10 136
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 136's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_136's page at Knotilus! Visit 10 136's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13 |
| Gauss code | -1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7 |
| Dowker-Thistlethwaite code | 4 8 10 -14 2 -18 -6 -20 -12 -16 |
| Conway Notation | [22,22,2-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 10, width is 5, Braid index is 4 | 
|  [{11, 2}, {1, 7}, {9, 5}, {7, 11}, {8, 10}, {2, 9}, {6, 4}, {5, 8}, {3, 6}, {4, 1}, {10, 3}] |
[edit Notes on presentations of 10 136] The knot 10_136 is the only knot in the Rolfsen Knot Table whose braid index is smaller than the width of its minimum braid.
The next such knot is K11n8.
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 136"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 8 10 -14 2 -18 -6 -20 -12 -16 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [22,22,2-] |
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,−2,1,−2,−3,2,2,4,−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, 10, 4 } |
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[{11, 2}, {1, 7}, {9, 5}, {7, 11}, {8, 10}, {2, 9}, {6, 4}, {5, 8}, {3, 6}, {4, 1}, {10, 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 | −t2 + 4t−5 + 4t−1−t−2 |
| Conway polynomial | 1−z4 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 15, 2 } |
| Jones polynomial | −q4 + 2q3−2q2 + 3q−2 + 2q−1−2q−2 + q−3 |
| HOMFLY-PT polynomial (db, data sources) | −z4 + a2z2 + 2z2a−2−3z2 + a2 + 3a−2−a−4−2 |
| Kauffman polynomial (db, data sources) | z8a−2 + z8 + 2az7 + 3z7a−1 + z7a−3 + a2z6−4z6a−2−3z6−9az5−14z5a−1−5z5a−3−4a2z4 + 2z4a−2−2z4 + 9az3 + 16z3a−1 + 7z3a−3 + 3a2z2 + 4z2a−2 + z2a−4 + 6z2−2az−4za−1−2za−3−a2−3a−2−a−4−2 |
| The A2 invariant | q10−q2 + q−4 + 2q−6 + q−8 + q−10−q−12−q−14 |
| The G2 invariant | q46−q44 + 2q42−2q40 + q38−2q34 + 6q32−4q30 + 3q28−2q24 + 3q22−2q20−q18 + 3q16−3q14 + 3q10−6q8 + 6q6−7q4 + 3−5q−2 + 4q−4−3q−6 + 3q−8 + q−12−q−14 + 2q−18 + q−20 + q−24 + 3q−26 + 4q−30−6q−32 + 6q−34−2q−36 + 4q−40−7q−42 + 6q−44−2q−48 + q−50−2q−52−2q−54 + 3q−56−3q−58 + q−60 + q−62−3q−64 + 3q−66−3q−68 + q−70 + q−72−2q−74 + q−76 |
Further Quantum Invariants
Further quantum knot invariants for 10_136.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q7−q5 + q−1 + q−3 + q−7−q−9 |
| 2 | q22−q20−2q18 + 2q16 + q14−q12 + q8 + q6−2q4 + 2−q−2 + 2q−6−q−8 + 2q−12 + q−14−q−16−q−18 + q−20−2q−24 + q−28 |
