K11n105
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n105's page at Knotilus! Visit K11n105's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X11,21,12,20 X13,1,14,22 X18,16,19,15 X8,18,9,17 X6,19,7,20 X21,13,22,12 |
| Gauss code | 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, -6, 11, -7, -3, 8, -4, 9, -8, 10, 6, -11, 7 |
| Dowker-Thistlethwaite code | 4 10 14 16 2 -20 -22 18 8 6 -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 | −t3 + 7t2−16t + 21−16t−1 + 7t−2−t−3 |
| Conway polynomial | −z6 + z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 69, 4 } |
| Jones polynomial | −q11 + 4q10−7q9 + 9q8−12q7 + 12q6−10q5 + 8q4−4q3 + 2q2 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−6 + 2z4a−4−3z4a−6 + 2z4a−8 + 5z2a−4−4z2a−6 + 3z2a−8−z2a−10 + 3a−4−2a−6 |
| Kauffman polynomial (db, data sources) | z9a−7 + z9a−9 + 2z8a−6 + 6z8a−8 + 4z8a−10 + z7a−5 + 3z7a−7 + 8z7a−9 + 6z7a−11−3z6a−6−10z6a−8−3z6a−10 + 4z6a−12 + z5a−5−5z5a−7−19z5a−9−12z5a−11 + z5a−13 + 3z4a−4 + 8z4a−6 + 8z4a−8−4z4a−10−7z4a−12−z3a−5 + 4z3a−7 + 11z3a−9 + 5z3a−11−z3a−13−6z2a−4−8z2a−6−2z2a−8 + z2a−10 + z2a−12−2za−5−za−7 + za−9 + 3a−4 + 2a−6 |
| The A2 invariant | 2q−6−q−8 + 3q−10 + q−12−q−14 + 3q−16−2q−18 + q−20−2q−22−2q−24 + q−26−2q−28 + 2q−30 + q−32−q−34 |
| The G2 invariant | Data:K11n105/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["K11n105"];
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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]=
| −t3 + 7t2−16t + 21−16t−1 + 7t−2−t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z6 + z4 + 3z2 + 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]=
| { 69, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q11 + 4q10−7q9 + 9q8−12q7 + 12q6−10q5 + 8q4−4q3 + 2q2 |
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−6 + 2z4a−4−3z4a−6 + 2z4a−8 + 5z2a−4−4z2a−6 + 3z2a−8−z2a−10 + 3a−4−2a−6 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−7 + z9a−9 + 2z8a−6 + 6z8a−8 + 4z8a−10 + z7a−5 + 3z7a−7 + 8z7a−9 + 6z7a−11−3z6a−6−10z6a−8−3z6a−10 + 4z6a−12 + z5a−5−5z5a−7−19z5a−9−12z5a−11 + z5a−13 + 3z4a−4 + 8z4a−6 + 8z4a−8−4z4a−10−7z4a−12−z3a−5 + 4z3a−7 + 11z3a−9 + 5z3a−11−z3a−13−6z2a−4−8z2a−6−2z2a−8 + z2a−10 + z2a−12−2za−5−za−7 + za−9 + 3a−4 + 2a−6 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_78, K11n98,}
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["K11n105"];
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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]=
| { −t3 + 7t2−16t + 21−16t−1 + 7t−2−t−3, −q11 + 4q10−7q9 + 9q8−12q7 + 12q6−10q5 + 8q4−4q3 + 2q2 } |
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]=
| {10_78, K11n98,} |
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 K11n105. 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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