10 7
From Knot Atlas
|
|
|
|
 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_7's page at Knotilus! Visit 10 7's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X11,20,12,1 X19,6,20,7 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 |
| Gauss code | -1, 4, -3, 1, -2, 6, -7, 10, -8, 9, -5, 3, -4, 2, -9, 8, -10, 7, -6, 5 |
| Dowker-Thistlethwaite code | 4 12 14 18 16 20 2 10 8 6 |
| Conway Notation | [5212] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{12, 7}, {1, 10}, {11, 8}, {7, 9}, {10, 12}, {6, 11}, {8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}] |
[edit Notes on presentations of 10 7]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["10 7"];
|
In[4]:=
| PD[K]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X11,20,12,1 X19,6,20,7 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 |
In[5]:=
| GaussCode[K]
|
Out[5]=
| -1, 4, -3, 1, -2, 6, -7, 10, -8, 9, -5, 3, -4, 2, -9, 8, -10, 7, -6, 5 |
In[6]:=
| DTCode[K]
|
Out[6]=
| 4 12 14 18 16 20 2 10 8 6 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
| ConwayNotation[K]
|
Out[8]=
| [5212] |
In[9]:=
| br = BR[K]
|
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
|
Out[9]=
| BR(5,{−1,−1,−2,1,−2,−3,2,−3,−3,4,−3,4}) |
In[10]:=
| {First[br], Crossings[br], BraidIndex[K]}
|
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
|
KnotTheory::loading: Loading precomputed data in IndianaData`.
|
Out[10]=
| { 5, 12, 5 } |
In[11]:=
| Show[BraidPlot[br]]
|
|
Out[11]=
| -Graphics- |
In[12]:=
| Show[DrawMorseLink[K]]
|
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
|
|
Out[12]=
| -Graphics- |
In[13]:=
| ap = ArcPresentation[K]
|
Out[13]=
| ArcPresentation[{12, 7}, {1, 10}, {11, 8}, {7, 9}, {10, 12}, {6, 11}, {8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}] |
In[14]:=
| Draw[ap]
|
|
|
Out[14]=
| -Graphics- |
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −3t2 + 11t−15 + 11t−1−3t−2 |
| Conway polynomial | −3z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 43, -2 } |
| Jones polynomial | q−2 + 4q−1−5q−2 + 7q−3−7q−4 + 6q−5−5q−6 + 3q−7−2q−8 + q−9 |
| HOMFLY-PT polynomial (db, data sources) | z2a8 + a8−z4a6−2z2a6−2a6−z4a4 + a4−z4a2−z2a2 + z2 + 1 |
| Kauffman polynomial (db, data sources) | z6a10−4z4a10 + 3z2a10 + 2z7a9−8z5a9 + 8z3a9−3za9 + 2z8a8−7z6a8 + 6z4a8−3z2a8 + a8 + z9a7−z7a7−6z5a7 + 10z3a7−5za7 + 4z8a6−15z6a6 + 20z4a6−10z2a6 + 2a6 + z9a5−z7a5−2z5a5 + 6z3a5−2za5 + 2z8a4−5z6a4 + 8z4a4−4z2a4 + a4 + 2z7a3−2z5a3 + z3a3 + 2z6a2−z4a2−2z2a2 + 2z5a−3z3a + z4−2z2 + 1 |
| The A2 invariant | q28 + q22−2q20−q18−q14 + q12 + q8 + q6−q4 + 2q2 + q−4 |
