10 160
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 160's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_160's page at Knotilus! Visit 10 160's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,4,13,3 X7,14,8,15 X9,19,10,18 X19,7,20,6 X5,17,6,16 X17,11,18,10 X13,8,14,9 X15,1,16,20 X2,12,3,11 |
| Gauss code | 1, -10, 2, -1, -6, 5, -3, 8, -4, 7, 10, -2, -8, 3, -9, 6, -7, 4, -5, 9 |
| Dowker-Thistlethwaite code | 4 12 -16 -14 -18 2 -8 -20 -10 -6 |
| Conway Notation | [-30:20:20] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{4, 10}, {3, 5}, {1, 4}, {7, 9}, {11, 8}, {10, 6}, {5, 7}, {6, 12}, {2, 11}, {12, 3}, {9, 2}, {8, 1}] |
[edit Notes on presentations of 10 160]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 160"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X4251 X12,4,13,3 X7,14,8,15 X9,19,10,18 X19,7,20,6 X5,17,6,16 X17,11,18,10 X13,8,14,9 X15,1,16,20 X2,12,3,11 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -10, 2, -1, -6, 5, -3, 8, -4, 7, 10, -2, -8, 3, -9, 6, -7, 4, -5, 9 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 12 -16 -14 -18 2 -8 -20 -10 -6 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [-30:20:20] |
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(4,{1,1,1,2,1,1,−3,2,−1,2,−3}) |
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]=
| { 4, 11, 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[{4, 10}, {3, 5}, {1, 4}, {7, 9}, {11, 8}, {10, 6}, {5, 7}, {6, 12}, {2, 11}, {12, 3}, {9, 2}, {8, 1}] |
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 | −t3 + 4t2−4t + 3−4t−1 + 4t−2−t−3 |
| Conway polynomial | −z6−2z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 21, 4 } |
| Jones polynomial | −2q7 + 3q6−3q5 + 4q4−3q3 + 3q2−2q + 1 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−4 + z4a−2−4z4a−4 + z4a−6 + 3z2a−2−3z2a−4 + 3z2a−6 + a−2 + a−6−a−8 |
| Kauffman polynomial (db, data sources) | z8a−4 + z8a−6 + 2z7a−3 + 3z7a−5 + z7a−7 + z6a−2−2z6a−4−3z6a−6−8z5a−3−11z5a−5−3z5a−7−4z4a−2−3z4a−4 + 2z4a−6 + z4a−8 + 7z3a−3 + 10z3a−5 + 3z3a−7 + 4z2a−2 + 3z2a−4 + z2a−8−za−3−3za−5 + 2za−9−a−2−a−6−a−8 |
| The A2 invariant | 1 + 2q−10 + 2q−14−q−22−q−26 |
| The G2 invariant | q−2−q−4 + 3q−6−4q−8 + 3q−10−4q−14 + 10q−16−8q−18 + 7q−20−q−22−6q−24 + 9q−26−8q−28 + q−30 + 5q−32−8q−34 + 6q−36−7q−40 + 13q−42−12q−44 + 5q−46 + 2q−48−7q−50 + 12q−52−8q−54 + 7q−56−q−58 + 2q−60 + 4q−62−6q−64 + 5q−66−2q−68 + q−70 + 3q−72−5q−74 + 3q−76 + 3q−78−8q−80 + 10q−82−11q−84 + 3q−86 + 6q−88−13q−90 + 11q−92−6q−94 + 5q−98−7q−100 + q−102 + q−104−3q−106 + 3q−108−2q−110−2q−112 + 3q−114−4q−116 + 2q−118 + 2q−120−2q−122 + q−124 |
Further Quantum Invariants
