10 76

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10_75

10_77

Contents

Image:10 76.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 76's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_76's page at Knotilus!

Visit 10 76's page at the original Knot Atlas!

[edit 10 76 Quick Notes]
[edit 10 76 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X14,10,15,9 X12,3,13,4 X2,13,3,14 X18,6,19,5 X20,8,1,7 X6,20,7,19 X8,18,9,17 X16,12,17,11 X10,16,11,15
Gauss code 1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6
Dowker-Thistlethwaite code 4 12 18 20 14 16 2 10 8 6
Conway Notation [3,3,2++]


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

Length is 11, width is 4,

Braid index is 4

Image:10 76_ML.gif Image:10 76_AP.gif
[{5, 12}, {6, 3}, {1, 5}, {4, 2}, {3, 7}, {2, 6}, {10, 4}, {9, 11}, {8, 10}, {7, 9}, {12, 8}, {11, 1}]

[edit Notes on presentations of 10 76]


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 76"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X14,10,15,9 X12,3,13,4 X2,13,3,14 X18,6,19,5 X20,8,1,7 X6,20,7,19 X8,18,9,17 X16,12,17,11 X10,16,11,15
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6
In[6]:= DTCode[K]
Out[6]= 4 12 18 20 14 16 2 10 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3,3,2++]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= BR(4,{1,1,1,1,2,−1,−3,2,2,2,−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:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.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 76_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{5, 12}, {6, 3}, {1, 5}, {4, 2}, {3, 7}, {2, 6}, {10, 4}, {9, 11}, {8, 10}, {7, 9}, {12, 8}, {11, 1}]
In[14]:= Draw[ap]
Image:10 76_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number {2,3}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-13]
Hyperbolic Volume 11.5129
A-Polynomial See Data:10 76/A-polynomial

[edit Notes for 10 76's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for 10 76's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −2t3 + 7t2−12t + 15−12t−1 + 7t−2−2t−3
Conway polynomial −2z6−5z4−2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 57, 4 }
Jones polynomial q10−3q9 + 6q8−8q7 + 9q6−10q5 + 8q4−6q3 + 4q2−q + 1
HOMFLY-PT polynomial (db, data sources) −z6a−4−z6a−6 + z4a−2−4z4a−4−3z4a−6 + z4a−8 + 4z2a−2−6z2a−4−2z2a−6 + 2z2a−8 + 4a−2−4a−4 + a−8
Kauffman polynomial (db, data sources) z9a−5 + z9a−7 + z8a−4 + 4z8a−6 + 3z8a−8 + z7a−3−2z7a−5 + 2z7a−7 + 5z7a−9 + z6a−2−9z6a−6−3z6a−8 + 5z6a−10−2z5a−3 + 7z5a−5−2z5a−7−8z5a−9 + 3z5a−11−5z4a−2−7z4a−4 + 10z4a−6 + 4z4a−8−7z4a−10 + z4a−12−2z3a−3−15z3a−5−3z3a−7 + 7z3a−9−3z3a−11 + 8z2a−2 + 9z2a−4−7z2a−6−4z2a−8 + 3z2a−10−z2a−12 + 4za−3 + 8za−5 + 2za−7−2za−9−4a−2−4a−4 + a−8
The A2 invariant 1 + q−2 + 2q−4 + 3q−6−q−8 + q−10−3q−12−2q−14−2q−18 + 2q−20−q−22 + q−24 + q−26−q−28 + q−30
The G2 invariant q−2 + 3q−6−2q−8 + 3q−10−q−12 + 7q−16−9q−18 + 14q−20−12q−22 + 12q−24 + q−26−12q−28 + 31q−30−40q−32 + 46q−34−33q−36 + q−38 + 33q−40−65q−42 + 80q−44−67q−46 + 28q−48 + 17q−50−59q−52 + 71q−54−63q−56 + 24q−58 + 17q−60−50q−62 + 46q−64−24q−66−19q−68 + 58q−70−75q−72 + 60q−74−21q−76−35q−78 + 87q−80−115q−82 + 106q−84−59q−86−3q−88 + 66q−90−101q−92 + 104q−94−68q−96 + 17q−98 + 32q−100−60q−102 + 55q−104−22q−106−19q−108 + 51q−110−52q−112 + 28q−114 + 11q−116−53q−118 + 76q−120−74q−122 + 48q−124−8q−126−33q−128 + 60q−130−64q−132 + 54q−134−28q−136 + 3q−138 + 15q−140−29q−142 + 28q−144−21q−146 + 12q−148−2q−150−3q−152 + 5q−154−6q−156 + 4q−158−2q−160 + q−162
Further Quantum Invariants
Further quantum knot invariants for 10_76.

