K11a83
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a83's page at Knotilus! Visit K11a83's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X12,5,13,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X20,16,21,15 X8,18,9,17 X14,20,15,19 X6,21,7,22 |
| Gauss code | 1, -5, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, -8, 11, -6 |
| Dowker-Thistlethwaite code | 4 10 12 16 2 22 18 20 8 14 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 | t4−5t3 + 14t2−23t + 27−23t−1 + 14t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6 + 4z4 + 4z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 113, 4 } |
| Jones polynomial | q10−4q9 + 8q8−13q7 + 16q6−18q5 + 18q4−14q3 + 11q2−6q + 3−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 6z6a−4−2z6a−6−4z4a−2 + 15z4a−4−8z4a−6 + z4a−8−5z2a−2 + 18z2a−4−11z2a−6 + 2z2a−8−2a−2 + 8a−4−6a−6 + a−8 |
| Kauffman polynomial (db, data sources) | z10a−4 + z10a−6 + 3z9a−3 + 8z9a−5 + 5z9a−7 + 3z8a−2 + 9z8a−4 + 16z8a−6 + 10z8a−8 + z7a−1−6z7a−3−13z7a−5 + 5z7a−7 + 11z7a−9−12z6a−2−41z6a−4−51z6a−6−14z6a−8 + 8z6a−10−4z5a−1−6z5a−3−15z5a−5−32z5a−7−15z5a−9 + 4z5a−11 + 16z4a−2 + 52z4a−4 + 50z4a−6 + 6z4a−8−7z4a−10 + z4a−12 + 5z3a−1 + 16z3a−3 + 31z3a−5 + 29z3a−7 + 7z3a−9−2z3a−11−9z2a−2−29z2a−4−24z2a−6−3z2a−8 + z2a−10−2za−1−7za−3−13za−5−9za−7−za−9 + 2a−2 + 8a−4 + 6a−6 + a−8 |
| The A2 invariant | −q2 + 1−2q−2 + q−4 + 2q−6−q−8 + 6q−10−q−12 + 3q−14−3q−18 + q−20−4q−22 + q−24−q−28 + q−30 |
| The G2 invariant | q12−2q10 + 6q8−11q6 + 14q4−16q2 + 5 + 17q−2−48q−4 + 80q−6−97q−8 + 78q−10−21q−12−76q−14 + 181q−16−251q−18 + 249q−20−156q−22−17q−24 + 210q−26−357q−28 + 399q−30−297q−32 + 94q−34 + 147q−36−324q−38 + 373q−40−270q−42 + 74q−44 + 147q−46−278q−48 + 274q−50−123q−52−99q−54 + 309q−56−392q−58 + 321q−60−100q−62−190q−64 + 437q−66−552q−68 + 482q−70−243q−72−79q−74 + 363q−76−518q−78 + 481q−80−289q−82 + 13q−84 + 217q−86−334q−88 + 287q−90−121q−92−84q−94 + 234q−96−259q−98 + 151q−100 + 31q−102−219q−104 + 327q−106−315q−108 + 199q−110−15q−112−164q−114 + 283q−116−310q−118 + 249q−120−130q−122−q−124 + 106q−126−168q−128 + 173q−130−136q−132 + 81q−134−17q−136−29q−138 + 53q−140−62q−142 + 51q−144−32q−146 + 15q−148 + q−150−8q−152 + 10q−154−10q−156 + 6q−158−3q−160 + q−162 |
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["K11a83"];
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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]=
| t4−5t3 + 14t2−23t + 27−23t−1 + 14t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 3z6 + 4z4 + 4z2 + 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]=
| { 113, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q10−4q9 + 8q8−13q7 + 16q6−18q5 + 18q4−14q3 + 11q2−6q + 3−q−1 |
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−4−z6a−2 + 6z6a−4−2z6a−6−4z4a−2 + 15z4a−4−8z4a−6 + z4a−8−5z2a−2 + 18z2a−4−11z2a−6 + 2z2a−8−2a−2 + 8a−4−6a−6 + a−8 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−4 + z10a−6 + 3z9a−3 + 8z9a−5 + 5z9a−7 + 3z8a−2 + 9z8a−4 + 16z8a−6 + 10z8a−8 + z7a−1−6z7a−3−13z7a−5 + 5z7a−7 + 11z7a−9−12z6a−2−41z6a−4−51z6a−6−14z6a−8 + 8z6a−10−4z5a−1−6z5a−3−15z5a−5−32z5a−7−15z5a−9 + 4z5a−11 + 16z4a−2 + 52z4a−4 + 50z4a−6 + 6z4a−8−7z4a−10 + z4a−12 + 5z3a−1 + 16z3a−3 + 31z3a−5 + 29z3a−7 + 7z3a−9−2z3a−11−9z2a−2−29z2a−4−24z2a−6−3z2a−8 + z2a−10−2za−1−7za−3−13za−5−9za−7−za−9 + 2a−2 + 8a−4 + 6a−6 + a−8 |
[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["K11a83"];
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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]=
| { t4−5t3 + 14t2−23t + 27−23t−1 + 14t−2−5t−3 + t−4, q10−4q9 + 8q8−13q7 + 16q6−18q5 + 18q4−14q3 + 11q2−6q + 3−q−1 } |
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 = 4 is the signature of K11a83. 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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