Further quantum knot invariants for 10_160.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q−q−1 + q−3 + q−7 + q−9 + q−13−2q−15 |
| 2 | q6−q4−2q2 + 3 + 2q−2−3q−4 + 3q−8−2q−12 + 2q−14 + 2q−16−2q−18 + q−20 + 3q−22−2q−24−q−26 + 2q−28−4q−32 + 2q−36−2q−38−q−40 + q−42 + q−44 |
| 3 | q15−q13−2q11 + 4q7 + 4q5−4q3−7q−q−1 + 7q−3 + 7q−5−3q−7−8q−9−3q−11 + 9q−13 + 10q−15−3q−17−12q−19−2q−21 + 12q−23 + 6q−25−10q−27−7q−29 + 8q−31 + 9q−33−6q−35−7q−37 + 6q−39 + 6q−41−5q−43−6q−45 + 2q−47 + 5q−49 + q−51−8q−53−8q−55 + 3q−57 + 13q−59 + q−61−14q−63−8q−65 + 15q−67 + 11q−69−7q−71−11q−73 + 9q−77 + 2q−79−2q−81−2q−83 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | 1 + 2q−10 + 2q−14−q−22−q−26 |
| 1,1 | q4−2q2 + 6−12q−2 + 17q−4−20q−6 + 24q−8−18q−10 + 11q−12 + 4q−14−14q−16 + 26q−18−34q−20 + 38q−22−32q−24 + 32q−26−21q−28 + 16q−30−4q−32−8q−34 + 13q−36−26q−38 + 24q−40−24q−42 + 18q−44−8q−46 + 6q−50−7q−52 + 2q−54−4q−56 + 4q−60−2q−62 + 2q−64 |
| 2,0 | q4−1−q−2 + q−4 + 2q−6 + 2q−12 + 2q−14−q−16−q−18 + q−22 + 2q−24 + 3q−28 + q−30 + 3q−32 + q−34−q−36−q−38−q−40−2q−42−4q−44−2q−46−q−48 + 2q−50−q−52−q−54 + q−56 + q−58 + q−60−q−62 + q−66 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | 1−q−2 + q−4 + q−6−q−8 + 2q−10 + q−14 + 2q−16 + q−20 + q−22 + q−24 + 2q−28 + 2q−32−q−34−q−36−q−38−4q−40−q−42−q−44−q−46 + q−48 + 2q−50 |
| 1,0,0 | q−1 + q−5−q−7 + q−9 + q−13 + q−15 + q−17 + q−19 + q−23−q−25 + q−27−q−29−q−33−q−35 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q−2 + q−8 + q−12 + q−14 + q−18 + 3q−20 + q−22 + 3q−26 + 3q−28 + q−30−2q−32 + 4q−34 + 2q−36−2q−38 + q−40 + 3q−42−3q−44−q−46 + q−48−3q−50−4q−52−q−54−4q−58−2q−60 + 2q−62 + q−64−2q−66 + 2q−68 + 2q−70 |
| 1,0,0,0 | q−2 + q−6 + q−12 + q−16 + 2q−20 + q−24 + q−28 + q−34−q−36−q−40−q−42−q−44 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | 1−q−2 + 3q−4−3q−6 + 3q−8−2q−10 + 2q−12−q−14 + 2q−18−3q−20 + 5q−22−5q−24 + 6q−26−4q−28 + 4q−30−2q−32 + q−34 + q−36−q−38 + 2q−40−3q−42 + 3q−44−3q−46 + q−48−2q−50 |
| 1,0 | q2−q−2−q−4 + 2q−6 + 2q−8−2q−10−2q−12 + 2q−14 + 3q−16−3q−20 + 4q−24 + 2q−26−2q−28−q−30 + q−32 + 2q−34−q−38 + 3q−42 + q−44−q−46−q−48 + 2q−50 + 3q−52−q−54−4q−56 + 2q−60−q−62−4q−64−2q−66 + q−68 + q−70−2q−72−2q−74 + q−76 + q−78 + 2q−80 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | q−2−q−4 + 2q−6−2q−8 + 4q−10−2q−12 + 2q−14−q−16 + 3q−18 + 2q−24−q−26 + 4q−28−2q−30 + 4q−32−4q−34 + 5q−36−3q−38 + 4q−40−2q−42 + 3q−44−q−46 + q−48−3q−52−4q−56−4q−60 + 2q−62−2q−64 + q−66 + 2q−70 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−2−q−4 + 3q−6−4q−8 + 3q−10−4q−14 + 10q−16−8q−18 + 7q−20−q−22−6q−24 + 9q−26−8q−28 + q−30 + 5q−32−8q−34 + 6q−36−7q−40 + 13q−42−12q−44 + 5q−46 + 2q−48−7q−50 + 12q−52−8q−54 + 7q−56−q−58 + 2q−60 + 4q−62−6q−64 + 5q−66−2q−68 + q−70 + 3q−72−5q−74 + 3q−76 + 3q−78−8q−80 + 10q−82−11q−84 + 3q−86 + 6q−88−13q−90 + 11q−92−6q−94 + 5q−98−7q−100 + q−102 + q−104−3q−106 + 3q−108−2q−110−2q−112 + 3q−114−4q−116 + 2q−118 + 2q−120−2q−122 + q−124 |