| The G2 invariant | q142−q140 + 2q138−3q136 + 2q134−3q132−q130 + 6q128−10q126 + 12q124−12q122 + 8q120 + 2q118−12q116 + 22q114−24q112 + 21q110−8q108−7q106 + 20q104−23q102 + 22q100−9q98−4q96 + 14q94−16q92 + 8q90 + 2q88−14q86 + 19q84−16q82 + q80 + 10q78−23q76 + 27q74−25q72 + 11q70 + 2q68−19q66 + 29q64−30q62 + 21q60−6q58−8q56 + 18q54−20q52 + 15q50−3q48−6q46 + 11q44−8q42 + q40 + 10q38−14q36 + 14q34−7q32−2q30 + 9q28−14q26 + 16q24−13q22 + 9q20−2q18−5q16 + 10q14−12q12 + 14q10−10q8 + 6q6−6q2 + 9−8q−2 + 7q−4−3q−6 + q−8 + 2q−10−3q−12 + 3q−14−q−16 + q−18 |
Further Quantum Invariants
Further quantum knot invariants for 10_7.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q19−q17 + q15−2q13 + q11−q9 + 2q5−q3 + 2q−q−1 + q−3 |
| 2 | q54−q52−q50 + 3q48−2q46−4q44 + 5q42 + q40−6q38 + 4q36 + 4q34−6q32 + q30 + 5q28−4q26−q24 + 3q22−4q18−q16 + 6q14−4q12−3q10 + 7q8−2q6−3q4 + 5q2−1−q−2 + 3q−4−q−6−q−8 + q−10 |
| 3 | q105−q103−q101 + q99 + 2q97−2q95−5q93 + 2q91 + 8q89−10q85−3q83 + 12q81 + 9q79−11q77−13q75 + 4q73 + 16q71 + q69−15q67−8q65 + 12q63 + 14q61−7q59−17q57 + 3q55 + 17q53−16q49−3q47 + 13q45 + 4q43−9q41−7q39 + 9q37 + 8q35−13q31−4q29 + 13q27 + 9q25−12q23−13q21 + 7q19 + 13q17−2q15−11q13−q11 + 6q9 + 4q7−q5−q3−2q + q−1 + 3q−3 + 2q−5−2q−7−3q−9 + q−11 + 3q−13−q−17−q−19 + q−21 |
| 4 | q172−q170−q168 + q166 + 2q162−4q160−3q158 + 4q156 + 3q154 + 8q152−8q150−13q148 + 2q146 + 8q144 + 23q142−4q140−26q138−16q136−2q134 + 39q132 + 20q130−15q128−29q126−32q124 + 24q122 + 34q120 + 21q118−5q116−47q114−18q112 + 3q110 + 36q108 + 44q106−16q104−42q102−48q100 + 13q98 + 72q96 + 31q94−28q92−73q90−22q88 + 62q86 + 55q84−4q82−66q80−36q78 + 38q76 + 47q74 + 7q72−43q70−32q68 + 17q66 + 36q64 + 12q62−22q60−30q58−10q56 + 29q54 + 30q52 + 12q50−34q48−53q46 + 8q44 + 46q42 + 58q40−11q38−82q36−38q34 + 27q32 + 85q30 + 34q28−63q26−62q24−20q22 + 62q20 + 60q18−14q16−42q14−46q12 + 18q10 + 45q8 + 15q6−6q4−38q2−8 + 18q−2 + 15q−4 + 11q−6−18q−8−10q−10 + q−12 + 5q−14 + 12q−16−4q−18−4q−20−3q−22−q−24 + 5q−26−q−32−q−34 + q−36 |
| 5 | q255−q253−q251 + q249−2q241−2q239 + 4q237 + 6q235 + q233−4q231−9q229−8q227 + 4q225 + 19q223 + 17q221−3q219−23q217−33q215−14q213 + 25q211 + 52q209 + 35q207−14q205−59q203−65q201−15q199 + 54q197 + 88q195 + 53q193−24q191−85q189−85q187−22q185 + 52q183 + 90q181 + 65q179−56q175−80q173−64q171−14q169 + 60q167 + 104q165 + 94q163 + 16q161−107q159−172q157−106q155 + 66q153 + 210q151 + 200q149 + 16q147−212q145−274q143−104q141 + 171q139 + 310q137 + 184q135−104q133−311q131−244q129 + 40q127 + 281q125 + 263q123 + 17q121−230q119−259q117−54q115 + 179q113 + 228q111 + 69q109−131q107−184q105−70q103 + 92q101 + 152q99 + 64q97−71q95−116q93−63q91 + 38q89 + 108q87 + 84q85−18q83−98q81−113q79−46q77 + 93q75 + 169q73 + 110q71−60q69−216q67−205q65 + 242q61 + 300q59 + 89q57−230q55−371q53−187q51 + 171q49 + 397q47 + 276q45−85q43−369q41−326q39−9q37 + 294q35 + 329q33 + 89q31−199q29−294q27−132q25 + 115q23 + 225q21 + 143q19−40q17−163q15−131q13 + 4q11 + 105q9 + 103q7 + 23q5−63q3−80q−30q−1 + 37q−3 + 58q−5 + 30q−7−16q−9−40q−11−27q−13 + 3q−15 + 27q−17 + 24q−19 + q−21−13q−23−16q−25−7q−27 + 5q−29 + 12q−31 + 5q−33−2q−35−3q−37−5q−39−q−41 + 3q−43 + 2q−45−q−51−q−53 + q−55 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q28 + q22−2q20−q18−q14 + q12 + q8 + q6−q4 + 2q2 + q−4 |
| 1,1 | q76−2q74 + 4q72−8q70 + 17q68−26q66 + 36q64−54q62 + 67q60−78q58 + 82q56−80q54 + 68q52−38q50 + 10q48 + 26q46−65q44 + 96q42−122q40 + 132q38−135q36 + 134q34−112q32 + 92q30−66q28 + 36q26−14q24−10q22 + 22q20−36q18 + 40q16−38q14 + 39q12−42q10 + 42q8−36q6 + 38q4−32q2 + 30−20q−2 + 15q−4−10q−6 + 6q−8−2q−10 + q−12 |
| 2,0 | q72 + q64−q62−4q60−q58 + 2q56 + q54−3q52 + 5q48 + 4q46−3q44−2q42 + 3q40 + q38−3q36−2q34 + 3q32 + q30−2q28 + q26−q24−3q22 + 2q20 + q18−4q16−2q14 + 5q12 + 2q10−5q8−q6 + 7q4 + 2q2−2 + q−2 + 2q−4−q−8 + q−12 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q60−q58 + q54−3q52 + q48−3q46 + 4q44 + 4q42−3q40 + 5q38 + 3q36−6q34−q32 + q30−5q28−2q26 + q24 + q22−2q20−q18 + 6q16−3q14−2q12 + 7q10−q8−4q6 + 5q4 + q2−2 + 3q−2 + q−4−q−6 + q−8 |
| 1,0,0 | q37 + q33 + q29−2q27−q25−2q23−q19 + q17 + q15 + q11 + q7−q5 + 2q3 + q−1 + q−5 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q60−q58 + 2q56−3q54 + 5q52−6q50 + 7q48−7q46 + 6q44−6q42 + 3q40−q38−3q36 + 6q34−9q32 + 11q30−13q28 + 14q26−13q24 + 11q22−8q20 + 5q18−2q16−q14 + 4q12−5q10 + 7q8−6q6 + 7q4−5q2 + 4−3q−2 + 3q−4−q−6 + q−8 |
| 1,0 | q98−q94−q92 + q90 + 2q88−q86−4q84−q82 + 4q80 + 3q78−4q76−6q74 + 2q72 + 8q70 + 4q68−6q66−5q64 + 4q62 + 8q60 + q58−6q56−3q54 + 3q52 + 2q50−3q48−4q46 + q44 + 3q42−2q40−5q38 + 5q34 + q32−5q30−3q28 + 5q26 + 5q24−3q22−6q20 + q18 + 7q16 + 4q14−4q12−5q10 + 6q6 + 3q4−2q2−3 + 3q−4 + 2q−6−q−8−q−10 + q−14 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q142−q140 + 2q138−3q136 + 2q134−3q132−q130 + 6q128−10q126 + 12q124−12q122 + 8q120 + 2q118−12q116 + 22q114−24q112 + 21q110−8q108−7q106 + 20q104−23q102 + 22q100−9q98−4q96 + 14q94−16q92 + 8q90 + 2q88−14q86 + 19q84−16q82 + q80 + 10q78−23q76 + 27q74−25q72 + 11q70 + 2q68−19q66 + 29q64−30q62 + 21q60−6q58−8q56 + 18q54−20q52 + 15q50−3q48−6q46 + 11q44−8q42 + q40 + 10q38−14q36 + 14q34−7q32−2q30 + 9q28−14q26 + 16q24−13q22 + 9q20−2q18−5q16 + 10q14−12q12 + 14q10−10q8 + 6q6−6q2 + 9−8q−2 + 7q−4−3q−6 + q−8 + 2q−10−3q−12 + 3q−14−q−16 + q−18 |
.
