10 35
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 35's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_35's page at Knotilus! Visit 10 35's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X7,10,8,11 X3948 X9,3,10,2 X5,16,6,17 X11,1,12,20 X13,19,14,18 X17,15,18,14 X19,13,20,12 X15,6,16,7 |
| Gauss code | -1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 9, -7, 8, -10, 5, -8, 7, -9, 6 |
| Dowker-Thistlethwaite code | 4 8 16 10 2 20 18 6 14 12 |
| Conway Notation | [2422] |
| 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}, {5, 10}, {7, 1}, {8, 6}, {2, 7}, {6, 12}, {1, 3}, {4, 2}, {3, 5}, {11, 4}] |
[edit Notes on presentations of 10 35]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 35"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X7,10,8,11 X3948 X9,3,10,2 X5,16,6,17 X11,1,12,20 X13,19,14,18 X17,15,18,14 X19,13,20,12 X15,6,16,7 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 9, -7, 8, -10, 5, -8, 7, -9, 6 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 8 16 10 2 20 18 6 14 12 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [2422] |
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,{−1,2,−1,2,3,−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}, {5, 10}, {7, 1}, {8, 6}, {2, 7}, {6, 12}, {1, 3}, {4, 2}, {3, 5}, {11, 4}] |
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 | 2t2−12t + 21−12t−1 + 2t−2 |
| Conway polynomial | 2z4−4z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 49, 0 } |
| Jones polynomial | q6−2q5 + 4q4−6q3 + 7q2−8q + 8−6q−1 + 4q−2−2q−3 + q−4 |
| HOMFLY-PT polynomial (db, data sources) | a4−2z2a2−a2 + z4 + 1 + z4a−2−2z2a−4−a−4 + a−6 |
| Kauffman polynomial (db, data sources) | z9a−1 + z9a−3 + 4z8a−2 + 2z8a−4 + 2z8 + 2az7 + 2z7a−5 + 2a2z6−11z6a−2−5z6a−4 + z6a−6−3z6 + 2a3z5−z5a−1−6z5a−3−7z5a−5 + a4z4 + 10z4a−2−4z4a−6 + 5z4−3a3z3−2az3 + 5z3a−3 + 6z3a−5−2a4z2−3a2z2−3z2a−2 + 3z2a−4 + 4z2a−6−3z2 + a3z + az + za−1−za−3−2za−5 + a4 + a2−a−4−a−6 + 1 |
| The A2 invariant | q14 + q12−q10 + q8−2q4 + 2q2 + q−2−q−6 + q−8−2q−10 + q−14−q−16 + q−18 + q−20 |
| The G2 invariant | Data:10 35/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
Further quantum knot invariants for 10_35.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q9−q7 + 2q5−2q3 + 2q−q−3 + q−5−2q−7 + 2q−9−q−11 + q−13 |
| 2 | q26−q24−q22 + 3q20−2q18−q16 + 6q14−6q12−2q10 + 10q8−7q6−5q4 + 10q2−1−5q−2 + 2q−4 + 4q−6−3q−8−5q−10 + 8q−12 + q−14−9q−16 + 7q−18 + 5q−20−10q−22 + 2q−24 + 7q−26−6q−28−2q−30 + 4q−32−q−34−q−36 + q−38 |
| 3 | q51−q49−q47 + 3q43−3q39−q37 + 2q35 + q33 + 2q29−2q27−8q25 + 5q23 + 16q21−4q19−25q17−3q15 + 33q13 + 11q11−33q9−19q7 + 25q5 + 26q3−14q−26q−1 + 4q−3 + 23q−5 + 10q−7−19q−9−18q−11 + 13q−13 + 25q−15−10q−17−30q−19 + 4q−21 + 34q−23 + 4q−25−35q−27−11q−29 + 32q−31 + 20q−33−25q−35−28q−37 + 13q−39 + 32q−41−2q−43−27q−45−10q−47 + 21q−49 + 16q−51−12q−53−15q−55 + 3q−57 + 12q−59 + q−61−7q−63−2q−65 + 3q−67 + q−69−q−71−q−73 + q−75 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q14 + q12−q10 + q8−2q4 + 2q2 + q−2−q−6 + q−8−2q−10 + q−14−q−16 + q−18 + q−20 |
