K11a170
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a170's page at Knotilus! Visit K11a170's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X18,5,19,6 X20,8,21,7 X2,10,3,9 X16,11,17,12 X6,14,7,13 X8,15,9,16 X22,17,1,18 X14,19,15,20 X12,22,13,21 |
| Gauss code | 1, -5, 2, -1, 3, -7, 4, -8, 5, -2, 6, -11, 7, -10, 8, -6, 9, -3, 10, -4, 11, -9 |
| Dowker-Thistlethwaite code | 4 10 18 20 2 16 6 8 22 14 12 |
| A Braid Representative |
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| A Morse Link Presentation | 
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[edit] Three dimensional invariants
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[edit] Four dimensional invariants
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[edit] Polynomial invariants
| Alexander polynomial | t4−6t3 + 20t2−40t + 51−40t−1 + 20t−2−6t−3 + t−4 |
| Conway polynomial | z8 + 2z6 + 4z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 185, 0 } |
| Jones polynomial | q6−6q5 + 13q4−20q3 + 27q2−30q + 30−25q−1 + 18q−2−10q−3 + 4q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | z8−a2z6−2z6a−2 + 5z6−3a2z4−5z4a−2 + z4a−4 + 11z4−4a2z2−3z2a−2 + 9z2−a2 + a−2−a−4 + 2 |
| Kauffman polynomial (db, data sources) | 4z10a−2 + 4z10 + 11az9 + 23z9a−1 + 12z9a−3 + 13a2z8 + 22z8a−2 + 13z8a−4 + 22z8 + 9a3z7−7az7−37z7a−1−15z7a−3 + 6z7a−5 + 4a4z6−20a2z6−65z6a−2−26z6a−4 + z6a−6−62z6 + a5z5−12a3z5−11az5 + 3z5a−1−7z5a−3−8z5a−5−4a4z4 + 17a2z4 + 41z4a−2 + 11z4a−4 + 51z4−a5z3 + 8a3z3 + 15az3 + 10z3a−1 + 4z3a−3 + a4z2−7a2z2−6z2a−2 + 2z2a−4−16z2−2a3z−4az−2za−1 + 2za−3 + 2za−5 + a2−a−2−a−4 + 2 |
| The A2 invariant | −q14 + 2q12−4q10 + 3q8 + 2q6−5q4 + 6q2−5 + 4q−2 + q−4−q−6 + 6q−8−5q−10 + 2q−12−3q−16 + q−18 |
| The G2 invariant | q80−3q78 + 7q76−13q74 + 17q72−19q70 + 12q68 + 10q66−43q64 + 89q62−133q60 + 152q58−131q56 + 45q54 + 111q52−308q50 + 508q48−630q46 + 572q44−291q42−223q40 + 849q38−1362q36 + 1527q34−1168q32 + 303q30 + 809q28−1756q26 + 2130q24−1712q22 + 614q20 + 723q18−1724q16 + 1928q14−1209q12−84q10 + 1365q8−2013q6 + 1679q4−493q2−1082 + 2370q−2−2805q−4 + 2187q−6−705q−8−1085q−10 + 2534q−12−3117q−14 + 2634q−16−1294q−18−417q−20 + 1865q−22−2504q−24 + 2151q−26−954q−28−523q−30 + 1645q−32−1927q−34 + 1239q−36 + 66q−38−1391q−40 + 2141q−42−1960q−44 + 941q−46 + 475q−48−1691q−50 + 2230q−52−1946q−54 + 1022q−56 + 97q−58−1000q−60 + 1400q−62−1274q−64 + 806q−66−224q−68−236q−70 + 457q−72−468q−74 + 331q−76−160q−78 + 29q−80 + 50q−82−69q−84 + 57q−86−35q−88 + 15q−90−5q−92 + q−94 |
Further Quantum Invariants
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["K11a170"];
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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]=
| t4−6t3 + 20t2−40t + 51−40t−1 + 20t−2−6t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 2z6 + 4z4 + 2z2 + 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]=
| { 185, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q6−6q5 + 13q4−20q3 + 27q2−30q + 30−25q−1 + 18q−2−10q−3 + 4q−4−q−5 |
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]=
| z8−a2z6−2z6a−2 + 5z6−3a2z4−5z4a−2 + z4a−4 + 11z4−4a2z2−3z2a−2 + 9z2−a2 + a−2−a−4 + 2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 4z10a−2 + 4z10 + 11az9 + 23z9a−1 + 12z9a−3 + 13a2z8 + 22z8a−2 + 13z8a−4 + 22z8 + 9a3z7−7az7−37z7a−1−15z7a−3 + 6z7a−5 + 4a4z6−20a2z6−65z6a−2−26z6a−4 + z6a−6−62z6 + a5z5−12a3z5−11az5 + 3z5a−1−7z5a−3−8z5a−5−4a4z4 + 17a2z4 + 41z4a−2 + 11z4a−4 + 51z4−a5z3 + 8a3z3 + 15az3 + 10z3a−1 + 4z3a−3 + a4z2−7a2z2−6z2a−2 + 2z2a−4−16z2−2a3z−4az−2za−1 + 2za−3 + 2za−5 + a2−a−2−a−4 + 2 |
[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["K11a170"];
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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]=
| { t4−6t3 + 20t2−40t + 51−40t−1 + 20t−2−6t−3 + t−4, q6−6q5 + 13q4−20q3 + 27q2−30q + 30−25q−1 + 18q−2−10q−3 + 4q−4−q−5 } |
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 K11a170. 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] Computer Talk
Much of the above data can be recomputed by Mathematica using the packageKnotTheory`. See A Sample KnotTheory` Session.
[edit] Modifying This Page
| Read me first: Modifying Knot Pages.
See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top. |
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