K11n45
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n45's page at Knotilus! Visit K11n45's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X12,5,13,6 X2837 X9,21,10,20 X11,18,12,19 X6,13,7,14 X15,10,16,11 X17,1,18,22 X19,14,20,15 X21,17,22,16 |
| Gauss code | 1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 6, -10, 5, -11, 9 |
| Dowker-Thistlethwaite code | 4 8 12 2 -20 -18 6 -10 -22 -14 -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 | 2t2−6t + 9−6t−1 + 2t−2 |
| Conway polynomial | 2z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 25, 0 } |
| Jones polynomial | q6−3q5 + 4q4−5q3 + 5q2−4q + 4−q−1 + q−3−q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−2 + z6−a2z4−5z4a−2 + z4a−4 + 7z4−4a2z2−9z2a−2 + 2z2a−4 + 13z2−3a2−5a−2 + a−4 + 8 |
| Kauffman polynomial (db, data sources) | z9a−1 + z9a−3 + a2z8 + 4z8a−2 + 3z8a−4 + 2z8 + a3z7 + az7−3z7a−1 + 3z7a−5−7a2z6−20z6a−2−11z6a−4 + z6a−6−15z6−6a3z5−10az5−7z5a−1−14z5a−3−11z5a−5 + 13a2z4 + 30z4a−2 + 8z4a−4−3z4a−6 + 32z4 + 9a3z3 + 20az3 + 23z3a−1 + 20z3a−3 + 8z3a−5−10a2z2−18z2a−2−3z2a−4 + z2a−6−24z2−4a3z−10az−12za−1−8za−3−2za−5 + 3a2 + 5a−2 + a−4 + 8 |
| The A2 invariant | −q12−q10−q8 + q6 + 3q2 + 3 + q−2 + 2q−4−2q−6−2q−10−q−12 + q−14−q−16 + q−18 |
| The G2 invariant | Data:K11n45/QuantumInvariant/G2/1,0 |
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["K11n45"];
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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]=
| 2t2−6t + 9−6t−1 + 2t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 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]=
| { 25, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q6−3q5 + 4q4−5q3 + 5q2−4q + 4−q−1 + q−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]=
| −z6a−2 + z6−a2z4−5z4a−2 + z4a−4 + 7z4−4a2z2−9z2a−2 + 2z2a−4 + 13z2−3a2−5a−2 + a−4 + 8 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−1 + z9a−3 + a2z8 + 4z8a−2 + 3z8a−4 + 2z8 + a3z7 + az7−3z7a−1 + 3z7a−5−7a2z6−20z6a−2−11z6a−4 + z6a−6−15z6−6a3z5−10az5−7z5a−1−14z5a−3−11z5a−5 + 13a2z4 + 30z4a−2 + 8z4a−4−3z4a−6 + 32z4 + 9a3z3 + 20az3 + 23z3a−1 + 20z3a−3 + 8z3a−5−10a2z2−18z2a−2−3z2a−4 + z2a−6−24z2−4a3z−10az−12za−1−8za−3−2za−5 + 3a2 + 5a−2 + a−4 + 8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {8_8, 10_129, K11n39, K11n50, K11n132,}
Same Jones Polynomial (up to mirroring, 
):
{K11n39,}
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["K11n45"];
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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]=
| { 2t2−6t + 9−6t−1 + 2t−2, q6−3q5 + 4q4−5q3 + 5q2−4q + 4−q−1 + q−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]=
| {8_8, 10_129, K11n39, K11n50, K11n132,} |
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]=
| {K11n39,} |
[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 = 0 is the signature of K11n45. 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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