K11a40
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a40's page at Knotilus! Visit K11a40's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X10,20,11,19 X12,22,13,21 |
| Gauss code | 1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 14 2 18 20 16 6 22 10 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 | t4−5t3 + 12t2−17t + 19−17t−1 + 12t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6 + 2z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 89, 4 } |
| Jones polynomial | q10−3q9 + 6q8−10q7 + 12q6−14q5 + 14q4−11q3 + 9q2−5q + 3−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 6z6a−4−2z6a−6−4z4a−2 + 14z4a−4−9z4a−6 + z4a−8−4z2a−2 + 16z2a−4−13z2a−6 + 3z2a−8−a−2 + 7a−4−7a−6 + 2a−8 |
| Kauffman polynomial (db, data sources) | z10a−4 + z10a−6 + 3z9a−3 + 6z9a−5 + 3z9a−7 + 3z8a−2 + 5z8a−4 + 7z8a−6 + 5z8a−8 + z7a−1−9z7a−3−14z7a−5 + 2z7a−7 + 6z7a−9−13z6a−2−30z6a−4−26z6a−6−4z6a−8 + 5z6a−10−4z5a−1 + 2z5a−3−16z5a−7−7z5a−9 + 3z5a−11 + 17z4a−2 + 42z4a−4 + 30z4a−6−z4a−8−5z4a−10 + z4a−12 + 4z3a−1 + 8z3a−3 + 14z3a−5 + 17z3a−7 + 4z3a−9−3z3a−11−8z2a−2−25z2a−4−21z2a−6−z2a−8 + 2z2a−10−z2a−12−za−1−3za−3−8za−5−8za−7−za−9 + za−11 + a−2 + 7a−4 + 7a−6 + 2a−8 |
| The A2 invariant | −q2 + 1−q−2 + q−4 + 2q−6 + 5q−10−q−12 + 2q−14−q−16−3q−18−3q−22 + q−24 + q−30 |
| The G2 invariant | q12−2q10 + 5q8−9q6 + 10q4−11q2 + 3 + 14q−2−36q−4 + 57q−6−65q−8 + 46q−10−4q−12−58q−14 + 118q−16−146q−18 + 129q−20−62q−22−37q−24 + 130q−26−180q−28 + 171q−30−98q−32 + q−34 + 95q−36−139q−38 + 127q−40−58q−42−22q−44 + 94q−46−112q−48 + 76q−50 + 5q−52−92q−54 + 161q−56−166q−58 + 111q−60−6q−62−115q−64 + 201q−66−231q−68 + 181q−70−73q−72−59q−74 + 157q−76−198q−78 + 162q−80−78q−82−24q−84 + 87q−86−106q−88 + 69q−90−6q−92−55q−94 + 86q−96−73q−98 + 23q−100 + 37q−102−92q−104 + 116q−106−103q−108 + 64q−110−6q−112−52q−114 + 97q−116−115q−118 + 106q−120−67q−122 + 20q−124 + 27q−126−63q−128 + 77q−130−70q−132 + 51q−134−20q−136−6q−138 + 23q−140−31q−142 + 28q−144−20q−146 + 11q−148−2q−150−4q−152 + 5q−154−6q−156 + 4q−158−2q−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["K11a40"];
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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 + 12t2−17t + 19−17t−1 + 12t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 3z6 + 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]=
| { 89, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q10−3q9 + 6q8−10q7 + 12q6−14q5 + 14q4−11q3 + 9q2−5q + 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 + 14z4a−4−9z4a−6 + z4a−8−4z2a−2 + 16z2a−4−13z2a−6 + 3z2a−8−a−2 + 7a−4−7a−6 + 2a−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 + 6z9a−5 + 3z9a−7 + 3z8a−2 + 5z8a−4 + 7z8a−6 + 5z8a−8 + z7a−1−9z7a−3−14z7a−5 + 2z7a−7 + 6z7a−9−13z6a−2−30z6a−4−26z6a−6−4z6a−8 + 5z6a−10−4z5a−1 + 2z5a−3−16z5a−7−7z5a−9 + 3z5a−11 + 17z4a−2 + 42z4a−4 + 30z4a−6−z4a−8−5z4a−10 + z4a−12 + 4z3a−1 + 8z3a−3 + 14z3a−5 + 17z3a−7 + 4z3a−9−3z3a−11−8z2a−2−25z2a−4−21z2a−6−z2a−8 + 2z2a−10−z2a−12−za−1−3za−3−8za−5−8za−7−za−9 + za−11 + a−2 + 7a−4 + 7a−6 + 2a−8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a330,}
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["K11a40"];
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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 + 12t2−17t + 19−17t−1 + 12t−2−5t−3 + t−4, q10−3q9 + 6q8−10q7 + 12q6−14q5 + 14q4−11q3 + 9q2−5q + 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]=
| {K11a330,} |
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 K11a40. 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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