| 3 | q45−q43−2q41 + 3q37 + 3q35−3q33−3q31 + 2q27 + 2q25−2q21−4q19 + q17 + 6q15 + 2q13−6q11−4q9 + 5q7 + 6q5−3q3−5q + 2q−1 + 6q−3−q−5−4q−7 + q−9 + 3q−11−q−13−2q−15 + q−17 + q−19 + 2q−21−4q−25−q−27 + 7q−29 + 5q−31−7q−33−10q−35 + 5q−37 + 8q−39−2q−41−7q−43−q−45 + 5q−47 + 2q−49−q−51−q−53 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q10−q2 + q−4 + 2q−6 + q−8 + q−10−q−12−q−14 |
| 2,0 | q28−q24−q22 + q18 + 2q8 + q6 + q2 + 1−2q−4−q−6−q−8−q−10 + 2q−14 + 4q−16 + 3q−18 + 4q−20−q−22−q−24−2q−26−2q−28−2q−30−2q−32 + q−34 + q−36 + q−38 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q20−q18 + q14 + q10 + q6−q4−q2−q−2 + q−6 + 3q−8 + q−10 + 3q−12 + q−14−q−18−q−20−q−24 |
| 1,0,0 | q13 + q9−q3−q−q−1 + q−5 + 2q−7 + 3q−9 + q−11 + q−13−q−15−q−17−q−19 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q20−q18 + 2q16−q14 + 2q12−q10−q6−q4 + q2−4 + 3q−2−2q−4 + 3q−6−q−8 + 3q−10 + q−12 + q−14 + 2q−16−q−18 + q−20−2q−22 + q−24−2q−26 |
| 1,0 | q34−q30−q28 + q26 + q24−q22−q20 + 2q18 + 2q16−q14−q12 + 2q8−2q4−q2 + 1−q−4 + 2q−10 + q−12 + 3q−18 + 2q−20−2q−24 + q−26 + q−28−q−30−2q−32 + q−36−q−40−q−42 + q−44 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q46−q44 + 2q42−2q40 + q38−2q34 + 6q32−4q30 + 3q28−2q24 + 3q22−2q20−q18 + 3q16−3q14 + 3q10−6q8 + 6q6−7q4 + 3−5q−2 + 4q−4−3q−6 + 3q−8 + q−12−q−14 + 2q−18 + q−20 + q−24 + 3q−26 + 4q−30−6q−32 + 6q−34−2q−36 + 4q−40−7q−42 + 6q−44−2q−48 + q−50−2q−52−2q−54 + 3q−56−3q−58 + q−60 + q−62−3q−64 + 3q−66−3q−68 + q−70 + q−72−2q−74 + q−76 |
.
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`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
| K = Knot["10 136"];
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In[4]:=
| Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| −t2 + 4t−5 + 4t−1−t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 1−z4 |
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]=
| { 15, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q4 + 2q3−2q2 + 3q−2 + 2q−1−2q−2 + q−3 |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
| −z4 + a2z2 + 2z2a−2−3z2 + a2 + 3a−2−a−4−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z8a−2 + z8 + 2az7 + 3z7a−1 + z7a−3 + a2z6−4z6a−2−3z6−9az5−14z5a−1−5z5a−3−4a2z4 + 2z4a−2−2z4 + 9az3 + 16z3a−1 + 7z3a−3 + 3a2z2 + 4z2a−2 + z2a−4 + 6z2−2az−4za−1−2za−3−a2−3a−2−a−4−2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {8_21,}
Same Jones Polynomial (up to mirroring, 
):
{K11n92,}
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["10 136"];
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In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
| { −t2 + 4t−5 + 4t−1−t−2, −q4 + 2q3−2q2 + 3q−2 + 2q−1−2q−2 + q−3 } |
In[5]:=
| DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
