10 10
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_10's page at Knotilus! Visit 10 10's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X19,7,20,6 X7,19,8,18 X9,17,10,16 X15,11,16,10 X17,9,18,8 X11,2,12,3 |
| Gauss code | -1, 10, -2, 1, -4, 5, -6, 9, -7, 8, -10, 2, -3, 4, -8, 7, -9, 6, -5, 3 |
| Dowker-Thistlethwaite code | 4 12 14 18 16 2 20 10 8 6 |
| Conway Notation | [51112] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{12, 5}, {1, 10}, {6, 11}, {10, 12}, {11, 4}, {5, 2}, {3, 1}, {4, 7}, {8, 6}, {7, 9}, {2, 8}, {9, 3}] |
[edit Notes on presentations of 10 10]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 10"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X19,7,20,6 X7,19,8,18 X9,17,10,16 X15,11,16,10 X17,9,18,8 X11,2,12,3 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 10, -2, 1, -4, 5, -6, 9, -7, 8, -10, 2, -3, 4, -8, 7, -9, 6, -5, 3 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 12 14 18 16 2 20 10 8 6 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [51112] |
In[9]:=
| br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
| BR(5,{−1,−1,2,−1,2,2,3,−2,3,4,−3,4}) |
In[10]:=
| {First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
| { 5, 12, 5 } |
In[11]:=
| Show[BraidPlot[br]]
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Out[11]=
| -Graphics- |
In[12]:=
| Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
| -Graphics- |
In[13]:=
| ap = ArcPresentation[K]
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Out[13]=
| ArcPresentation[{12, 5}, {1, 10}, {6, 11}, {10, 12}, {11, 4}, {5, 2}, {3, 1}, {4, 7}, {8, 6}, {7, 9}, {2, 8}, {9, 3}] |
In[14]:=
| Draw[ap]
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Out[14]=
| -Graphics- |
[edit] Three dimensional invariants
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[edit] Four dimensional invariants
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[edit] Polynomial invariants
| Alexander polynomial | 3t2−11t + 17−11t−1 + 3t−2 |
| Conway polynomial | 3z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 45, 0 } |
| Jones polynomial | −q7 + 2q6−3q5 + 5q4−6q3 + 7q2−7q + 6−4q−1 + 3q−2−q−3 |
| HOMFLY-PT polynomial (db, data sources) | z4a−2 + z4a−4 + z4−a2z2 + 2z2a−4−z2a−6 + z2−a−2 + 2a−4−a−6 + 1 |
| Kauffman polynomial (db, data sources) | z9a−3 + z9a−5 + 3z8a−2 + 5z8a−4 + 2z8a−6 + 4z7a−1 + 2z7a−3−z7a−5 + z7a−7−7z6a−2−21z6a−4−10z6a−6 + 4z6 + 4az5−7z5a−1−16z5a−3−10z5a−5−5z5a−7 + 3a2z4 + 5z4a−2 + 26z4a−4 + 15z4a−6−3z4 + a3z3−3az3 + 3z3a−1 + 17z3a−3 + 17z3a−5 + 7z3a−7−2a2z2−4z2a−2−12z2a−4−8z2a−6−2z2−za−1−4za−3−6za−5−3za−7 + a−2 + 2a−4 + a−6 + 1 |
| The A2 invariant | −q10 + q8 + q6−q4 + 2q2−q−6 + q−8 + q−12 + 2q−14−q−16−q−22 |
