K11a248
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a248's page at Knotilus! Visit K11a248's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X20,11,21,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X14,19,15,20 X12,21,13,22 |
| Gauss code | 1, -9, 2, -8, 3, -1, 4, -2, 5, -3, 6, -11, 7, -10, 8, -5, 9, -4, 10, -6, 11, -7 |
| Dowker-Thistlethwaite code | 6 8 10 18 16 20 22 4 2 14 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 + 6t3−18t2 + 34t−41 + 34t−1−18t−2 + 6t−3−t−4 |
| Conway polynomial | −z8−2z6−2z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 159, -2 } |
| Jones polynomial | q3−4q2 + 9q−15 + 22q−1−25q−2 + 26q−3−23q−4 + 17q−5−11q−6 + 5q−7−q−8 |
| HOMFLY-PT polynomial (db, data sources) | −a2z8 + 2a4z6−5a2z6 + z6−a6z4 + 6a4z4−10a2z4 + 3z4−a6z2 + 5a4z2−7a2z2 + 3z2 + 1 |
| Kauffman polynomial (db, data sources) | 3a4z10 + 3a2z10 + 9a5z9 + 17a3z9 + 8az9 + 13a6z8 + 17a4z8 + 12a2z8 + 8z8 + 11a7z7−2a5z7−33a3z7−16az7 + 4z7a−1 + 5a8z6−17a6z6−44a4z6−44a2z6 + z6a−2−21z6 + a9z5−15a7z5−14a5z5 + 21a3z5 + 10az5−9z5a−1−4a8z4 + 4a6z4 + 32a4z4 + 45a2z4−2z4a−2 + 19z4 + 4a7z3 + 3a5z3−7a3z3−2az3 + 4z3a−1−a6z2−9a4z2−15a2z2−7z2 + a7z + 3a5z + 3a3z + az + 1 |
| The A2 invariant | −q24 + 2q22−2q18 + 4q16−5q14 + q12−2q8 + 6q6−4q4 + 5q2−1−2q−2 + 3q−4−2q−6 + q−8 |
| The G2 invariant | Data:K11a248/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["K11a248"];
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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 + 6t3−18t2 + 34t−41 + 34t−1−18t−2 + 6t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−2z6−2z4 + 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]=
| { 159, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q3−4q2 + 9q−15 + 22q−1−25q−2 + 26q−3−23q−4 + 17q−5−11q−6 + 5q−7−q−8 |
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]=
| −a2z8 + 2a4z6−5a2z6 + z6−a6z4 + 6a4z4−10a2z4 + 3z4−a6z2 + 5a4z2−7a2z2 + 3z2 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 3a4z10 + 3a2z10 + 9a5z9 + 17a3z9 + 8az9 + 13a6z8 + 17a4z8 + 12a2z8 + 8z8 + 11a7z7−2a5z7−33a3z7−16az7 + 4z7a−1 + 5a8z6−17a6z6−44a4z6−44a2z6 + z6a−2−21z6 + a9z5−15a7z5−14a5z5 + 21a3z5 + 10az5−9z5a−1−4a8z4 + 4a6z4 + 32a4z4 + 45a2z4−2z4a−2 + 19z4 + 4a7z3 + 3a5z3−7a3z3−2az3 + 4z3a−1−a6z2−9a4z2−15a2z2−7z2 + a7z + 3a5z + 3a3z + az + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a71,}
Same Jones Polynomial (up to mirroring, 
):
{K11a71,}
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["K11a248"];
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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 + 6t3−18t2 + 34t−41 + 34t−1−18t−2 + 6t−3−t−4, q3−4q2 + 9q−15 + 22q−1−25q−2 + 26q−3−23q−4 + 17q−5−11q−6 + 5q−7−q−8 } |
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]=
| {K11a71,} |
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]=
| {K11a71,} |
[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 K11a248. 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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