K11a179
From Knot Atlas
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 (Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
| Planar diagram presentation | X4251 X12,3,13,4 X14,6,15,5 X16,8,17,7 X18,10,19,9 X20,11,21,12 X2,13,3,14 X6,16,7,15 X8,18,9,17 X22,20,1,19 X10,21,11,22 |
| Gauss code | 1, -7, 2, -1, 3, -8, 4, -9, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -10 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 20 2 6 8 22 10 |
| 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 | 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^4-5 t^3+9 t^2-9 t+9-9 t^{-1} +9 t^{-2} -5 t^{-3} + t^{-4} } |
| 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^8+3 z^6-z^4-2 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 | { 57, 4 } |
| Jones 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 -q^9+3 q^8-4 q^7+6 q^6-8 q^5+8 q^4-8 q^3+7 q^2-5 q+4-2 q^{-1} + q^{-2} } |
| 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^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +12 z^4 a^{-4} -4 z^4 a^{-6} +z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -3 z^2 a^{-6} +4 z^2-4 a^{-2} +2 a^{-4} +3} |
| 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^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} -z^8 a^{-2} +2 z^8 a^{-4} +4 z^8 a^{-6} +z^8-11 z^7 a^{-1} -23 z^7 a^{-3} -8 z^7 a^{-5} +4 z^7 a^{-7} -15 z^6 a^{-2} -22 z^6 a^{-4} -9 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6+18 z^5 a^{-1} +28 z^5 a^{-3} +2 z^5 a^{-5} -4 z^5 a^{-7} +4 z^5 a^{-9} +35 z^4 a^{-2} +31 z^4 a^{-4} +3 z^4 a^{-6} -2 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-9 z^3 a^{-1} -7 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -3 z^2 a^{-8} -2 z^2 a^{-10} -10 z^2-z a^{-3} +z a^{-5} +3 z a^{-7} +z a^{-9} +4 a^{-2} +2 a^{-4} +3} |
| The A2 invariant | 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^6+q^4+1+ q^{-4} - q^{-8} -3 q^{-12} + q^{-14} + q^{-18} + q^{-20} + q^{-24} - q^{-26} } |
| The G2 invariant | 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^{26}-q^{24}+4 q^{22}-5 q^{20}+6 q^{18}-4 q^{16}+11 q^{12}-18 q^{10}+24 q^8-21 q^6+10 q^4+8 q^2-25+39 q^{-2} -35 q^{-4} +23 q^{-6} -2 q^{-8} -17 q^{-10} +27 q^{-12} -29 q^{-14} +18 q^{-16} -4 q^{-18} -9 q^{-20} +15 q^{-22} -10 q^{-24} +12 q^{-28} -18 q^{-30} +17 q^{-32} -11 q^{-34} -6 q^{-36} +18 q^{-38} -32 q^{-40} +34 q^{-42} -25 q^{-44} +8 q^{-46} +9 q^{-48} -26 q^{-50} +31 q^{-52} -29 q^{-54} +15 q^{-56} -2 q^{-58} -10 q^{-60} +14 q^{-62} -10 q^{-64} +5 q^{-66} +3 q^{-68} -2 q^{-70} +2 q^{-72} +4 q^{-74} -6 q^{-76} +10 q^{-78} -6 q^{-80} +3 q^{-82} +3 q^{-84} -3 q^{-86} +6 q^{-88} -7 q^{-90} +7 q^{-92} -7 q^{-94} +9 q^{-96} -8 q^{-98} -3 q^{-100} +6 q^{-102} -11 q^{-104} +16 q^{-106} -16 q^{-108} +11 q^{-110} -2 q^{-112} -7 q^{-114} +13 q^{-116} -18 q^{-118} +14 q^{-120} -10 q^{-122} +5 q^{-124} +2 q^{-126} -8 q^{-128} +12 q^{-130} -9 q^{-132} +8 q^{-134} -3 q^{-136} + q^{-140} -4 q^{-142} +3 q^{-144} -2 q^{-146} + q^{-148} } |
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["K11a179"];
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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^4-5 t^3+9 t^2-9 t+9-9 t^{-1} +9 t^{-2} -5 t^{-3} + t^{-4} } |
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^8+3 z^6-z^4-2 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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{ 57, 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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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^6-8 q^5+8 q^4-8 q^3+7 q^2-5 q+4-2 q^{-1} + q^{-2} } |
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^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +12 z^4 a^{-4} -4 z^4 a^{-6} +z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -3 z^2 a^{-6} +4 z^2-4 a^{-2} +2 a^{-4} +3} |
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^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} -z^8 a^{-2} +2 z^8 a^{-4} +4 z^8 a^{-6} +z^8-11 z^7 a^{-1} -23 z^7 a^{-3} -8 z^7 a^{-5} +4 z^7 a^{-7} -15 z^6 a^{-2} -22 z^6 a^{-4} -9 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6+18 z^5 a^{-1} +28 z^5 a^{-3} +2 z^5 a^{-5} -4 z^5 a^{-7} +4 z^5 a^{-9} +35 z^4 a^{-2} +31 z^4 a^{-4} +3 z^4 a^{-6} -2 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-9 z^3 a^{-1} -7 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -3 z^2 a^{-8} -2 z^2 a^{-10} -10 z^2-z a^{-3} +z a^{-5} +3 z a^{-7} +z a^{-9} +4 a^{-2} +2 a^{-4} +3} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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}} ): {}
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["K11a179"];
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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^4-5 t^3+9 t^2-9 t+9-9 t^{-1} +9 t^{-2} -5 t^{-3} + t^{-4} } , 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^6-8 q^5+8 q^4-8 q^3+7 q^2-5 q+4-2 q^{-1} + q^{-2} } } |
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, -2) |
| 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 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 r} ). 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=} 4 is the signature of K11a179. 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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