| The G2 invariant | q52−2q50 + 3q48−4q46 + q44−3q40 + 8q38−10q36 + 11q34−7q32 + q30 + 5q28−10q26 + 14q24−14q22 + 11q20−6q18−2q16 + 9q14−10q12 + 13q10−11q8 + 8q6−3q4−3q2 + 9−10q−2 + 9q−4−3q−6−q−8 + 6q−10−7q−12 + 3q−14 + 5q−16−13q−18 + 15q−20−14q−22 + q−24 + 14q−26−26q−28 + 28q−30−23q−32 + 10q−34 + 6q−36−22q−38 + 29q−40−26q−42 + 18q−44−2q−46−10q−48 + 19q−50−13q−52 + 12q−54−10q−58 + 14q−60−9q−62 + 13q−66−22q−68 + 24q−70−15q−72−q−74 + 15q−76−27q−78 + 27q−80−21q−82 + 6q−84 + 5q−86−15q−88 + 17q−90−14q−92 + 8q−94−q−96−3q−98 + 3q−100−4q−102 + 3q−104−q−106 + q−108 |
Further Quantum Invariants
Further quantum knot invariants for 10_10.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q7 + 2q5−q3 + 2q−q−1 + q−5−q−7 + 2q−9−q−11 + q−13−q−15 |
| 2 | q20−2q18−q16 + 4q14−3q12 + 4q8−5q6 + q4 + 5q2−4 + 5q−4−q−6−3q−8 + 2q−10 + 3q−12−3q−14−2q−16 + 5q−18−2q−20−5q−22 + 5q−24 + q−26−6q−28 + 4q−30 + 4q−32−5q−34 + 3q−38−2q−40−q−42 + q−44 |
| 3 | −q39 + 2q37 + q35−2q33−3q31 + q29 + 6q27−2q25−4q23−2q21 + 4q19 + 2q17−5q13−4q11 + 6q9 + 9q7−2q5−11q3 + 12q−1 + 6q−3−8q−5−8q−7 + 11q−11 + 6q−13−10q−15−10q−17 + 7q−19 + 13q−21−6q−23−14q−25 + 3q−27 + 14q−29−12q−33−2q−35 + 11q−37 + 7q−39−9q−41−11q−43 + 4q−45 + 13q−47 + q−49−14q−51−9q−53 + 12q−55 + 13q−57−5q−59−15q−61 + q−63 + 14q−65 + 4q−67−10q−69−7q−71 + 5q−73 + 6q−75−2q−77−4q−79 + 2q−83 + q−85−q−87 |
| 4 | q64−2q62−q60 + 2q58 + q56 + 5q54−7q52−5q50 + q48 + 4q46 + 18q44−11q42−15q40−8q38 + 7q36 + 37q34−4q32−28q30−33q28 + 2q26 + 63q24 + 23q22−32q20−65q18−21q16 + 77q14 + 61q12−12q10−85q8−59q6 + 57q4 + 79q2 + 31−57q−2−78q−4 + 2q−6 + 49q−8 + 55q−10 + 2q−12−49q−14−37q−16−9q−18 + 38q−20 + 47q−22 + q−24−41q−26−42q−28 + 11q−30 + 55q−32 + 27q−34−38q−36−51q−38 + 56q−42 + 35q−44−40q−46−53q−48−6q−50 + 56q−52 + 48q−54−30q−56−55q−58−27q−60 + 38q−62 + 58q−64−q−66−31q−68−42q−70−4q−72 + 39q−74 + 22q−76 + 13q−78−27q−80−35q−82−4q−84 + 8q−86 + 42q−88 + 15q−90−22q−92−30q−94−29q−96 + 28q−98 + 38q−100 + 15q−102−15q−104−45q−106−6q−108 + 20q−110 + 27q−112 + 12q−114−25q−116−17q−118−4q−120 + 12q−122 + 16q−124−4q−126−6q−128−6q−130 + 6q−134 + q−136−2q−140−q−142 + q−144 |
| 5 | −q95 + 2q93 + q91−2q89−q87−3q85 + q83 + 6q81 + 6q79−4q77−10q75−10q73 + 2q71 + 19q69 + 19q67−32q63−32q61 + 2q59 + 41q57 + 55q55 + 9q53−68q51−83q49−11q47 + 81q45 + 114q43 + 34q41−112q39−160q37−42q35 + 137q33 + 203q31 + 69q29−161q27−261q25−103q23 + 180q21 + 316q19 + 152q17−172q15−354q13−219q11 + 135q9 + 373q7 + 281q5−61q3−345q−325q−1−34q−3 + 268q−5 + 333q−7 + 131q−9−158q−11−292q−13−193q−15 + 30q−17 + 210q−19 + 224q−21 + 77q−23−114q−25−207q−27−145q−29 + 19q−31 + 172q−33 + 180q−35 + 37q−37−137q−39−184q−41−65q−43 + 111q−45 + 183q−47 + 74q−49−114q−51−185q−53−69q−55 + 124q−57 + 205q−59 + 80q−61−137q−63−235q−65−107q−67 + 139q−69 + 264q−71 + 147q−73−116q−75−283q−77−199q−79 + 71q−81 + 282q−83 + 241q−85−6q−87−248q−89−273q−91−66q−93 + 190q−95 + 