K11n139
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n139's page at Knotilus! Visit K11n139's page at the original Knot Atlas! |
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K11n139 is also known as the pretzel knot P(5,3,-3). |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,3,13,4 X16,6,17,5 X14,8,15,7 X9,21,10,20 X11,19,12,18 X2,13,3,14 X6,16,7,15 X17,22,18,1 X19,11,20,10 X21,9,22,8 |
| Gauss code | 1, -7, 2, -1, 3, -8, 4, 11, -5, 10, -6, -2, 7, -4, 8, -3, -9, 6, -10, 5, -11, 9 |
| Dowker-Thistlethwaite code | 4 12 16 14 -20 -18 2 6 -22 -10 -8 |
| 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 | −2t + 5−2t−1 |
| Conway polynomial | 1−2z2 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 9, 0 } |
| Jones polynomial | q8−q7 + q6−2q5 + q4−q3 + q2 + 1 |
| HOMFLY-PT polynomial (db, data sources) | −z2a−4−z2a−6 + a−2−a−4−a−6 + a−8 + 1 |
| Kauffman polynomial (db, data sources) | z9a−5 + z9a−7 + z8a−4 + 2z8a−6 + z8a−8−7z7a−5−7z7a−7−6z6a−4−13z6a−6−7z6a−8 + z5a−3 + 17z5a−5 + 16z5a−7 + 10z4a−4 + 25z4a−6 + 15z4a−8−5z3a−3−20z3a−5−15z3a−7−5z2a−4−15z2a−6−10z2a−8 + 4za−3 + 10za−5 + 6za−7−a−2−a−4 + a−6 + a−8 + 1 |
| The A2 invariant | Data:K11n139/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n139/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["K11n139"];
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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]=
| −2t + 5−2t−1 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 1−2z2 |
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]=
| { 9, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q8−q7 + q6−2q5 + q4−q3 + q2 + 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]=
| −z2a−4−z2a−6 + a−2−a−4−a−6 + a−8 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−5 + z9a−7 + z8a−4 + 2z8a−6 + z8a−8−7z7a−5−7z7a−7−6z6a−4−13z6a−6−7z6a−8 + z5a−3 + 17z5a−5 + 16z5a−7 + 10z4a−4 + 25z4a−6 + 15z4a−8−5z3a−3−20z3a−5−15z3a−7−5z2a−4−15z2a−6−10z2a−8 + 4za−3 + 10za−5 + 6za−7−a−2−a−4 + a−6 + a−8 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {6_1, 9_46, K11n67, K11n97,}
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["K11n139"];
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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]=
| { −2t + 5−2t−1, q8−q7 + q6−2q5 + q4−q3 + q2 + 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]=
| {6_1, 9_46, K11n67, K11n97,} |
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 = 0 is the signature of K11n139. 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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