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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 35"];
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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]=
| 2t2−12t + 21−12t−1 + 2t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 2z4−4z2 + 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]=
| { 49, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q6−2q5 + 4q4−6q3 + 7q2−8q + 8−6q−1 + 4q−2−2q−3 + q−4 |
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]=
| a4−2z2a2−a2 + z4 + 1 + z4a−2−2z2a−4−a−4 + a−6 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−1 + z9a−3 + 4z8a−2 + 2z8a−4 + 2z8 + 2az7 + 2z7a−5 + 2a2z6−11z6a−2−5z6a−4 + z6a−6−3z6 + 2a3z5−z5a−1−6z5a−3−7z5a−5 + a4z4 + 10z4a−2−4z4a−6 + 5z4−3a3z3−2az3 + 5z3a−3 + 6z3a−5−2a4z2−3a2z2−3z2a−2 + 3z2a−4 + 4z2a−6−3z2 + a3z + az + za−1−za−3−2za−5 + a4 + a2−a−4−a−6 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{10_22,}
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 35"];
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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]=
| { 2t2−12t + 21−12t−1 + 2t−2, q6−2q5 + 4q4−6q3 + 7q2−8q + 8−6q−1 + 4q−2−2q−3 + q−4 } |
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]=
| {10_22,} |
[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 35. 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 | q18−2q17 + 6q15−8q14−4q13 + 19q12−13q11−16q10 + 34q9−11q8−32q7 + 44q6−4q5−45q4 + 46q3 + 3q2−47q + 39 + 7q−1−36q−2 + 24q−3 + 5q−4−19q−5 + 12q−6 + q−7−7q−8 + 5q−9−2q−11 + q−12 |
| 3 | q36−2q35 + 2q33 + 3q32−7q31−5q30 + 10q29 + 14q28−16q27−23q26 + 13q25 + 42q24−11q23−54q22−4q21 + 67q20 + 23q19−73q18−45q17 + 70q16 + 68q15−61q14−88q13 + 46q12 + 107q11−31q10−118q9 + 12q8 + 127q7 + 4q6−130q5−19q4 + 126q3 + 33q2−117q−38 + 96q−1 + 45q−2−77q−3−39q−4 + 52q−5 + 31q−6−33q−7−17q−8 + 16q−9 + 9q−10−12q−11 + 3q−12 + 5q−13−4q−14−6q−15 + 7q−16 + 3q−17−3q−18−5q−19 + 4q−20 + q−21−2q−23 + q−24 |
| 4 | q60−2q59 + 2q57−q56 + 4q55−9q54−q53 + 10q52 + 14q50−30q49−15q48 + 23q47 + 12q46 + 52q45−57q44−56q43 + 9q42 + 15q41 + 137q40−47q39−91q38−47q37−48q36 + 217q35 + 16q34−46q33−87q32−189q31 + 209q30 + 66q29 + 89q28−27q27−330q26 + 94q25 + 25q24 + 247q23 + 133q22−398q21−57q20−103q19 + 357q18 + 325q17−385q16−186q15−259q14 + 412q13 + 490q12−333q11−276q10−392q9 + 417q8 + 604q7−251q6−320q5−491q4 + 359q3 + 646q2−128q−283−537q−1 + 220q−2 + 577q−3 + 7q−4−155q−5−484q−6 + 53q−7 + 393q−8 + 76q−9 + 5q−10−331q−11−54q−12 + 185q−13 + 58q−14 + 94q−15−162q−16−64q−17 + 51q−18 + 6q−19 + 94q−20−53q−21−33q−22 + 2q−23−20q−24 + 56q−25−11q−26−8q−27−4q−28−19q−29 + 23q−30−q−31 + q−32−q−33−9q−34 + 6q−35 + q−37−2q−39 + q−40 |