273q−97 + 126q−99−108q−101−237q−103−166q−105 + 21q−107 + 170q−109 + 173q−111 + 46q−113−80q−115−131q−117−90q−119−2q−121 + 69q−123 + 84q−125 + 53q−127 + 10q−129−41q−131−74q−133−66q−135−19q−137 + 43q−139 + 87q−141 + 77q−143 + 7q−145−71q−147−101q−149−58q−151 + 28q−153 + 92q−155 + 90q−157 + 21q−159−58q−161−90q−163−52q−165 + 16q−167 + 63q−169 + 62q−171 + 16q−173−34q−175−51q−177−27q−179 + 7q−181 + 29q−183 + 28q−185 + 6q−187−14q−189−17q−191−7q−193 + 2q−195 + 8q−197 + 8q−199−4q−203−3q−205−q−207 + 2q−211 + q−213−q−215 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q10 + q8 + q6−q4 + 2q2−q−6 + q−8 + q−12 + 2q−14−q−16−q−22 |
| 1,1 | q28−4q26 + 8q24−12q22 + 18q20−28q18 + 34q16−36q14 + 39q12−42q10 + 42q8−34q6 + 32q4−32q2 + 30−24q−2 + 21q−4−12q−6 + 22q−10−44q−12 + 76q−14−102q−16 + 124q−18−137q−20 + 146q−22−140q−24 + 116q−26−91q−28 + 54q−30−20q−32−22q−34 + 56q−36−74q−38 + 92q−40−92q−42 + 85q−44−70q−46 + 52q−48−38q−50 + 21q−52−12q−54 + 6q−56−2q−58 + q−60 |
| 2,0 | q26−q24−2q22 + q20 + 2q18−q16−3q14 + 4q12 + 4q10−5q8−4q6 + 5q4 + 2q2−3 + 4q−4 + 2q−6−q−8 + 2q−10 + q−12−q−14 + 3q−16 + q−18−4q−20−2q−22 + 2q−24−4q−28−2q−30 + 4q−32 + 3q−34−2q−36−q−38 + 3q−40 + 2q−42−2q−44−3q−46 + q−48 + q−50−q−52−q−54 + q−58 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q22−2q20−q18 + 4q16−3q14−2q12 + 6q10−q8−4q6 + 6q4 + q2−4 + 3q−2 + q−4−3q−6−q−8 + 2q−10 + 2q−12−3q−14 + 2q−16 + 5q−18−3q−20 + q−22 + 5q−24−3q−26 + 2q−30−4q−32−q−34 + q−36−3q−38 + q−40 + q−42−q−44 + q−46 |
| 1,0,0 | −q13 + q11 + q7−q5 + 2q3 + q−1−q−7−q−9 + q−11 + 2q−15 + q−17 + 2q−19−q−21−q−25−q−29 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q22 + 2q20−3q18 + 4q16−5q14 + 6q12−6q10 + 7q8−6q6 + 6q4−3q2 + 3q−2−5q−4 + 9q−6−11q−8 + 14q−10−14q−12 + 13q−14−12q−16 + 9q−18−7q−20 + 3q−22 + q−24−3q−26 + 6q−28−6q−30 + 8q−32−7q−34 + 7q−36−5q−38 + 3q−40−3q−42 + q−44−q−46 |
| 1,0 | q36−2q32−2q30 + q28 + 4q26 + q24−4q22−4q20 + 2q18 + 6q16 + 3q14−4q12−5q10 + q8 + 7q6 + 2q4−5q2−4 + 3q−2 + 5q−4−q−6−6q−8−q−10 + 5q−12 + 3q−14−3q−16−2q−18 + 3q−20 + 3q−22−2q−24−3q−26 + 3q−28 + 5q−30−q−32−7q−34−2q−36 + 7q−38 + 6q−40−3q−42−8q−44 + 7q−48 + 4q−50−5q−52−6q−54 + q−56 + 5q−58−4q−62−2q−64 + 2q−66 + 2q−68−q−70−q−72 + q−76 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q52−2q50 + 3q48−4q46 + q44−3q40 + 8q38−10q36 + 11q34−7q32 + q30 + 5q28−10q26 + 14q24−14q22 + 11q20−6q18−2q16 + 9q14−10q12 + 13q10−11q8 + 8q6−3q4−3q2 + 9−10q−2 + 9q−4−3q−6−q−8 + 6q−10−7q−12 + 3q−14 + 5q−16−13q−18 + 15q−20−14q−22 + q−24 + 14q−26−26q−28 + 28q−30−23q−32 + 10q−34 + 6q−36−22q−38 + 29q−40−26q−42 + 18q−44−2q−46−10q−48 + 19q−50−13q−52 + 12q−54−10q−58 + 14q−60−9q−62 + 13q−66−22q−68 + 24q−70−15q−72−q−74 + 15q−76−27q−78 + 27q−80−21q−82 + 6q−84 + 5q−86−15q−88 + 17q−90−14q−92 + 8q−94−q−96−3q−98 + 3q−100−4q−102 + 3q−104−q−106 + q−108 |
.