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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["10 160"];
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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]=
| −t3 + 4t2−4t + 3−4t−1 + 4t−2−t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z6−2z4 + 3z2 + 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]=
| { 21, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −2q7 + 3q6−3q5 + 4q4−3q3 + 3q2−2q + 1 |
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]=
| −z6a−4 + z4a−2−4z4a−4 + z4a−6 + 3z2a−2−3z2a−4 + 3z2a−6 + a−2 + a−6−a−8 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z8a−4 + z8a−6 + 2z7a−3 + 3z7a−5 + z7a−7 + z6a−2−2z6a−4−3z6a−6−8z5a−3−11z5a−5−3z5a−7−4z4a−2−3z4a−4 + 2z4a−6 + z4a−8 + 7z3a−3 + 10z3a−5 + 3z3a−7 + 4z2a−2 + 3z2a−4 + z2a−8−za−3−3za−5 + 2za−9−a−2−a−6−a−8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n118,}
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`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 160"];
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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]=
| { −t3 + 4t2−4t + 3−4t−1 + 4t−2−t−3, −2q7 + 3q6−3q5 + 4q4−3q3 + 3q2−2q + 1 } |
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]=
| {K11n118,} |
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]=
| {} |
[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 10 160. 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 | q21−2q19 + 4q17−4q16−4q15 + 8q14−2q13−7q12 + 7q11 + 3q10−9q9 + 4q8 + 7q7−9q6 + 9q4−6q3−3q2 + 6q−1−2q−1 + q−2 |
| 3 | −2q40 + 4q38 + 7q37−11q36−11q35 + 8q34 + 25q33−7q32−34q31 + 2q30 + 40q29 + 5q28−44q27−9q26 + 40q25 + 14q24−40q23−12q22 + 32q21 + 15q20−29q19−12q18 + 19q17 + 16q16−14q15−13q14 + 4q13 + 13q12 + 2q11−7q10−10q9 + 3q8 + 11q7 + 6q6−11q5−9q4 + 6q3 + 11q2−q−9−2q−1 + 5q−2 + 2q−3−q−4−2q−5 + q−6 |
| 4 | q66 + 2q64−4q63−8q62 + 7q60 + 27q59 + 2q58−38q57−35q56−2q55 + 82q54 + 54q53−53q52−101q51−58q50 + 120q49 + 131q48−24q47−142q46−124q45 + 107q44 + 173q43 + 17q42−138q41−154q40 + 79q39 + 173q38 + 33q37−114q36−150q35 + 53q34 + 157q33 + 40q32−86q31−139q30 + 22q29 + 136q28 + 52q27−47q26−125q25−21q24 + 99q23 + 63q22 + 4q21−90q20−57q19 + 43q18 + 47q17 + 45q16−31q15−55q14−2q13 + q12 + 41q11 + 16q10−16q9−3q8−33q7 + 5q6 + 15q5 + 9q4 + 21q3−21q2−12q−7 + q−1 + 22q−2−q−3−2q−4−7q−5−6q−6 + 6q−7 + q−8 + 2q−9−q−10−2q−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. |
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