A1 Invariants.

Weight Invariant
1 q + 3q−3−2q−5 + 2q−7−2q−9−q−11 + q−13−2q−15 + 3q−17−2q−19 + q−21
2 q6 + 3−q−2−3q−4 + 8q−6−2q−8−11q−10 + 12q−12 + 3q−14−18q−16 + 9q−18 + 9q−20−14q−22 + 2q−24 + 10q−26−3q−28−7q−30 + 4q−32 + 10q−34−12q−36−4q−38 + 18q−40−11q−42−9q−44 + 15q−46−4q−48−7q−50 + 6q−52−2q−56 + q−58
3 q15 + 2q7−q5−q3 + q + 6q−1−3q−3−10q−5 + 21q−9 + 3q−11−29q−13−18q−15 + 36q−17 + 37q−19−36q−21−54q−23 + 25q−25 + 73q−27−9q−29−78q−31−12q−33 + 79q−35 + 25q−37−67q−39−43q−41 + 58q−43 + 47q−45−37q−47−50q−49 + 17q−51 + 50q−53 + 4q−55−46q−57−31q−59 + 38q−61 + 52q−63−23q−65−74q−67 + 10q−69 + 83q−71 + 9q−73−82q−75−22q−77 + 70q−79 + 33q−81−53q−83−34q−85 + 34q−87 + 26q−89−17q−91−20q−93 + 9q−95 + 13q−97−5q−99−6q−101 + q−103 + 3q−105−2q−109 + q−111
4 q28−q20 + 2q18−q16 + 3q12−2q10 + 3q8−6q6−5q4 + 8q2 + 6 + 15q−2−17q−4−32q−6−5q−8 + 22q−10 + 71q−12 + 3q−14−76q−16−79q−18−15q−20 + 156q−22 + 118q−24−45q−26−187q−28−182q−30 + 141q−32 + 267q−34 + 135q−36−175q−38−382q−40−48q−42 + 278q−44 + 354q−46 + 10q−48−433q−50−266q−52 + 125q−54 + 433q−56 + 213q−58−319q−60−360q−62−48q−64 + 365q−66 + 301q−68−162q−70−333q−72−148q−74 + 237q−76 + 295q−78−9q−80−255q−82−214q−84 + 84q−86 + 257q−88 + 167q−90−131q−92−277q−94−132q−96 + 171q−98 + 360q−100 + 64q−102−282q−104−362q−106−6q−108 + 447q−110 + 278q−112−145q−114−456q−116−212q−118 + 341q−120 + 350q−122 + 53q−124−333q−126−287q−128 + 136q−130 + 234q−132 + 143q−134−135q−136−194q−138 + 18q−140 + 76q−142 + 98q−144−26q−146−78q−148 + q−150 + 8q−152 + 39q−154−5q−156−25q−158 + 5q−160−3q−162 + 12q−164−7q−168 + 2q−170−2q−172 + 3q−174−2q−178 + q−180
5 q45−q37−q35 + 2q33 + 3q27−5q23−2q19−q17 + 10q15 + 10q13−4q11−9q9−20q7−18q5 + 15q3 + 43q + 39q−1 + 6q−3−60q−5−103q−7−50q−9 + 64q−11 + 165q−13 + 168q−15−4q−17−237q−19−315q−21−143q−23 + 213q−25 + 505q−27 + 420q−29−80q−31−615q−33−751q−35−264q−37 + 569q−39 + 1085q−41 + 751q−43−298q−45−1258q−47−1298q−49−243q−51 + 1179q−53 + 1780q−55 + 920q−57−819q−59−2018q−61−1629q−63 + 196q−65 + 1996q−67 + 2188q−69 + 519q−71−1668q−73−2508q−75−1203q−77 + 1184q−79 + 2542q−81 + 1713q−83−602q−85−2384q−87−2002q−89 + 130q−91 + 2039q−93 + 2073q−95 + 269q−97−1690q−99−2005q−101−479q−103 + 1337q−105 + 1828q−107 + 644q−109−1045q−111−1666q−113−744q−115 + 779q−117 + 1507q−119 + 879q−121−482q−123−1387q−125−1092q−127 + 127q−129 + 1251q−131 + 1354q−133 + 348q−135−1024q−137−1665q−139−924q−141 + 686q−143 + 1876q−145 + 1573q−147−168q−149−1950q−151−2164q−153−479q−155 + 1770q−157 + 2602q−159 + 1171q−161−1359q−163−2742q−165−1762q−167 + 739q−169 + 2579q−171 + 2132q−173−104q−175−2104q−177−2191q−179−454q−181 + 1477q−183 + 1971q−185 + 780q−187−854q−189−1530q−191−867q−193 + 345q−195 + 1034q−197 + 772q−199−47q−201−607q−203−545q−205−93q−207 + 295q−209 + 339q−211 + 114q−213−131q−215−179q−217−69q−219 + 45q−221 + 78q−223 + 38q−225−13q−227−40q−229−14q−231 + 13q−233 + 13q−235 + 3q−237−4q−239−4q−241−7q−243 + 3q−245 + 7q−247−q−249−2q−251 + q−253−q−255−2q−257 + 3q−259−2q−263 + q−265