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.
|
In[3]:=
| K = Knot["10 10"];
|
In[4]:=
| Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| 3t2−11t + 17−11t−1 + 3t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 3z4 + z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
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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.
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Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 45, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q7 + 2q6−3q5 + 5q4−6q3 + 7q2−7q + 6−4q−1 + 3q−2−q−3 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z4a−2 + z4a−4 + z4−a2z2 + 2z2a−4−z2a−6 + z2−a−2 + 2a−4−a−6 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z9a−3 + z9a−5 + 3z8a−2 + 5z8a−4 + 2z8a−6 + 4z7a−1 + 2z7a−3−z7a−5 + z7a−7−7z6a−2−21z6a−4−10z6a−6 + 4z6 + 4az5−7z5a−1−16z5a−3−10z5a−5−5z5a−7 + 3a2z4 + 5z4a−2 + 26z4a−4 + 15z4a−6−3z4 + a3z3−3az3 + 3z3a−1 + 17z3a−3 + 17z3a−5 + 7z3a−7−2a2z2−4z2a−2−12z2a−4−8z2a−6−2z2−za−1−4za−3−6za−5−3za−7 + a−2 + 2a−4 + a−6 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_164,}
Same Jones Polynomial (up to mirroring, 
):
{}
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.
|
In[3]:=
| K = Knot["10 10"];
|
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]=
| { 3t2−11t + 17−11t−1 + 3t−2, −q7 + 2q6−3q5 + 5q4−6q3 + 7q2−7q + 6−4q−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]=
| {10_164,} |
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: | (1, 2) |
| V2,1 through V6,9: |
|
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 = 0 is the signature of 10 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
[edit] The Coloured Jones Polynomials
| n | Jn |
| 2 | q21−2q20−q19 + 6q18−5q17−6q16 + 15q15−5q14−16q13 + 22q12−q11−26q10 + 25q9 + 6q8−33q7 + 24q6 + 12q5−34q4 + 19q3 + 14q2−28q + 14 + 10q−1−19q−2 + 10q−3 + 4q−4−10q−5 + 6q−6 + q−7−3q−8 + q−9 |
| 3 | −q42 + 2q41 + q40−2q39−5q38 + 4q37 + 9q36−3q35−17q34 + q33 + 23q32 + 7q31−30q30−15q29 + 33q28 + 25q27−31q26−36q25 + 28q24 + 40q23−19q22−45q21 + 13q20 + 42q19−3q18−41q17 + 32q15 + 9q14−27q13−11q12 + 15q11 + 17q10−8q9−17q8−2q7 + 17q6 + 8q5−12q4−13q3 + 9q2 + 8q + 2−7q−1−3q−2−3q−3 + 11q−4 + 4q−5−6q−6−13q−7 + 10q−8 + 9q−9−4q−10−11q−11 + 4q−12 + 7q−13−2q−14−3q−15−q−16 + 3q−17−q−18 |
| 4 | q70−2q69−q68 + 2q67 + q66 + 6q65−8q64−7q63 + 2q62 + 3q61 + 26q60−12q59−23q58−11q57−5q56 + 63q55 + 3q54−30q53−37q52−44q51 + 93q50 + 33q49−7q48−47q47−101q46 + 92q45 + 41q44 + 30q43−20q42−135q41 + 80q40 + 10q39 + 38q38 + 20q37−126q36 + 97q35−33q34 + 31q32−96q31 + 156q30−53q29−65q28 + 3q27−71q26 + 234q25−45q24−127q23−44q22−58q21 + 309q20−24q19−183q18−95q17−45q16 + 374q15 + 4q14−227q13−148q12−44q11 + 416q10 + 50q9−236q8−195q7−72q6 + 404q5 + 101q4−183q3−201q2−119q + 324 + 122q−1−95q−2−153q−3−141q−4 + 208q−5 + 96q−6−22q−7−80q−8−125q−9 + 110q−10 + 52q−11 + 11q−12−25q−13−85q−14 + 49q−15 + 17q−16 + 16q−17−q−18−44q−19 + 19q−20 + 2q−21 + 9q−22 + 3q−23−15q−24 + 5q−25−q−26 + 3q−27 + q−28−3q−29 + q−30 |
[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. |
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