K11a68
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a68's page at Knotilus! Visit K11a68's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X10,18,11,17 X12,20,13,19 X6,21,7,22 |
| Gauss code | 1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8 |
| Dowker-Thistlethwaite code | 4 8 16 14 2 18 20 22 10 12 6 |
| 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 + 6t3−14t2 + 20t−21 + 20t−1−14t−2 + 6t−3−t−4 |
| Conway polynomial | −z8−2z6 + 2z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 103, 2 } |
| Jones polynomial | q7−4q6 + 7q5−11q4 + 15q3−16q2 + 16q−13 + 10q−1−6q−2 + 3q−3−q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−8z4a−2 + 3z4a−4 + 8z4−3a2z2−4z2a−2 + z2a−4 + 8z2−a2 + a−2−a−4 + 2 |
| Kauffman polynomial (db, data sources) | 2z10a−2 + 2z10 + 4az9 + 10z9a−1 + 6z9a−3 + 3a2z8 + 6z8a−2 + 8z8a−4 + z8 + a3z7−14az7−33z7a−1−10z7a−3 + 8z7a−5−12a2z6−30z6a−2−11z6a−4 + 7z6a−6−24z6−4a3z5 + 12az5 + 32z5a−1 + 5z5a−3−7z5a−5 + 4z5a−7 + 14a2z4 + 30z4a−2 + 2z4a−4−7z4a−6 + z4a−8 + 34z4 + 4a3z3−2az3−10z3a−1−4z3a−3−3z3a−5−3z3a−7−7a2z2−7z2a−2 + 2z2a−4 + z2a−6−15z2−a3z−az + za−1 + 3za−3 + 2za−5 + a2−a−2−a−4 + 2 |
| The A2 invariant | −q12 + q8−q6 + 2q4−2q2 + 1 + 2q−2−q−4 + 5q−6−2q−8 + 2q−10−q−12−2q−14 + q−16−2q−18 + q−20 |
| The G2 invariant | q60−2q58 + 5q56−9q54 + 11q52−13q50 + 6q48 + 11q46−36q44 + 65q42−85q40 + 75q38−31q36−52q34 + 146q32−211q30 + 217q28−139q26−7q24 + 171q22−287q20 + 306q18−209q16 + 35q14 + 141q12−251q10 + 246q8−135q6−24q4 + 167q2−218 + 159q−2−20q−4−145q−6 + 262q−8−285q−10 + 203q−12−34q−14−159q−16 + 317q−18−370q−20 + 309q−22−137q−24−74q−26 + 250q−28−329q−30 + 286q−32−138q−34−38q−36 + 174q−38−207q−40 + 137q−42−7q−44−119q−46 + 176q−48−148q−50 + 46q−52 + 72q−54−166q−56 + 201q−58−166q−60 + 85q−62 + 10q−64−97q−66 + 141q−68−155q−70 + 134q−72−86q−74 + 29q−76 + 33q−78−81q−80 + 104q−82−100q−84 + 75q−86−37q−88−3q−90 + 32q−92−49q−94 + 47q−96−32q−98 + 18q−100−2q−102−6q−104 + 9q−106−10q−108 + 6q−110−3q−112 + q−114 |
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["K11a68"];
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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 + 6t3−14t2 + 20t−21 + 20t−1−14t−2 + 6t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−2z6 + 2z4 + 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]=
| { 103, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q7−4q6 + 7q5−11q4 + 15q3−16q2 + 16q−13 + 10q−1−6q−2 + 3q−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]=
| −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−8z4a−2 + 3z4a−4 + 8z4−3a2z2−4z2a−2 + z2a−4 + 8z2−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]=
| 2z10a−2 + 2z10 + 4az9 + 10z9a−1 + 6z9a−3 + 3a2z8 + 6z8a−2 + 8z8a−4 + z8 + a3z7−14az7−33z7a−1−10z7a−3 + 8z7a−5−12a2z6−30z6a−2−11z6a−4 + 7z6a−6−24z6−4a3z5 + 12az5 + 32z5a−1 + 5z5a−3−7z5a−5 + 4z5a−7 + 14a2z4 + 30z4a−2 + 2z4a−4−7z4a−6 + z4a−8 + 34z4 + 4a3z3−2az3−10z3a−1−4z3a−3−3z3a−5−3z3a−7−7a2z2−7z2a−2 + 2z2a−4 + z2a−6−15z2−a3z−az + za−1 + 3za−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, 
):
{K11a111,}
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["K11a68"];
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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 + 6t3−14t2 + 20t−21 + 20t−1−14t−2 + 6t−3−t−4, q7−4q6 + 7q5−11q4 + 15q3−16q2 + 16q−13 + 10q−1−6q−2 + 3q−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]=
| {K11a111,} |
[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 K11a68. 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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