A2 Invariants.

Weight Invariant
1,0 1 + q−2 + 2q−4 + 3q−6−q−8 + q−10−3q−12−2q−14−2q−18 + 2q−20−q−22 + q−24 + q−26−q−28 + q−30
1,1 q4 + 6−4q−2 + 18q−4−20q−6 + 42q−8−62q−10 + 93q−12−140q−14 + 182q−16−240q−18 + 272q−20−302q−22 + 288q−24−244q−26 + 164q−28−46q−30−92q−32 + 246q−34−378q−36 + 502q−38−576q−40 + 612q−42−587q−44 + 512q−46−402q−48 + 254q−50−107q−52−40q−54 + 160q−56−244q−58 + 292q−60−300q−62 + 282q−64−246q−66 + 195q−68−144q−70 + 102q−72−66q−74 + 38q−76−20q−78 + 10q−80−4q−82 + q−84
2,0 q4 + q2 + 1 + 2q−2 + 5q−4 + 3q−6 + q−10 + 5q−12−4q−14−10q−16−2q−18−9q−22−6q−24 + 6q−26 + 5q−28−2q−30 + 4q−32 + 9q−34−3q−36−q−38 + 7q−40−6q−44 + 2q−46 + 4q−48−6q−50−5q−52 + 5q−54 + 3q−56−6q−58−q−60 + 4q−62−q−64−2q−66 + 2q−70−q−74 + q−76

A3 Invariants.

Weight Invariant
0,1,0 1 + 3q−4 + 4q−6 + 2q−8 + 6q−10 + 7q−12−7q−14 + 2q−16−20q−20 + 4q−24−14q−26 + 5q−28 + 12q−30−4q−32 + q−34 + 5q−36 + 3q−38−5q−40−3q−42 + 11q−44−5q−46−9q−48 + 14q−50−4q−52−11q−54 + 11q−56−q−58−6q−60 + 5q−62−2q−66 + q−68
1,0,0 q−1 + q−3 + 3q−5 + 2q−7 + 4q−9−q−11 + q−13−4q−15−2q−17−3q−19−q−21−q−25 + 2q−27−q−29 + 2q−31−q−33 + 2q−35−q−37 + q−39

A4 Invariants.

