10 13
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_13's page at Knotilus! Visit 10 13's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,18,6,19 X13,1,14,20 X19,15,20,14 X7,16,8,17 X15,8,16,9 X17,6,18,7 |
| Gauss code | -1, 4, -3, 1, -5, 10, -8, 9, -2, 3, -4, 2, -6, 7, -9, 8, -10, 5, -7, 6 |
| Dowker-Thistlethwaite code | 4 10 18 16 12 2 20 8 6 14 |
| Conway Notation | [4222] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||||
Length is 11, width is 6, Braid index is 6 | 
|  [{12, 9}, {10, 8}, {9, 11}, {3, 10}, {7, 1}, {8, 6}, {5, 7}, {6, 4}, {2, 5}, {4, 12}, {1, 3}, {11, 2}] |
[edit Notes on presentations of 10 13]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 13"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,18,6,19 X13,1,14,20 X19,15,20,14 X7,16,8,17 X15,8,16,9 X17,6,18,7 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 4, -3, 1, -5, 10, -8, 9, -2, 3, -4, 2, -6, 7, -9, 8, -10, 5, -7, 6 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 10 18 16 12 2 20 8 6 14 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [4222] |
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(6,{−1,−1,−2,1,3,−2,−4,3,5,−4,5}) |
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]=
| { 6, 11, 6 } |
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, 9}, {10, 8}, {9, 11}, {3, 10}, {7, 1}, {8, 6}, {5, 7}, {6, 4}, {2, 5}, {4, 12}, {1, 3}, {11, 2}] |
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 | 2t2−13t + 23−13t−1 + 2t−2 |
| Conway polynomial | 2z4−5z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 53, 0 } |
| Jones polynomial | q4−2q3 + 5q2−7q + 8−9q−1 + 8q−2−6q−3 + 4q−4−2q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | a6−2z2a4−a4 + z4a2 + a2 + z4−z2−1−2z2a−2 + a−4 |
| Kauffman polynomial (db, data sources) | a3z9 + az9 + 2a4z8 + 4a2z8 + 2z8 + 2a5z7 + az7 + 3z7a−1 + a6z6−5a4z6−9a2z6 + 3z6a−2−7a5z5−4a3z5−2az5−3z5a−1 + 2z5a−3−4a6z4 + a4z4 + 6a2z4−3z4a−2 + z4a−4−3z4 + 6a5z3 + a3z3 + 3z3a−1−2z3a−3 + 4a6z2 + 2a4z2−a2z2 + z2a−2−2z2a−4 + 4z2−a5z + a3z−2za−1−a6−a4−a2 + a−4−1 |
| The A2 invariant | q20 + q18−q16 + q14−2q10 + 2q8−2 + q−2−2q−4 + q−6 + 2q−8−q−10 + q−12 + q−14 |
| The G2 invariant | Data:10 13/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
Further quantum knot invariants for 10_13.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q13−q11 + 2q9−2q7 + 2q5−q3−q + q−1−2q−3 + 3q−5−q−7 + q−9 |
| 2 | q38−q36−q34 + 4q32−2q30−6q28 + 8q26 + 2q24−11q22 + 7q20 + 6q18−13q16 + 3q14 + 9q12−8q10−2q8 + 7q6 + 2q4−6q2 + 11q−2−7q−4−7q−6 + 13q−8−5q−10−8q−12 + 9q−14−2q−16−4q−18 + 4q−20−q−24 + q−26 |
| 3 | q75−q73−q71 + q69 + 3q67−2q65−7q63 + q61 + 13q59 + 3q57−17q55−12q53 + 20q51 + 23q49−17q47−34q45 + 6q43 + 41q41 + 6q39−42q37−19q35 + 39q33 + 31q31−31q29−37q27 + 23q25 + 40q23−13q21−42q19 + 7q17 + 37q15 + 5q13−32q11−16q9 + 22q7 + 26q5−10q3−36q−5q−1 + 40q−3 + 22q−5−40q−7−29q−9 + 32q−11 + 36q−13−24q−15−30q−17 + 14q−19 + 26q−21−9q−23−18q−25 + 7q−27 + 9q−29−4q−31−7q−33 + 4q−35 + 4q−37−3q−39−3q−41 + 2q−43 + q−45−q−49 + q−51 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q20 + q18−q16 + q14−2q10 + 2q8−2 + q−2−2q−4 + q−6 + 2q−8−q−10 + q−12 + q−14 |
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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 13"];
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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]=
| 2t2−13t + 23−13t−1 + 2t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 2z4−5z2 + 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]=
| { 53, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q4−2q3 + 5q2−7q + 8−9q−1 + 8q−2−6q−3 + 4q−4−2q−5 + q−6 |
