K11a301
From Knot Atlas
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 (Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
| Planar diagram presentation | X6271 X10,3,11,4 X18,5,19,6 X14,7,15,8 X16,10,17,9 X4,11,5,12 X22,13,1,14 X20,16,21,15 X12,18,13,17 X2,19,3,20 X8,21,9,22 |
| Gauss code | 1, -10, 2, -6, 3, -1, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 6 10 18 14 16 4 22 20 12 2 8 |
| A Braid Representative |
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| A Morse Link Presentation | 
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Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -t^4+7 t^3-22 t^2+43 t-53+43 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math] |
| Conway polynomial | [math]\displaystyle{ -z^8-z^6+2 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 199, -2 } |
| Jones polynomial | [math]\displaystyle{ q^3-5 q^2+12 q-20+28 q^{-1} -32 q^{-2} +33 q^{-3} -28 q^{-4} +21 q^{-5} -13 q^{-6} +5 q^{-7} - q^{-8} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -a^2 z^8+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+5 a^4 z^4-6 a^2 z^4+2 z^4-a^6 z^2+4 a^4 z^2-2 a^2 z^2+z^2-a^6+a^4+a^2 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ 5 a^4 z^{10}+5 a^2 z^{10}+15 a^5 z^9+27 a^3 z^9+12 a z^9+19 a^6 z^8+26 a^4 z^8+18 a^2 z^8+11 z^8+13 a^7 z^7-11 a^5 z^7-47 a^3 z^7-18 a z^7+5 z^7 a^{-1} +5 a^8 z^6-29 a^6 z^6-70 a^4 z^6-59 a^2 z^6+z^6 a^{-2} -22 z^6+a^9 z^5-15 a^7 z^5-13 a^5 z^5+13 a^3 z^5+2 a z^5-8 z^5 a^{-1} -2 a^8 z^4+16 a^6 z^4+46 a^4 z^4+43 a^2 z^4-z^4 a^{-2} +14 z^4+6 a^7 z^3+10 a^5 z^3+5 a^3 z^3+4 a z^3+3 z^3 a^{-1} -5 a^6 z^2-10 a^4 z^2-8 a^2 z^2-3 z^2-2 a^7 z-2 a^5 z+a^6+a^4-a^2 }[/math] |
| The A2 invariant | [math]\displaystyle{ -q^{24}+2 q^{22}-q^{20}-4 q^{18}+5 q^{16}-5 q^{14}+3 q^{12}+2 q^{10}-3 q^8+7 q^6-6 q^4+6 q^2-1-3 q^{-2} +4 q^{-4} -3 q^{-6} + q^{-8} }[/math] |
| The G2 invariant | Data:K11a301/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]:=
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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]:=
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K = Knot["K11a301"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ -t^4+7 t^3-22 t^2+43 t-53+43 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -z^8-z^6+2 z^2+1 }[/math] |
In[6]:=
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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]=
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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 199, -2 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ q^3-5 q^2+12 q-20+28 q^{-1} -32 q^{-2} +33 q^{-3} -28 q^{-4} +21 q^{-5} -13 q^{-6} +5 q^{-7} - q^{-8} }[/math] |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+5 a^4 z^4-6 a^2 z^4+2 z^4-a^6 z^2+4 a^4 z^2-2 a^2 z^2+z^2-a^6+a^4+a^2 }[/math] |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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[math]\displaystyle{ 5 a^4 z^{10}+5 a^2 z^{10}+15 a^5 z^9+27 a^3 z^9+12 a z^9+19 a^6 z^8+26 a^4 z^8+18 a^2 z^8+11 z^8+13 a^7 z^7-11 a^5 z^7-47 a^3 z^7-18 a z^7+5 z^7 a^{-1} +5 a^8 z^6-29 a^6 z^6-70 a^4 z^6-59 a^2 z^6+z^6 a^{-2} -22 z^6+a^9 z^5-15 a^7 z^5-13 a^5 z^5+13 a^3 z^5+2 a z^5-8 z^5 a^{-1} -2 a^8 z^4+16 a^6 z^4+46 a^4 z^4+43 a^2 z^4-z^4 a^{-2} +14 z^4+6 a^7 z^3+10 a^5 z^3+5 a^3 z^3+4 a z^3+3 z^3 a^{-1} -5 a^6 z^2-10 a^4 z^2-8 a^2 z^2-3 z^2-2 a^7 z-2 a^5 z+a^6+a^4-a^2 }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}
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]:=
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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]:=
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K = Knot["K11a301"];
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In[4]:=
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{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]=
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{ [math]\displaystyle{ -t^4+7 t^3-22 t^2+43 t-53+43 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-5 q^2+12 q-20+28 q^{-1} -32 q^{-2} +33 q^{-3} -28 q^{-4} +21 q^{-5} -13 q^{-6} +5 q^{-7} - q^{-8} }[/math] } |
In[5]:=
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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]=
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{} |
In[6]:=
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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]=
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{} |
Vassiliev invariants
| V2 and V3: | (2, -3) |
| V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
| The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of K11a301. 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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Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.
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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