Weight Invariant
0,1,0,0 q−2 + q−4 + 3q−6 + 5q−8 + 7q−10 + 7q−12 + 9q−14 + 8q−16 + 4q−18−4q−20−4q−22−10q−24−21q−26−16q−28−4q−30−10q−32−9q−34 + 12q−36 + 15q−38 + 3q−40 + 4q−42 + 17q−44 + 3q−46−9q−48 + 4q−50 + 6q−52−11q−54−2q−56 + 10q−58−5q−60−8q−62 + 5q−64 + 4q−66−9q−68−3q−70 + 8q−72 + q−74−6q−76 + 2q−78 + 4q−80−2q−82−q−84 + q−86
1,0,0,0 q−2 + q−4 + 3q−6 + 3q−8 + 3q−10 + 4q−12−q−14 + q−16−4q−18−3q−20−3q−22−3q−24−q−26−q−28 + q−30−q−32 + 2q−34−q−36 + 2q−38 + 2q−44−q−46 + q−48

B2 Invariants.

Weight Invariant
0,1 1 + 3q−4−2q−6 + 6q−8−6q−10 + 11q−12−11q−14 + 14q−16−14q−18 + 12q−20−10q−22 + 2q−24 + 2q−26−11q−28 + 16q−30−24q−32 + 27q−34−29q−36 + 27q−38−23q−40 + 17q−42−9q−44 + 3q−46 + 5q−48−10q−50 + 14q−52−15q−54 + 15q−56−13q−58 + 10q−60−7q−62 + 4q−64−2q−66 + q−68
1,0 q2 + 3q−6 + 3q−8 + q−10−2q−12 + q−14 + 6q−16 + 8q−18−3q−20−10q−22−3q−24 + 11q−26 + 7q−28−12q−30−18q−32−2q−34 + 14q−36 + 4q−38−13q−40−11q−42 + 7q−44 + 12q−46−9q−50 + 2q−52 + 11q−54 + 3q−56−9q−58−3q−60 + 9q−62 + 7q−64−8q−66−10q−68 + 6q−70 + 12q−72−2q−74−15q−76−4q−78 + 14q−80 + 11q−82−9q−84−15q−86 + 13q−90 + 6q−92−7q−94−8q−96 + q−98 + 6q−100 + 2q−102−2q−104−2q−106 + q−110

D4 Invariants.

Weight Invariant
1,0,0,0 q−2 + 3q−6 + q−8 + 7q−10 + q−12 + 10q−14−q−16 + 12q−18−9q−20 + 8q−22−15q−24 + 5q−26−18q−28−11q−32−6q−38 + 12q−40−8q−42 + 20q−44−19q−46 + 22q−48−20q−50 + 23q−52−20q−54 + 16q−56−14q−58 + 13q−60−4q−62 + q−64−5q−68 + 10q−70−11q−72 + 9q−74−13q−76 + 13q−78−9q−80 + 8q−82−8q−84 + 6q−86−3q−88 + 2q−90−2q−92 + q−94

G2 Invariants.

Weight Invariant
1,0 q−2 + 3q−6−2q−8 + 3q−10−q−12 + 7q−16−9q−18 + 14q−20−12q−22 + 12q−24 + q−26−12q−28 + 31q−30−40q−32 + 46q−34−33q−36 + q−38 + 33q−40−65q−42 + 80q−44−67q−46 + 28q−48 + 17q−50−59q−52 + 71q−54−63q−56 + 24q−58 + 17q−60−50q−62 + 46q−64−24q−66−19q−68 + 58q−70−75q−72 + 60q−74−21q−76−35q−78 + 87q−80−115q−82 + 106q−84−59q−86−3q−88 + 66q−90−101q−92 + 104q−94−68q−96 + 17q−98 + 32q−100−60q−102 + 55q−104−22q−106−19q−108 + 51q−110−52q−112 + 28q−114 + 11q−116−53q−118 + 76q−120−74q−122 + 48q−124−8q−126−33q−128 + 60q−130−64q−132 + 54q−134−28q−136 + 3q−138 + 15q−140−29q−142 + 28q−144−21q−146 + 12q−148−2q−150−3q−152 + 5q−154−6q−156 + 4q−158−2q−160 + q−162

.

