K11n54
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![]() (Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
Planar diagram presentation | X4251 X8493 X14,6,15,5 X2837 X9,16,10,17 X11,19,12,18 X6,14,7,13 X15,22,16,1 X17,21,18,20 X19,13,20,12 X21,10,22,11 |
Gauss code | 1, -4, 2, -1, 3, -7, 4, -2, -5, 11, -6, 10, 7, -3, -8, 5, -9, 6, -10, 9, -11, 8 |
Dowker-Thistlethwaite code | 4 8 14 2 -16 -18 6 -22 -20 -12 -10 |
A Braid Representative |
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A Morse Link Presentation | ![]() |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
Alexander polynomial | 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 t^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} } |
Conway polynomial | 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 z^6+2 z^4+3 z^2+1} |
2nd Alexander ideal (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 \{1\}} |
Determinant and Signature | { 43, 2 } |
Jones polynomial | |
HOMFLY-PT 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 z^6 a^{-4} -z^4 a^{-2} +5 z^4 a^{-4} -2 z^4 a^{-6} -2 z^2 a^{-2} +10 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +6 a^{-4} -5 a^{-6} + a^{-8} } |
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 z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +z^7 a^{-3} +2 z^7 a^{-7} +3 z^7 a^{-9} -8 z^6 a^{-4} -19 z^6 a^{-6} -10 z^6 a^{-8} +z^6 a^{-10} -2 z^5 a^{-3} -9 z^5 a^{-5} -18 z^5 a^{-7} -11 z^5 a^{-9} +3 z^4 a^{-2} +18 z^4 a^{-4} +25 z^4 a^{-6} +7 z^4 a^{-8} -3 z^4 a^{-10} +z^3 a^{-1} +6 z^3 a^{-3} +19 z^3 a^{-5} +23 z^3 a^{-7} +9 z^3 a^{-9} -4 z^2 a^{-2} -16 z^2 a^{-4} -15 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -9 z a^{-7} -2 z a^{-9} + a^{-2} +6 a^{-4} +5 a^{-6} + a^{-8} } |
The A2 invariant | Data:K11n54/QuantumInvariant/A2/1,0 |
The G2 invariant | Data:K11n54/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["K11n54"];
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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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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 t^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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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 z^6+2 z^4+3 z^2+1} |
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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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 \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 43, 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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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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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 z^6 a^{-4} -z^4 a^{-2} +5 z^4 a^{-4} -2 z^4 a^{-6} -2 z^2 a^{-2} +10 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +6 a^{-4} -5 a^{-6} + a^{-8} } |
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 z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +z^7 a^{-3} +2 z^7 a^{-7} +3 z^7 a^{-9} -8 z^6 a^{-4} -19 z^6 a^{-6} -10 z^6 a^{-8} +z^6 a^{-10} -2 z^5 a^{-3} -9 z^5 a^{-5} -18 z^5 a^{-7} -11 z^5 a^{-9} +3 z^4 a^{-2} +18 z^4 a^{-4} +25 z^4 a^{-6} +7 z^4 a^{-8} -3 z^4 a^{-10} +z^3 a^{-1} +6 z^3 a^{-3} +19 z^3 a^{-5} +23 z^3 a^{-7} +9 z^3 a^{-9} -4 z^2 a^{-2} -16 z^2 a^{-4} -15 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -9 z a^{-7} -2 z a^{-9} + a^{-2} +6 a^{-4} +5 a^{-6} + a^{-8} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_151, K11n129,}
Same Jones Polynomial (up to mirroring, 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 q\leftrightarrow q^{-1}} ): {}
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["K11n54"];
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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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{ 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 t^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} } , 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 q^9-3 q^8+4 q^7-6 q^6+7 q^5-7 q^4+7 q^3-4 q^2+3 q-1} } |
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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{10_151, K11n129,} |
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: | (3, 4) |
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 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 t^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where 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 j-2r=s+1} or 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 j-2r=s-1} , where 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 s=} 2 is the signature of K11n54. 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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