10 132
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 132's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_132's page at Knotilus! Visit 10 132's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837 |
| Gauss code | 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7 |
| Dowker-Thistlethwaite code | 4 8 -12 2 -16 -6 -20 -18 -10 -14 |
| Conway Notation | [23,3,2-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 8}, {7, 9}, {8, 5}, {4, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}] |
[edit Notes on presentations of 10 132]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 132"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 8 -12 2 -16 -6 -20 -18 -10 -14 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [23,3,2-] |
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(4,{1,1,1,−2,−1,−1,−2,−3,2,−3,−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[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 8}, {7, 9}, {8, 5}, {4, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}] |
In[14]:=
| Draw[ap]
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Out[14]=
| -Graphics- |
[edit] Three dimensional invariants
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[edit Notes for 10 132's three dimensional invariants] 10 132 is a very interesting knot from the point of view of contact geometry. In particular, it is a transversely nonsimple knot, and it was the last knot with at most 10 crossings for which the maximal Thurston-Bennequin number was calculated. |
[edit] Four dimensional invariants
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[edit] Polynomial invariants
| Alexander polynomial | t2−t + 1−t−1 + t−2 |
| Conway polynomial | z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 5, 0 } |
| Jones polynomial | q−2 + q−4−q−5 + q−6−q−7 |
| HOMFLY-PT polynomial (db, data sources) | −z2a6−2a6 + z4a4 + 4z2a4 + 3a4 |
| Kauffman polynomial (db, data sources) | a6z8 + a4z8 + a7z7 + 2a5z7 + a3z7−6a6z6−6a4z6−6a7z5−12a5z5−6a3z5 + 10a6z4 + 10a4z4 + 10a7z3 + 19a5z3 + 9a3z3−6a6z2−7a4z2−a2z2−5a7z−8a5z−4a3z−az + 2a6 + 3a4 |
| The A2 invariant | −q22−q20−q18 + q14 + q12 + 2q10 + q8 + q6 |
| The G2 invariant | q108 + q104−q100−q92−q90−q86−q84−q82−2q80−q78−q76−2q74−q68−q64 + q62 + q60 + q58 + q56 + q54 + 2q52 + 3q50 + q48 + q46 + 2q44 + q42 + 2q40 + q38 + q34 + q32−q28 + q26−q24−q18 + q16−q12 + q4−q2 + 1 + q−6−q−8 |
Further Quantum Invariants
Further quantum knot invariants for 10_132.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q15 + q7 + q5 + q3 |
| 2 | q44−q40−q30−q24 + q16 + q14 + q10 + q6 + 2−q−4 |
| 3 | −q87 + q83 + q81−q77 + q67−q63 + q59 + q57−q55−q53−q45−q43 + q39−q35 + q31−q27−q25 + q17 + 2q15 + 2q13 + q7 + 2q5 + q3−q−q−1 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q22−q20−q18 + q14 + q12 + 2q10 + q8 + q6 |
| 1,1 | q60 + 2q56−2q50−2q48−2q44 + 2q36−2q34−2q30−q28−2q24 + 2q22−q20 + 4q18 + 6q14 + q12 + 4q10 + 2q8−2q6 + 2q4−2q2 + 2−2q−2 |
| 2,0 | q58 + q56 + q54−q48−q46−2q44−2q42−2q40−q38 + q36−q34 + q28 + q24 + 2q22 + 2q20 + q18 + q16 + q14 + q10 + q8 + q4 + q2 + 1−q−2−q−4 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q46 + q42 + q40−2q36−2q34−4q32−5q30−2q28−q26 + 2q24 + 3q22 + 6q20 + 4q18 + 3q16 + 2q14 + q12−q10−q8−q4 + 1 |