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["10 7"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −3t2 + 11t−15 + 11t−1−3t−2 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −3z4−z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 43, -2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q−2 + 4q−1−5q−2 + 7q−3−7q−4 + 6q−5−5q−6 + 3q−7−2q−8 + q−9 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z2a8 + a8−z4a6−2z2a6−2a6−z4a4 + a4−z4a2−z2a2 + z2 + 1 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z6a10−4z4a10 + 3z2a10 + 2z7a9−8z5a9 + 8z3a9−3za9 + 2z8a8−7z6a8 + 6z4a8−3z2a8 + a8 + z9a7−z7a7−6z5a7 + 10z3a7−5za7 + 4z8a6−15z6a6 + 20z4a6−10z2a6 + 2a6 + z9a5−z7a5−2z5a5 + 6z3a5−2za5 + 2z8a4−5z6a4 + 8z4a4−4z2a4 + a4 + 2z7a3−2z5a3 + z3a3 + 2z6a2−z4a2−2z2a2 + 2z5a−3z3a + z4−2z2 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a59, K11n3,}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["10 7"];
|
In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
| { −3t2 + 11t−15 + 11t−1−3t−2, q−2 + 4q−1−5q−2 + 7q−3−7q−4 + 6q−5−5q−6 + 3q−7−2q−8 + q−9 } |
In[5]:=
| DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
| {K11a59, K11n3,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {} |
[edit] Vassiliev invariants
| V2 and V3: | (-1, 3) |
| V2,1 through V6,9: |
|
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 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
[edit] The Coloured Jones Polynomials
| n | Jn |
| 2 | q4−2q3 + 5q−6 + 11q−2−14q−3 + q−4 + 20q−5−24q−6 + 30q−8−31q−9−3q−10 + 34q−11−28q−12−7q−13 + 31q−14−19q−15−11q−16 + 24q−17−9q−18−11q−19 + 14q−20−2q−21−7q−22 + 5q−23−2q−25 + q−26 |
| 3 | q9−2q8 + q6 + 4q5−4q4−4q3 + 2q2 + 8q−3−6q−1−q−2 + 9q−3−3q−4−q−5 + q−6 + 2q−7−13q−8 + 8q−9 + 16q−10−4q−11−33q−12 + 9q−13 + 37q−14−50q−16 + 50q−18 + 8q−19−49q−20−16q−21 + 48q−22 + 21q−23−40q−24−32q−25 + 35q−26 + 37q−27−23q−28−46q−29 + 15q−30 + 47q−31−2q−32−48q−33−5q−34 + 40q−35 + 14q−36−33q−37−17q−38 + 23q−39 + 16q−40−13q−41−14q−42 + 8q−43 + 9q−44−3q−45−6q−46 + 2q−47 + 2q−48−2q−50 + q−51 |
| 4 | q16−2q15 + q13 + 6q11−8q10−2q9 + 22q6−15q5−6q4−11q3−8q2 + 51q−11−3q−1−37q−2−38q−3 + 83q−4 + 10q−5 + 27q−6−64q−7−102q−8 + 87q−9 + 38q−10 + 101q−11−62q−12−184q−13 + 45q−14 + 37q−15 + 198q−16−11q−17−242q−18−20q−19−7q−20 + 269q−21 + 58q−22−254q−23−58q−24−68q−25 + 288q−26 + 104q−27−236q−28−59q−29−107q−30 + 268q−31 + 112q−32−202q−33−35q−34−126q−35 + 219q−36 + 101q−37−152q−38 + 5q−39−135q−40 + 145q−41 + 71q−42−90q−43 + 64q−44−128q−45 + 61q−46 + 20q−47−45q−48 + 123q−49−87q−50 + 2q−51−41q−52−39q−53 + 149q−54−27q−55−6q−56−74q−57−60q−58 + 120q−59 + 15q−60 + 20q−61−61q−62−70q−63 + 64q−64 + 18q−65 + 34q−66−26q−67−51q−68 + 23q−69 + 4q−70 + 24q−71−4q−72−24q−73 + 8q−74−2q−75 + 9q−76 + q−77−8q−78 + 3q−79−q−80 + 2q−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. |
|