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 76"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −2t3 + 7t2−12t + 15−12t−1 + 7t−2−2t−3
In[5]:= Conway[K][z]
Out[5]= −2z6−5z4−2z2 + 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]= { 57, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q10−3q9 + 6q8−8q7 + 9q6−10q5 + 8q4−6q3 + 4q2−q + 1
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z6a−4−z6a−6 + z4a−2−4z4a−4−3z4a−6 + z4a−8 + 4z2a−2−6z2a−4−2z2a−6 + 2z2a−8 + 4a−2−4a−4 + a−8
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z9a−5 + z9a−7 + z8a−4 + 4z8a−6 + 3z8a−8 + z7a−3−2z7a−5 + 2z7a−7 + 5z7a−9 + z6a−2−9z6a−6−3z6a−8 + 5z6a−10−2z5a−3 + 7z5a−5−2z5a−7−8z5a−9 + 3z5a−11−5z4a−2−7z4a−4 + 10z4a−6 + 4z4a−8−7z4a−10 + z4a−12−2z3a−3−15z3a−5−3z3a−7 + 7z3a−9−3z3a−11 + 8z2a−2 + 9z2a−4−7z2a−6−4z2a−8 + 3z2a−10−z2a−12 + 4za−3 + 8za−5 + 2za−7−2za−9−4a−2−4a−4 + a−8

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

Same Alexander/Conway Polynomial: {}

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

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

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

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

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

In[3]:= K = Knot["10 76"];
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 + 7t2−12t + 15−12t−1 + 7t−2−2t−3, q10−3q9 + 6q8−8q7 + 9q6−10q5 + 8q4−6q3 + 4q2−q + 1 }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

[edit] Vassiliev invariants

V2 and V3: (-2, -6)

[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 = 4 is the signature of 10 76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-1012345678χ
21          11
19         2 -2
17        41 3
15       42  -2
13      54   1
11     54    -1
9    35     -2
7   35      2
5  13       -2
3 14        3
1           0
-11          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 3 i = 5
r = −2 {\mathbb Z}
r = −1 {\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z}^{4} {\mathbb Z}
r = 1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 8 {\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 q28−3q27 + 2q26 + 7q25−16q24 + 5q23 + 26q22−40q21 + 3q20 + 55q19−62q18−5q17 + 77q16−68q15−16q14 + 81q13−55q12−24q11 + 65q10−32q9−24q8 + 38q7−11q6−15q5 + 15q4−2q3−5q2 + 4q−q−1 + q−2
3 q54−3q53 + 2q52 + 3q51−q50−10q49 + 3q48 + 21q47−5q46−39q45 + 6q44 + 64q43 + 3q42−107q41−13q40 + 150q39 + 40q38−199q37−73q36 + 241q35 + 114q34−272q33−157q32 + 292q31 + 189q30−286q29−226q28 + 277q27 + 239q26−240q25−259q24 + 210q23 + 252q22−156q21−248q20 + 109q19 + 228q18−64q17−194q16 + 18q15 + 162q14 + 5q13−112q12−30q11 + 83q10 + 23q9−39q8−31q7 + 29q6 + 12q5−7q4−13q3 + 8q2 + 2q−4q−1 + 3q−2−q−5 + q−6
4 q88−3q87 + 2q86 + 3q85−5q84 + 5q83−12q82 + 9q81 + 15q80−20q79 + 13q78−42q77 + 29q76 + 59q75−51q74 + 6q73−121q72 + 81q71 + 183q70−73q69−52q68−333q67 + 140q66 + 461q65 + 18q64−150q63−756q62 + 94q61 + 847q60 + 315q59−159q58−1309q57−150q56 + 1158q55 + 738q54 + 10q53−1762q52−506q51 + 1238q50 + 1084q49 + 306q48−1951q47−809q46 + 1093q45 + 1230q44 + 604q43−1861q42−982q41 + 795q40 + 1189q39 + 850q38−1557q37−1040q36 + 410q35 + 1004q34 + 1021q33−1094q32−976q31−3q30 + 692q29 + 1062q28−562q27−756q26−311q25 + 301q24 + 895q23−119q22−412q21−387q20−25q19 + 561q18 + 88q17−102q16−255q15−151q14 + 238q13 + 83q12 + 40q11−92q10−113q9 + 67q8 + 19q7 + 43q6−13q5−45q4 + 18q3−8q2 + 16q + 2−13q−1 + 9q−2−6q−3 + 3q−4 + q−5−4q−6 + 4q−7−q−8−q−11 + q−12

[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_75

10_77

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