K11a40
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 X8493 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X10,20,11,19 X12,22,13,21 |
| Gauss code | 1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 14 2 18 20 16 6 22 10 12 |
| 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+12 t^2-17 t+19-17 t^{-1} +12 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+2 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 | { 89, 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^{10}-3 q^9+6 q^8-10 q^7+12 q^6-14 q^5+14 q^4-11 q^3+9 q^2-5 q+3- q^{-1} } |
| 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} -z^6 a^{-2} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +14 z^4 a^{-4} -9 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +16 z^2 a^{-4} -13 z^2 a^{-6} +3 z^2 a^{-8} - a^{-2} +7 a^{-4} -7 a^{-6} +2 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^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +6 z^9 a^{-5} +3 z^9 a^{-7} +3 z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +5 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -14 z^7 a^{-5} +2 z^7 a^{-7} +6 z^7 a^{-9} -13 z^6 a^{-2} -30 z^6 a^{-4} -26 z^6 a^{-6} -4 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} +2 z^5 a^{-3} -16 z^5 a^{-7} -7 z^5 a^{-9} +3 z^5 a^{-11} +17 z^4 a^{-2} +42 z^4 a^{-4} +30 z^4 a^{-6} -z^4 a^{-8} -5 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +8 z^3 a^{-3} +14 z^3 a^{-5} +17 z^3 a^{-7} +4 z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -21 z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} -z a^{-1} -3 z a^{-3} -8 z a^{-5} -8 z a^{-7} -z a^{-9} +z a^{-11} + a^{-2} +7 a^{-4} +7 a^{-6} +2 a^{-8} } |
| 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^2+1- q^{-2} + q^{-4} +2 q^{-6} +5 q^{-10} - q^{-12} +2 q^{-14} - q^{-16} -3 q^{-18} -3 q^{-22} + q^{-24} + q^{-30} } |
| 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^{12}-2 q^{10}+5 q^8-9 q^6+10 q^4-11 q^2+3+14 q^{-2} -36 q^{-4} +57 q^{-6} -65 q^{-8} +46 q^{-10} -4 q^{-12} -58 q^{-14} +118 q^{-16} -146 q^{-18} +129 q^{-20} -62 q^{-22} -37 q^{-24} +130 q^{-26} -180 q^{-28} +171 q^{-30} -98 q^{-32} + q^{-34} +95 q^{-36} -139 q^{-38} +127 q^{-40} -58 q^{-42} -22 q^{-44} +94 q^{-46} -112 q^{-48} +76 q^{-50} +5 q^{-52} -92 q^{-54} +161 q^{-56} -166 q^{-58} +111 q^{-60} -6 q^{-62} -115 q^{-64} +201 q^{-66} -231 q^{-68} +181 q^{-70} -73 q^{-72} -59 q^{-74} +157 q^{-76} -198 q^{-78} +162 q^{-80} -78 q^{-82} -24 q^{-84} +87 q^{-86} -106 q^{-88} +69 q^{-90} -6 q^{-92} -55 q^{-94} +86 q^{-96} -73 q^{-98} +23 q^{-100} +37 q^{-102} -92 q^{-104} +116 q^{-106} -103 q^{-108} +64 q^{-110} -6 q^{-112} -52 q^{-114} +97 q^{-116} -115 q^{-118} +106 q^{-120} -67 q^{-122} +20 q^{-124} +27 q^{-126} -63 q^{-128} +77 q^{-130} -70 q^{-132} +51 q^{-134} -20 q^{-136} -6 q^{-138} +23 q^{-140} -31 q^{-142} +28 q^{-144} -20 q^{-146} +11 q^{-148} -2 q^{-150} -4 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} } |
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["K11a40"];
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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+12 t^2-17 t+19-17 t^{-1} +12 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+2 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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{ 89, 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^{10}-3 q^9+6 q^8-10 q^7+12 q^6-14 q^5+14 q^4-11 q^3+9 q^2-5 q+3- q^{-1} } |
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} -z^6 a^{-2} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +14 z^4 a^{-4} -9 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +16 z^2 a^{-4} -13 z^2 a^{-6} +3 z^2 a^{-8} - a^{-2} +7 a^{-4} -7 a^{-6} +2 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^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +6 z^9 a^{-5} +3 z^9 a^{-7} +3 z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +5 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -14 z^7 a^{-5} +2 z^7 a^{-7} +6 z^7 a^{-9} -13 z^6 a^{-2} -30 z^6 a^{-4} -26 z^6 a^{-6} -4 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} +2 z^5 a^{-3} -16 z^5 a^{-7} -7 z^5 a^{-9} +3 z^5 a^{-11} +17 z^4 a^{-2} +42 z^4 a^{-4} +30 z^4 a^{-6} -z^4 a^{-8} -5 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +8 z^3 a^{-3} +14 z^3 a^{-5} +17 z^3 a^{-7} +4 z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -21 z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} -z a^{-1} -3 z a^{-3} -8 z a^{-5} -8 z a^{-7} -z a^{-9} +z a^{-11} + a^{-2} +7 a^{-4} +7 a^{-6} +2 a^{-8} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a330,}
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["K11a40"];
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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+12 t^2-17 t+19-17 t^{-1} +12 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^{10}-3 q^9+6 q^8-10 q^7+12 q^6-14 q^5+14 q^4-11 q^3+9 q^2-5 q+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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{K11a330,} |
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, 1) |
| 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 K11a40. 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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