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]=
| a6−2z2a4−a4 + z4a2 + a2 + z4−z2−1−2z2a−2 + a−4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a3z9 + az9 + 2a4z8 + 4a2z8 + 2z8 + 2a5z7 + az7 + 3z7a−1 + a6z6−5a4z6−9a2z6 + 3z6a−2−7a5z5−4a3z5−2az5−3z5a−1 + 2z5a−3−4a6z4 + a4z4 + 6a2z4−3z4a−2 + z4a−4−3z4 + 6a5z3 + a3z3 + 3z3a−1−2z3a−3 + 4a6z2 + 2a4z2−a2z2 + z2a−2−2z2a−4 + 4z2−a5z + a3z−2za−1−a6−a4−a2 + a−4−1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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 13"];
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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]=
| { 2t2−13t + 23−13t−1 + 2t−2, q4−2q3 + 5q2−7q + 8−9q−1 + 8q−2−6q−3 + 4q−4−2q−5 + q−6 } |
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]=
| {} |
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 = 0 is the signature of 10 13. 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 | q12−2q11 + q10 + 5q9−10q8 + 3q7 + 16q6−27q5 + 6q4 + 34q3−47q2 + 6q + 52−58q−1 + 60q−3−53q−4−9q−5 + 54q−6−36q−7−15q−8 + 38q−9−17q−10−14q−11 + 20q−12−4q−13−8q−14 + 6q−15−2q−17 + q−18 |
| 3 | q24−2q23 + q22 + q21 + 2q20−7q19 + q18 + 8q17 + 2q16−18q15 + 4q14 + 21q13−43q11 + 13q10 + 56q9−12q8−87q7 + 19q6 + 116q5−16q4−148q3 + 8q2 + 178q + 2−193q−1−23q−2 + 204q−3 + 38q−4−197q−5−61q−6 + 188q−7 + 75q−8−165q−9−91q−10 + 139q−11 + 104q−12−112q−13−108q−14 + 79q−15 + 110q−16−50q−17−100q−18 + 21q−19 + 87q−20−2q−21−65q−22−14q−23 + 47q−24 + 15q−25−25q−26−17q−27 + 15q−28 + 10q−29−5q−30−7q−31 + 3q−32 + 2q−33−2q−35 + q−36 |
| 4 | q40−2q39 + q38 + q37−2q36 + 5q35−9q34 + 3q33 + 6q32−7q31 + 16q30−25q29 + 7q28 + 13q27−23q26 + 40q25−42q24 + 25q23 + 17q22−73q21 + 61q20−56q19 + 96q18 + 50q17−179q16 + 21q15−92q14 + 251q13 + 170q12−303q11−114q10−214q9 + 435q8 + 399q7−351q6−286q5−434q4 + 541q3 + 644q2−283q−374−665q−1 + 517q−2 + 782q−3−156q−4−334q−5−803q−6 + 401q−7 + 773q−8−29q−9−206q−10−834q−11 + 240q−12 + 664q−13 + 87q−14−42q−15−781q−16 + 53q−17 + 484q−18 + 189q−19 + 145q−20−653q−21−127q−22 + 253q−23 + 228q−24 + 310q−25−440q−26−222q−27 + 18q−28 + 160q−29 + 377q−30−196q−31−187q−32−125q−33 + 30q−34 + 306q−35−22q−36−74q−37−134q−38−59q−39 + 165q−40 + 30q−41 + 9q−42−68q−43−64q−44 + 57q−45 + 15q−46 + 24q−47−17q−48−31q−49 + 15q−50 + 10q−52−q−53−9q−54 + 4q−55−q−56 + 2q−57−2q−59 + q−60 |
| 5 | q60−2q59 + q58 + q57−2q56 + q55 + 3q54−7q53 + q52 + 7q51−4q50 + 2q49 + 7q48−18q47−4q46 + 11q45 + 2q44 + 11q43 + 20q42−24q41−31q40−12q39−8q38 + 45q37 + 80q36 + 13q35−61q34−115q33−112q32 + 55q31 + 236q30 + 201q29−q28−291q27−450q26−128q25 + 426q24 + 676q23 + 369q22−409q21−1059q20−748q19 + 402q18 + 1378q17 + 1226q16−177q15−1719q14−1811q13−121q12 + 1936q11 + 2388q10 + 568q9−2021q8−2925q7−1073q6 + 1978q5 + 3332q4 + 1559q3−1786q2−3585q−2009 + 1546q−1 + 3683q−2 + 2315q−3−1236q−4−3642q−5−2550q−6 + 980q−7 + 3493q−8 + 2629q−9−682q−10−3283q−11−2679q−12 + 453q−13 + 3017q−14 + 2624q−15−173q−16−2696q−17−2585q−18−85q−19 + 2338q−20 + 2471q−21 + 374q−22−1902q−23−2326q−24−672q−25 + 1431q−26 + 2109q−27 + 921q−28−909q−29−1790q−30−1127q−31 + 382q−32 + 1420q−33 + 1197q−34 + 89q−35−948q−36−1161q−37−475q−38 + 491q−39 + 975q−40 + 710q−41−46q−42−708q−43−788q−44−285q−45 + 374q−46 + 722q−47 + 505q−48−91q−49−542q−50−550q−51−165q−52 + 328q−53 + 518q−54 + 270q−55−128q−56−366q−57−320q−58−23q−59 + 246q−60 + 258q−61 + 93q−62−104q−63−195q−64−110q−65 + 37q−66 + 106q−67 + 90q−68 + 15q−69−63q−70−57q−71−12q−72 + 14q−73 + 32q−74 + 21q−75−11q−76−17q−77−q−78−3q−79 + 3q−80 + 9q−81−2q−82−5q−83 + 2q−84−q−86 + 2q−87−2q−89 + q−90 |
[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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