K11a185
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a185's page at Knotilus! Visit K11a185's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X22,12,1,11 X2,13,3,14 X20,15,21,16 X10,17,11,18 X8,19,9,20 X6,22,7,21 |
| Gauss code | 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -4, 9, -5, 10, -8, 11, -6 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 22 2 20 10 8 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 | −2t3 + 11t2−25t + 33−25t−1 + 11t−2−2t−3 |
| Conway polynomial | −2z6−z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 109, 0 } |
| Jones polynomial | q4−4q3 + 8q2−12q + 16−17q−1 + 17q−2−14q−3 + 10q−4−6q−5 + 3q−6−q−7 |
| HOMFLY-PT polynomial (db, data sources) | −z2a6−a6 + 2z4a4 + 4z2a4 + 2a4−z6a2−2z4a2−2z2a2−a2−z6−2z4−z2 + 1 + z4a−2 + z2a−2 |
| Kauffman polynomial (db, data sources) | a4z10 + a2z10 + 3a5z9 + 7a3z9 + 4az9 + 3a6z8 + 7a4z8 + 12a2z8 + 8z8 + a7z7−7a5z7−11a3z7 + 7az7 + 10z7a−1−12a6z6−31a4z6−32a2z6 + 8z6a−2−5z6−4a7z5−2a5z5−9a3z5−27az5−12z5a−1 + 4z5a−3 + 15a6z4 + 34a4z4 + 21a2z4−8z4a−2 + z4a−4−7z4 + 5a7z3 + 11a5z3 + 15a3z3 + 16az3 + 5z3a−1−2z3a−3−6a6z2−13a4z2−7a2z2 + 3z2a−2 + 3z2−2a7z−4a5z−4a3z−3az−za−1 + a6 + 2a4 + a2 + 1 |
| The A2 invariant | Data:K11a185/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a185/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["K11a185"];
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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]=
| −2t3 + 11t2−25t + 33−25t−1 + 11t−2−2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −2z6−z4 + z2 + 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]=
| { 109, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q4−4q3 + 8q2−12q + 16−17q−1 + 17q−2−14q−3 + 10q−4−6q−5 + 3q−6−q−7 |
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]=
| −z2a6−a6 + 2z4a4 + 4z2a4 + 2a4−z6a2−2z4a2−2z2a2−a2−z6−2z4−z2 + 1 + z4a−2 + z2a−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a4z10 + a2z10 + 3a5z9 + 7a3z9 + 4az9 + 3a6z8 + 7a4z8 + 12a2z8 + 8z8 + a7z7−7a5z7−11a3z7 + 7az7 + 10z7a−1−12a6z6−31a4z6−32a2z6 + 8z6a−2−5z6−4a7z5−2a5z5−9a3z5−27az5−12z5a−1 + 4z5a−3 + 15a6z4 + 34a4z4 + 21a2z4−8z4a−2 + z4a−4−7z4 + 5a7z3 + 11a5z3 + 15a3z3 + 16az3 + 5z3a−1−2z3a−3−6a6z2−13a4z2−7a2z2 + 3z2a−2 + 3z2−2a7z−4a5z−4a3z−3az−za−1 + a6 + 2a4 + a2 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a56, K11a265,}
Same Jones Polynomial (up to mirroring, 
):
{K11a265,}
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["K11a185"];
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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]=
| { −2t3 + 11t2−25t + 33−25t−1 + 11t−2−2t−3, q4−4q3 + 8q2−12q + 16−17q−1 + 17q−2−14q−3 + 10q−4−6q−5 + 3q−6−q−7 } |
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]=
| {K11a56, K11a265,} |
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]=
| {K11a265,} |
[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 K11a185. 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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