| {8_21,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
| {K11n92,} |
[edit] Vassiliev invariants
| V2 and V3: | (0, 1) |
| V2,1 through V6,9: |
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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 10 136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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[edit] The Coloured Jones Polynomials
| n | Jn |
| 2 | q13−q12−2q11 + 3q10−4q8 + 3q7 + 2q6−3q5 + q4 + q3−q + 3q−1−3q−2−2q−3 + 6q−4−3q−5−3q−6 + 5q−7−q−8−2q−9 + q−10 |
| 3 | −q25 + 3q23 + 3q22−7q21−6q20 + 8q19 + 13q18−10q17−21q16 + 11q15 + 25q14−8q13−29q12 + 8q11 + 29q10−6q9−30q8 + 8q7 + 26q6−5q5−26q4 + 6q3 + 21q2−2q−19 + 2q−1 + 14q−2−10q−4 + q−5 + 5q−6−2q−7−2q−8 + 5q−9−7q−11 + 7q−13 + 2q−14−7q−15−2q−16 + 4q−17 + 2q−18−q−19−2q−20 + q−21 |
| 4 | q42−q41−2q39−2q38 + 5q37 + q36 + 8q35−7q34−15q33 + q32 + 2q31 + 33q30 + 2q29−31q28−22q27−13q26 + 62q25 + 26q24−33q23−42q22−36q21 + 71q20 + 45q19−22q18−48q17−51q16 + 67q15 + 51q14−16q13−45q12−54q11 + 57q10 + 52q9−9q8−39q7−55q6 + 41q5 + 53q4 + 4q3−29q2−58q + 17 + 50q−1 + 20q−2−13q−3−54q−4−9q−5 + 36q−6 + 27q−7 + 6q−8−37q−9−23q−10 + 18q−11 + 17q−12 + 13q−13−14q−14−17q−15 + 11q−16 + 4q−18−4q−19−7q−20 + 16q−21−3q−22−3q−23−7q−24−6q−25 + 15q−26 + q−27−5q−29−6q−30 + 5q−31 + q−32 + 2q−33−q−34−2q−35 + q−36 |
| 5 | −q62 + q60 + 3q59 + 2q58−7q56−11q55−4q54 + 10q53 + 22q52 + 19q51−4q50−37q49−44q48−12q47 + 45q46 + 73q45 + 45q44−37q43−106q42−86q41 + 19q40 + 124q39 + 128q38 + 12q37−130q36−165q35−43q34 + 126q33 + 184q32 + 69q31−113q30−191q29−87q28 + 103q27 + 188q26 + 95q25−93q24−184q23−94q22 + 86q21 + 176q20 + 94q19−81q18−171q17−90q16 + 75q15 + 159q14 + 90q13−64q12−149q11−89q10 + 49q9 + 131q8 + 92q7−28q6−110q5−92q4 + 4q3 + 86q2 + 87q + 21−52q−1−80q−2−44q−3 + 22q−4 + 60q−5 + 54q−6 + 19q−7−37q−8−61q−9−40q−10 + 3q−11 + 48q−12 + 61q−13 + 24q−14−30q−15−58q−16−46q−17 + 4q−18 + 51q−19 + 51q−20 + 14q−21−31q−22−45q−23−23q−24 + 13q−25 + 32q−26 + 20q−27−5q−28−16q−29−10q−30 + 11q−32 + 4q−33−10q−34−6q−35 + 2q−36 + 8q−37 + 10q−38 + 2q−39−12q−40−10q−41−2q−42 + 6q−43 + 8q−44 + 4q−45−7q−47−4q−48 + q−49 + 2q−50 + q−51 + 2q−52−q−53−2q−54 + q−55 |
| 6 | q87−q86−2q83−2q82−q81 + 5q80 + 4q79 + 10q78 + 5q77−6q76−22q75−25q74−14q73 + 4q72 + 54q71 + 61q70 + 41q69−30q68−86q67−123q66−88q65 + 64q64 + 167q63 + 213q62 + 90q61−80q60−290q59−322q58−83q57 + 187q56 + 422q55 + 346q54 + 98q53−358q52−560q51−339q50 + 53q49 + 506q48 + 562q47 + 341q46−284q45−649q44−522q43−113q42 + 461q41 + 633q40 + 489q39−193q38−625q37−574q36−196q35 + 403q34 + 620q33 + 528q32−159q31−588q30−566q29−214q28 + 378q27 + 599q26 + 523q25−151q24−557q23−548q22−221q21 + 347q20 + 571q19 + 516q18−119q17−497q16−527q15−251q14 + 273q13 + 511q12 + 516q11−37q10−384q9−487q8−307q7 + 141q6 + 402q5 + 503q4 + 81q3−210q2−398q−352−28q−1 + 231q−2 + 427q−3 + 177q−4−4q−5−233q−6−315q−7−155q−8 + 31q−9 + 253q−10 + 166q−11 + 142q−12−33q−13−166q−14−150q−15−93q−16 + 48q−17 + 35q−18 + 132q−19 + 77q−20 + 4q−21−21q−22−63q−23−47q−24−92q−25 + 8q−26 + 33q−27 + 52q−28 + 86q−29 + 42q−30−2q−31−90q−32−65q−33−51q−34−10q−35 + 73q−36 + 69q−37 + 50q−38−16q−39−34q−40−53q−41−45q−42 + 25q−43 + 22q−44 + 30q−45 + 7q−46−q−47−18q−48−22q−49 + 24q−50 + 7q−52−6q−53−6q−54−14q−55−11q−56 + 28q−57 + 5q−58 + 9q−59−2q−60−5q−61−15q−62−12q−63 + 11q−64 + 2q−65 + 8q−66 + 3q−67 + 3q−68−7q−69−6q−70 + 3q−71−2q−72 + 2q−73 + q−74 + 2q−75−q−76−2q−77 + q−78 |
| 7 | −q115 + q113 + q112 + 2q111 + 2q110−q108−9q107−10q106−6q105−2q104 + 12q103 + 23q102 + 32q101 + 29q100−5q99−45q98−64q97−82q96−48q95 + 22q94 + 109q93 + 184q92 + 154q91 + 46q90−99q89−272q88−336q87−235q86 + 346q84 + 539q83 + 497q82 + 225q81−284q80−708q79−840q78−577q77 + 103q76 + 791q75 + 1145q74 + 994q73 + 211q72−728q71−1362q70−1414q69−601q68 + 552q67 + 1476q66 + 1729q65 + 965q64−298q63−1434q62−1937q61−1286q60 + 46q59 + 1340q58 + 2021q57 + 1470q56 + 169q55−1195q54−2016q53−1585q52−312q51 + 1087q50 + 1972q49 + 1607q48 + 387q47−988q46−1922q45−1614q44−415q43 + 951q42 + 1875q41 + 1580q40 + 430q39−907q38−1850q37−1583q36−422q35 + 907q34 + 1811q33 + 1544q32 + 440q31−852q30−1783q29−1560q28−452q27 + 830q26 + 1720q25 + 1518q24 + 504q23−719q22−1652q21−1534q20−564q19 + 630q18 + 1538q17 + 1490q16 + 666q15−447q14−1400q13−1488q12−772q11 + 265q10 + 1210q9 + 1418q8 + 897q7−11q6−978q5−1338q4−1001q3−232q2 + 689q + 1174 + 1068q−1 + 489q−2−368q−3−962q−4−1049q−5−692q−6 + 30q−7 + 672q−8 + 952q−9 + 815q−10 + 263q−11−345q−12−745q−13−825q−14−492q−15 + 34q−16 + 478q−17 + 709q−18 + 591q−19 + 220q−20−184q−21−508q−22−556q−23−359q−24−59q−25 + 255q−26 + 417q−27 + 370q−28 + 200q−29−35q−30−223q−31−265q−32−228q−33−100q−34 + 46q−35 + 117q−36 + 150q−37 + 124q−38 + 60q−39 + 21q−40−26q−41−73q−42−75q−43−92q−44−71q−45−18q−46 + 20q−47 + 91q−48 + 123q−49 + 85q−50 + 37q−51−40q−52−104q−53−99q−54−88q−55−18q−56 + 66q−57 + 82q−58 + 86q−59 + 38q−60−23q−61−35q−62−59q−63−50q−64 + 5q−65 + 18q−66 + 36q−67 + 23q−68−14q−69−q−70−12q−71−16q−72 + 12q−73 + 11q−74 + 17q−75 + 10q−76−24q−77−12q−78−13q−79−12q−80 + 13q−81 + 10q−82 + 15q−83 + 16q−84−4q−85−8q−86−12q−87−14q−88 + 4q−89 + 3q−91 + 10q−92 + 3q−93 + 2q−94−4q−95−6q−96 + q−97−2q−99 + 2q−100 + q−101 + 2q−102−q−103−2q−104 + q−105 |
[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. |
|