| 5 | q90−2q89 + 2q87−q86 + 2q84−5q83−2q82 + 9q81 + 3q80−3q79−2q78−16q77−9q76 + 20q75 + 30q74 + 11q73−13q72−49q71−52q70 + 14q69 + 73q68 + 83q67 + 27q66−79q65−140q64−75q63 + 54q62 + 160q61 + 163q60 + 5q59−166q58−199q57−104q56 + 82q55 + 227q54 + 198q53 + 22q52−142q51−237q50−199q49−10q48 + 208q47 + 327q46 + 246q45−44q44−408q43−531q42−213q41 + 384q40 + 791q39 + 569q38−237q37−1005q36−976q35−4q34 + 1129q33 + 1376q32 + 340q31−1166q30−1751q29−708q28 + 1124q27 + 2062q26 + 1094q25−1034q24−2322q23−1437q22 + 902q21 + 2518q20 + 1763q19−774q18−2679q17−2026q16 + 642q15 + 2783q14 + 2269q13−501q12−2866q11−2471q10 + 356q9 + 2881q8 + 2637q7−156q6−2829q5−2786q4−57q3 + 2684q2 + 2832q + 339−2418q−1−2830q−2−597q−3 + 2062q−4 + 2667q−5 + 843q−6−1607q−7−2417q−8−997q−9 + 1135q−10 + 2040q−11 + 1068q−12−701q−13−1613q−14−1013q−15 + 334q−16 + 1172q−17 + 904q−18−87q−19−808q−20−699q−21−68q−22 + 484q−23 + 537q−24 + 131q−25−288q−26−356q−27−136q−28 + 129q−29 + 244q−30 + 119q−31−63q−32−138q−33−96q−34 + 10q−35 + 93q−36 + 66q−37−4q−38−36q−39−48q−40−20q−41 + 34q−42 + 28q−43 + 3q−44−2q−45−16q−46−17q−47 + 9q−48 + 10q−49 + 4q−51−3q−52−7q−53 + 2q−54 + 2q−55 + q−57−2q−59 + q−60 |
| 6 | q126−2q125 + 2q123−q122−2q120 + 6q119−6q118−3q117 + 11q116−q115−2q114−13q113 + 12q112−14q111−8q110 + 36q109 + 14q108 + 5q107−43q106 + 7q105−55q104−38q103 + 80q102 + 72q101 + 75q100−55q99 + 6q98−162q97−170q96 + 53q95 + 131q94 + 235q93 + 50q92 + 153q91−233q90−378q89−160q88−6q87 + 284q86 + 157q85 + 522q84−21q83−336q82−310q81−304q80−39q79−181q78 + 653q77 + 281q76 + 174q75 + 151q74−79q73−315q72−1049q71−79q70−218q69 + 454q68 + 1126q67 + 1289q66 + 531q65−1460q64−1367q63−2050q62−695q61 + 1387q60 + 3221q59 + 2903q58−157q57−1803q56−4397q55−3511q54−267q53 + 4179q52 + 5792q51 + 2917q50−285q49−5705q48−6862q47−3692q46 + 3198q45 + 7698q44 + 6564q43 + 2897q42−5203q41−9364q40−7647q39 + 696q38 + 8017q37 + 9509q36 + 6528q35−3394q34−10533q33−10960q32−2188q31 + 7252q30 + 11336q29 + 9566q28−1301q27−10792q26−13248q25−4584q24 + 6227q23 + 12356q22 + 11709q21 + 433q20−10720q19−14752q18−6362q17 + 5315q16 + 12954q15 + 13222q14 + 1892q13−10414q12−15733q11−7912q10 + 4177q9 + 13030q8 + 14349q7 + 3582q6−9333q5−15934q4−9452q3 + 2209q2 + 11893q + 14709 + 5625q−1−6869q−2−14545q−3−10393q−4−540q−5 + 8990q−6 + 13375q−7 + 7173q−8−3324q−9−11136q−10−9671q−11−2930q−12 + 4926q−13 + 10062q−14 + 7055q−15−147q−16−6647q−17−7105q−18−3691q−19 + 1384q−20 + 5925q−21 + 5217q−22 + 1360q−23−2894q−24−3925q−25−2838q−26−416q−27 + 2671q−28 + 2875q−29 + 1304q−30−883q−31−1584q−32−1495q−33−728q−34 + 984q−35 + 1215q−36 + 704q−37−223q−38−474q−39−579q−40−484q−41 + 375q−42 + 442q−43 + 284q−44−94q−45−117q−46−197q−47−262q−48 + 181q−49 + 163q−50 + 121q−51−53q−52−24q−53−73q−54−147q−55 + 87q−56 + 59q−57 + 62q−58−20q−59 + 5q−60−27q−61−76q−62 + 33q−63 + 13q−64 + 29q−65−6q−66 + 11q−67−6q−68−31q−69 + 11q−70−2q−71 + 10q−72−2q−73 + 5q−74−9q−76 + 4q−77−2q−78 + 2q−79 + q−81−2q−83 + q−84 |
[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. |
|
