 + q−2−17q−3 + 9q−4 + q−5−3q−6 + q−7
3 −q45 + 3q44−2q43−4q42 + q41 + 16q40−3q39−37q38−4q37 + 76q36 + 24q35−121q34−83q33 + 187q32 + 159q31−229q30−284q29 + 264q28 + 423q27−262q26−578q25 + 235q24 + 722q23−181q22−841q21 + 99q20 + 941q19−26q18−980q17−77q16 + 1003q15 + 146q14−946q13−242q12 + 880q11 + 290q10−746q9−340q8 + 611q7 + 339q6−445q5−327q4 + 312q3 + 270q2−187q−212 + 104q−1 + 148q−2−51q−3−93q−4 + 21q−5 + 54q−6−8q−7−28q−8 + 2q−9 + 14q−10−2q−11−4q−12−q−13 + 3q−14−q−15
4 q74−3q73 + 2q72 + 4q71−6q70 + 3q69−15q68 + 14q67 + 32q66−24q65−9q64−86q63 + 40q62 + 165q61 + 7q60−41q59−367q58−42q57 + 457q56 + 306q55 + 117q54−972q53−571q52 + 672q51 + 1024q50 + 906q49−1618q48−1741q47 + 273q46 + 1854q45 + 2550q44−1717q43−3213q42−992q41 + 2220q40 + 4630q39−998q38−4334q37−2753q36 + 1857q35 + 6434q34 + 235q33−4745q32−4371q31 + 981q30 + 7489q29 + 1508q28−4466q27−5445q26−101q25 + 7652q24 + 2556q23−3597q22−5814q21−1235q20 + 6865q19 + 3243q18−2210q17−5366q16−2232q15 + 5199q14 + 3327q13−620q12−4081q11−2700q10 + 3081q9 + 2656q8 + 573q7−2366q6−2372q5 + 1288q4 + 1535q3 + 941q2−934q−1509 + 320q−1 + 575q−2 + 686q−3−196q−4−697q−5 + 35q−6 + 102q−7 + 316q−8 + 9q−9−243q−10 + 10q−11−14q−12 + 102q−13 + 18q−14−70q−15 + 12q−16−13q−17 + 24q−18 + 6q−19−17q−20 + 5q−21−3q−22 + 4q−23 + q−24−3q−25 + q−26
5 −q110 + 3q109−2q108−4q107 + 6q106 + 2q105−4q104 + 4q103−9q102−20q101 + 18q100 + 39q99 + 12q98−5q97−72q96−107q95 + q94 + 177q93 + 233q92 + 88q91−253q90−550q89−355q88 + 315q87 + 985q86 + 953q85−130q84−1563q83−1956q82−529q81 + 1978q80 + 3504q79 + 1967q78−2082q77−5264q76−4325q75 + 1138q74 + 7133q73 + 7721q72 + 868q71−8382q70−11715q69−4551q68 + 8635q67 + 16102q66 + 9450q65−7391q64−20027q63−15524q62 + 4559q61 + 23142q60 + 22010q59−280q58−24937q57−28407q56−5039q55 + 25357q54 + 34133q53 + 10867q52−24534q51−38763q50−16734q49 + 22677q48 + 42328q47 + 22081q46−20190q45−44518q44−26909q43 + 17195q42 + 45882q41 + 30809q40−14055q39−45895q38−34176q37 + 10486q36 + 45346q35 + 36615q34−6845q33−43341q32−38528q31 + 2651q30 + 40641q29 + 39418q28 + 1558q27−36357q26−39401q25−6048q24 + 31277q23 + 37958q22 + 10054q21−24954q20−35247q19−13438q18 + 18417q17 + 31021q16 + 15490q15−11689q14−25873q13−16211q12 + 5995q11 + 20010q10 + 15330q9−1317q8−14308q7−13391q6−1630q5 + 9231q4 + 10599q3 + 3223q2−5233q−7728−3548q−1 + 2461q−2 + 5105q−3 + 3120q−4−790q−5−3052q−6−2356q−7−38q−8 + 1651q−9 + 1565q−10 + 312q−11−786q−12−933q−13−315q−14 + 326q−15 + 498q−16 + 234q−17−120q−18−256q−19−119q−20 + 39q−21 + 96q−22 + 76q−23−7q−24−62q−25−21q−26 + 15q−27 + 5q−28 + 14q−29 + 6q−30−19q−31−2q−32 + 9q−33−2q−34 + 3q−36−4q−37−q−38 + 3q−39−q−40

[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.

10_56

10_58

Retrieved from "http://katlas.math.toronto.edu/wiki/10_57"
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