K11a71
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a71's page at Knotilus! Visit K11a71's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X12,5,13,6 X14,8,15,7 X2,10,3,9 X22,11,1,12 X18,13,19,14 X20,16,21,15 X6,18,7,17 X8,19,9,20 X16,22,17,21 |
| Gauss code | 1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -6 |
| Dowker-Thistlethwaite code | 4 10 12 14 2 22 18 20 6 8 16 |
| 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 + 6t3−18t2 + 34t−41 + 34t−1−18t−2 + 6t−3−t−4 |
| Conway polynomial | −z8−2z6−2z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 159, 2 } |
| Jones polynomial | −q8 + 5q7−11q6 + 17q5−23q4 + 26q3−25q2 + 22q−15 + 9q−1−4q−2 + q−3 |
| HOMFLY-PT polynomial (db, data sources) | −z8a−2−5z6a−2 + 2z6a−4 + z6−10z4a−2 + 6z4a−4−z4a−6 + 3z4−7z2a−2 + 5z2a−4−z2a−6 + 3z2 + 1 |
| Kauffman polynomial (db, data sources) | 2z10a−2 + 2z10a−4 + 6z9a−1 + 14z9a−3 + 8z9a−5 + 17z8a−2 + 23z8a−4 + 13z8a−6 + 7z8 + 4az7−3z7a−1−13z7a−3 + 5z7a−5 + 11z7a−7 + a2z6−47z6a−2−53z6a−4−17z6a−6 + 5z6a−8−15z6−9az5−14z5a−1−18z5a−3−29z5a−5−15z5a−7 + z5a−9−2a2z4 + 39z4a−2 + 35z4a−4 + 5z4a−6−4z4a−8 + 11z4 + 6az3 + 14z3a−1 + 20z3a−3 + 17z3a−5 + 5z3a−7 + a2z2−12z2a−2−8z2a−4−z2a−6−4z2−az−3za−1−3za−3−za−5 + 1 |
| The A2 invariant | q8−2q6 + 3q4−2q2−1 + 5q−2−4q−4 + 6q−6−2q−8 + q−12−5q−14 + 4q−16−2q−18 + 2q−22−q−24 |
| The G2 invariant | q46−3q44 + 8q42−16q40 + 22q38−25q36 + 14q34 + 18q32−65q30 + 127q28−176q26 + 178q24−110q22−51q20 + 277q18−495q16 + 619q14−551q12 + 254q10 + 221q8−733q6 + 1094q4−1121q2 + 759−102q−2−637q−4 + 1158q−6−1252q−8 + 878q−10−176q−12−542q−14 + 971q−16−920q−18 + 406q−20 + 341q−22−974q−24 + 1199q−26−873q−28 + 94q−30 + 840q−32−1553q−34 + 1765q−36−1353q−38 + 450q−40 + 627q−42−1496q−44 + 1839q−46−1553q−48 + 761q−50 + 212q−52−989q−54 + 1279q−56−1014q−58 + 349q−60 + 402q−62−898q−64 + 913q−66−467q−68−244q−70 + 900q−72−1206q−74 + 1053q−76−495q−78−230q−80 + 847q−82−1157q−84 + 1079q−86−685q−88 + 156q−90 + 320q−92−610q−94 + 664q−96−522q−98 + 289q−100−49q−102−126q−104 + 202q−106−205q−108 + 147q−110−76q−112 + 22q−114 + 16q−116−28q−118 + 27q−120−20q−122 + 10q−124−4q−126 + q−128 |
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["K11a71"];
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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 + 6t3−18t2 + 34t−41 + 34t−1−18t−2 + 6t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−2z6−2z4 + 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]=
| { 159, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q8 + 5q7−11q6 + 17q5−23q4 + 26q3−25q2 + 22q−15 + 9q−1−4q−2 + q−3 |
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]=
| −z8a−2−5z6a−2 + 2z6a−4 + z6−10z4a−2 + 6z4a−4−z4a−6 + 3z4−7z2a−2 + 5z2a−4−z2a−6 + 3z2 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2z10a−2 + 2z10a−4 + 6z9a−1 + 14z9a−3 + 8z9a−5 + 17z8a−2 + 23z8a−4 + 13z8a−6 + 7z8 + 4az7−3z7a−1−13z7a−3 + 5z7a−5 + 11z7a−7 + a2z6−47z6a−2−53z6a−4−17z6a−6 + 5z6a−8−15z6−9az5−14z5a−1−18z5a−3−29z5a−5−15z5a−7 + z5a−9−2a2z4 + 39z4a−2 + 35z4a−4 + 5z4a−6−4z4a−8 + 11z4 + 6az3 + 14z3a−1 + 20z3a−3 + 17z3a−5 + 5z3a−7 + a2z2−12z2a−2−8z2a−4−z2a−6−4z2−az−3za−1−3za−3−za−5 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a248,}
Same Jones Polynomial (up to mirroring, 
):
{K11a248,}
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["K11a71"];
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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 + 6t3−18t2 + 34t−41 + 34t−1−18t−2 + 6t−3−t−4, −q8 + 5q7−11q6 + 17q5−23q4 + 26q3−25q2 + 22q−15 + 9q−1−4q−2 + q−3 } |
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]=
| {K11a248,} |
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]=
| {K11a248,} |
[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 K11a71. 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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