K11a81
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a81's page at Knotilus! Visit K11a81's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,13,19,14 X20,16,21,15 X8,17,9,18 X6,20,7,19 X16,22,17,21 |
| Gauss code | 1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5 |
| Dowker-Thistlethwaite code | 4 10 12 14 22 2 18 20 8 6 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 + 6t3−16t2 + 26t−29 + 26t−1−16t−2 + 6t−3−t−4 |
| Conway polynomial | −z8−2z6 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 127, 2 } |
| Jones polynomial | q7−4q6 + 8q5−13q4 + 18q3−20q2 + 20q−17 + 13q−1−8q−2 + 4q−3−q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−9z4a−2 + 3z4a−4 + 7z4−2a2z2−6z2a−2 + 2z2a−4 + 6z2 + 1 |
| Kauffman polynomial (db, data sources) | 2z10a−2 + 2z10 + 5az9 + 12z9a−1 + 7z9a−3 + 4a2z8 + 15z8a−2 + 11z8a−4 + 8z8 + a3z7−13az7−26z7a−1−z7a−3 + 11z7a−5−14a2z6−50z6a−2−14z6a−4 + 8z6a−6−42z6−3a3z5 + 3az5−23z5a−3−13z5a−5 + 4z5a−7 + 15a2z4 + 39z4a−2 + 2z4a−4−7z4a−6 + z4a−8 + 44z4 + 3a3z3 + 9az3 + 18z3a−1 + 19z3a−3 + 5z3a−5−2z3a−7−5a2z2−9z2a−2 + z2a−4 + 2z2a−6−13z2−a3z−4az−6za−1−4za−3−za−5 + 1 |
| The A2 invariant | −q12 + q10 + q8−q6 + 3q4−3q2 + 1 + q−2−2q−4 + 5q−6−3q−8 + 3q−10−q−12−2q−14 + 2q−16−2q−18 + q−20 |
| The G2 invariant | q60−3q58 + 9q56−19q54 + 28q52−32q50 + 15q48 + 29q46−92q44 + 157q42−186q40 + 140q38−15q36−174q34 + 360q32−449q30 + 392q28−169q26−152q24 + 452q22−607q20 + 545q18−277q16−87q14 + 395q12−526q10 + 425q8−144q6−183q4 + 416q2−444 + 250q−2 + 81q−4−420q−6 + 618q−8−586q−10 + 326q−12 + 84q−14−495q−16 + 758q−18−764q−20 + 514q−22−89q−24−345q−26 + 628q−28−662q−30 + 448q−32−89q−34−249q−36 + 431q−38−389q−40 + 158q−42 + 136q−44−356q−46 + 405q−48−270q−50 + 10q−52 + 254q−54−425q−56 + 448q−58−321q−60 + 111q−62 + 110q−64−276q−66 + 338q−68−310q−70 + 215q−72−84q−74−35q−76 + 125q−78−169q−80 + 165q−82−127q−84 + 72q−86−16q−88−29q−90 + 53q−92−62q−94 + 51q−96−31q−98 + 15q−100 + q−102−8q−104 + 10q−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["K11a81"];
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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 + 6t3−16t2 + 26t−29 + 26t−1−16t−2 + 6t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−2z6 + 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]=
| { 127, 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 + 8q5−13q4 + 18q3−20q2 + 20q−17 + 13q−1−8q−2 + 4q−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−a2z4−9z4a−2 + 3z4a−4 + 7z4−2a2z2−6z2a−2 + 2z2a−4 + 6z2 + 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 + 2z10 + 5az9 + 12z9a−1 + 7z9a−3 + 4a2z8 + 15z8a−2 + 11z8a−4 + 8z8 + a3z7−13az7−26z7a−1−z7a−3 + 11z7a−5−14a2z6−50z6a−2−14z6a−4 + 8z6a−6−42z6−3a3z5 + 3az5−23z5a−3−13z5a−5 + 4z5a−7 + 15a2z4 + 39z4a−2 + 2z4a−4−7z4a−6 + z4a−8 + 44z4 + 3a3z3 + 9az3 + 18z3a−1 + 19z3a−3 + 5z3a−5−2z3a−7−5a2z2−9z2a−2 + z2a−4 + 2z2a−6−13z2−a3z−4az−6za−1−4za−3−za−5 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a282,}
Same Jones Polynomial (up to mirroring, 
):
{K11a282,}
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["K11a81"];
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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 + 6t3−16t2 + 26t−29 + 26t−1−16t−2 + 6t−3−t−4, q7−4q6 + 8q5−13q4 + 18q3−20q2 + 20q−17 + 13q−1−8q−2 + 4q−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]=
| {K11a282,} |
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]=
| {K11a282,} |
[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 K11a81. 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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