K11a320
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(Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
Planar diagram presentation | X6271 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X4,12,5,11 X20,14,21,13 X2,16,3,15 X10,18,11,17 X8,20,9,19 X14,22,15,21 |
Gauss code | 1, -8, 2, -6, 3, -1, 4, -10, 5, -9, 6, -2, 7, -11, 8, -3, 9, -5, 10, -7, 11, -4 |
Dowker-Thistlethwaite code | 6 12 16 22 18 4 20 2 10 8 14 |
A Braid Representative | {{{braid_table}}} |
A Morse Link Presentation |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
Alexander polynomial | |
Conway polynomial | |
2nd Alexander ideal (db, data sources) | |
Determinant and Signature | { 109, 4 } |
Jones polynomial | |
HOMFLY-PT polynomial (db, data sources) | |
Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^{10} a^{-10} +2 z^{10} a^{-12} +6 z^9 a^{-9} +10 z^9 a^{-11} +4 z^9 a^{-13} +9 z^8 a^{-8} +8 z^8 a^{-10} +2 z^8 a^{-12} +3 z^8 a^{-14} +8 z^7 a^{-7} -6 z^7 a^{-9} -27 z^7 a^{-11} -12 z^7 a^{-13} +z^7 a^{-15} +6 z^6 a^{-6} -17 z^6 a^{-8} -33 z^6 a^{-10} -21 z^6 a^{-12} -11 z^6 a^{-14} +3 z^5 a^{-5} -10 z^5 a^{-7} -8 z^5 a^{-9} +17 z^5 a^{-11} +8 z^5 a^{-13} -4 z^5 a^{-15} +z^4 a^{-4} -7 z^4 a^{-6} +16 z^4 a^{-8} +32 z^4 a^{-10} +20 z^4 a^{-12} +12 z^4 a^{-14} -2 z^3 a^{-5} +5 z^3 a^{-7} +11 z^3 a^{-9} -2 z^3 a^{-11} -z^3 a^{-13} +5 z^3 a^{-15} -z^2 a^{-4} +5 z^2 a^{-6} -9 z^2 a^{-8} -15 z^2 a^{-10} -4 z^2 a^{-12} -4 z^2 a^{-14} -4 z a^{-9} -z a^{-11} +z a^{-13} -2 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10} } |
The A2 invariant | Data:K11a320/QuantumInvariant/A2/1,0 |
The G2 invariant | Data:K11a320/QuantumInvariant/G2/1,0 |
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["K11a320"];
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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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In[5]:=
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Conway[K][z]
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Out[5]=
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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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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 109, 4 } |
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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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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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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^{10} a^{-10} +2 z^{10} a^{-12} +6 z^9 a^{-9} +10 z^9 a^{-11} +4 z^9 a^{-13} +9 z^8 a^{-8} +8 z^8 a^{-10} +2 z^8 a^{-12} +3 z^8 a^{-14} +8 z^7 a^{-7} -6 z^7 a^{-9} -27 z^7 a^{-11} -12 z^7 a^{-13} +z^7 a^{-15} +6 z^6 a^{-6} -17 z^6 a^{-8} -33 z^6 a^{-10} -21 z^6 a^{-12} -11 z^6 a^{-14} +3 z^5 a^{-5} -10 z^5 a^{-7} -8 z^5 a^{-9} +17 z^5 a^{-11} +8 z^5 a^{-13} -4 z^5 a^{-15} +z^4 a^{-4} -7 z^4 a^{-6} +16 z^4 a^{-8} +32 z^4 a^{-10} +20 z^4 a^{-12} +12 z^4 a^{-14} -2 z^3 a^{-5} +5 z^3 a^{-7} +11 z^3 a^{-9} -2 z^3 a^{-11} -z^3 a^{-13} +5 z^3 a^{-15} -z^2 a^{-4} +5 z^2 a^{-6} -9 z^2 a^{-8} -15 z^2 a^{-10} -4 z^2 a^{-12} -4 z^2 a^{-14} -4 z a^{-9} -z a^{-11} +z a^{-13} -2 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, ): {}
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["K11a320"];
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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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{ , } |
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: | (9, 26) |
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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 4 is the signature of K11a320. 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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