| 1,0,0 | −q29−q27−2q25−q23 + q19 + 2q17 + 2q15 + 2q13 + q11 + q9 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q60 + q58 + 2q56 + 2q54 + 2q52 + q50−q48−4q46−6q44−8q42−9q40−8q38−5q36 + 4q32 + 8q30 + 11q28 + 11q26 + 8q24 + 6q22 + q20−q18−3q16−2q14−2q12−q10 + q6 + 1 |
| 1,0,0,0 | −q36−q34−2q32−2q30−q28 + q24 + 2q22 + 3q20 + 2q18 + 2q16 + q14 + q12 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q46−q42−q40 + q30 + q26 + q22 + q16 + q12 + q10 + q8 + q4−1 |
| 1,0 | q76 + q68−q58−2q56−q54−q52−2q50−2q48−q46−q44 + q38 + q36 + 2q34 + 2q32 + 2q30 + q28 + 2q26 + 2q24 + q22 + q18−q12−q4 + 1 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | q62 + q58 + q56 + q54−q50−2q48−4q46−4q44−5q42−4q40−3q38 + q34 + 4q32 + 5q30 + 6q28 + 5q26 + 4q24 + 3q22 + q20 + q18−q16−q14−q12−q8 + 1 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q108 + q104−q100−q92−q90−q86−q84−q82−2q80−q78−q76−2q74−q68−q64 + q62 + q60 + q58 + q56 + q54 + 2q52 + 3q50 + q48 + q46 + 2q44 + q42 + 2q40 + q38 + q34 + q32−q28 + q26−q24−q18 + q16−q12 + q4−q2 + 1 + q−6−q−8 |
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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["10 132"];
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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−t + 1−t−1 + t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z4 + 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]=
| { 5, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q−2 + q−4−q−5 + q−6−q−7 |
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]=
| −z2a6−2a6 + z4a4 + 4z2a4 + 3a4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a6z8 + a4z8 + a7z7 + 2a5z7 + a3z7−6a6z6−6a4z6−6a7z5−12a5z5−6a3z5 + 10a6z4 + 10a4z4 + 10a7z3 + 19a5z3 + 9a3z3−6a6z2−7a4z2−a2z2−5a7z−8a5z−4a3z−az + 2a6 + 3a4 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {5_1,}
Same Jones Polynomial (up to mirroring, 
):
{5_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.
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In[3]:=
| K = Knot["10 132"];
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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−t + 1−t−1 + t−2, q−2 + q−4−q−5 + q−6−q−7 } |
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]=
| {5_1,} |
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]=
| {5_1,} |
[edit] Vassiliev invariants
| V2 and V3: | (3, -5) |
| 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 = 0 is the signature of 10 132. 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 | −q + 1 + 2q−1−3q−2 + q−3 + 3q−4−4q−5 + 2q−6 + 2q−7−3q−8 + 2q−9 + q−10−3q−11 + 2q−12−2q−14 + 2q−15−q−16−q−17 + 2q−18−q−19−q−20 + q−21 |
| 3 | −q−1 + 2q−3 + q−4−2q−5−q−6 + 2q−7 + 3q−8−2q−9−2q−10 + q−11 + 3q−12−2q−13−3q−14 + q−15 + 4q−16−q−17−4q−18 + 5q−20−5q−22−q−23 + 5q−24 + q−25−5q−26−q−27 + 4q−28 + q−29−3q−30−q−31 + 3q−32−2q−34 + 2q−36−2q−38 + q−40 + q−41−q−42 |
| 4 | q5−q4−q3−q2 + 5−q−2−6q−3−2q−4 + 9q−5 + 2q−6−9q−8−3q−9 + 10q−10 + 3q−11 + q−12−10q−13−3q−14 + 11q−15 + 3q−16−q−17−10q−18−3q−19 + 11q−20 + 2q−21−2q−22−9q−23−2q−24 + 10q−25 + 2q−26−2q−27−7q−28−2q−29 + 8q−30 + 2q−31−2q−32−5q−33−q−34 + 6q−35 + q−36−3q−37−3q−38 + q−39 + 5q−40−5q−42−2q−43 + 2q−44 + 6q−45−6q−47−2q−48 + q−49 + 6q−50 + q−51−4q−52−2q−53−q−54 + 5q−55−2q−57−q−58−q−59 + 4q−60−q−61−q−62−q−63−q−64 + 3q−65−q−68−q−69 + q−70 |
[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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