K11a176
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 X12,4,13,3 X14,5,15,6 X16,8,17,7 X18,10,19,9 X2,12,3,11 X22,13,1,14 X20,15,21,16 X8,18,9,17 X6,20,7,19 X10,21,11,22 |
| Gauss code | 1, -6, 2, -1, 3, -10, 4, -9, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 2 22 20 8 6 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^3-13 t^2+23 t-27+23 t^{-1} -13 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^6-3 z^4+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 | { 111, 2 } |
| 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^8+4 q^7-8 q^6+12 q^5-16 q^4+18 q^3-17 q^2+15 q-10+6 q^{-1} -3 q^{-2} + q^{-3} } |
| 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^{-2} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-14 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} +4 z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -2 z^2 a^{-6} +5 z^2-3 a^{-2} +3 a^{-4} - a^{-6} +2} |
| 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} +3 z^9 a^{-1} +7 z^9 a^{-3} +4 z^9 a^{-5} +8 z^8 a^{-2} +11 z^8 a^{-4} +7 z^8 a^{-6} +4 z^8+3 a z^7-2 z^7 a^{-1} -10 z^7 a^{-3} +2 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-26 z^6 a^{-2} -29 z^6 a^{-4} -10 z^6 a^{-6} +4 z^6 a^{-8} -10 z^6-9 a z^5-7 z^5 a^{-1} +z^5 a^{-3} -14 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+32 z^4 a^{-2} +31 z^4 a^{-4} +4 z^4 a^{-6} -6 z^4 a^{-8} +8 z^4+7 a z^3+7 z^3 a^{-1} +6 z^3 a^{-3} +12 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +2 a^2 z^2-19 z^2 a^{-2} -15 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} -5 z^2-a z-2 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2} |
| 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^8-q^6+2 q^4-q^2-1+3 q^{-2} -3 q^{-4} +4 q^{-6} - q^{-8} + q^{-12} -3 q^{-14} +3 q^{-16} - q^{-18} + q^{-22} - q^{-24} } |
| 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^{46}-2 q^{44}+5 q^{42}-9 q^{40}+10 q^{38}-10 q^{36}+2 q^{34}+15 q^{32}-33 q^{30}+54 q^{28}-64 q^{26}+54 q^{24}-22 q^{22}-37 q^{20}+107 q^{18}-165 q^{16}+190 q^{14}-155 q^{12}+60 q^{10}+81 q^8-220 q^6+320 q^4-326 q^2+227-48 q^{-2} -161 q^{-4} +313 q^{-6} -357 q^{-8} +279 q^{-10} -98 q^{-12} -98 q^{-14} +232 q^{-16} -254 q^{-18} +142 q^{-20} +45 q^{-22} -225 q^{-24} +306 q^{-26} -245 q^{-28} +52 q^{-30} +202 q^{-32} -413 q^{-34} +503 q^{-36} -419 q^{-38} +184 q^{-40} +124 q^{-42} -401 q^{-44} +544 q^{-46} -504 q^{-48} +309 q^{-50} -29 q^{-52} -222 q^{-54} +359 q^{-56} -340 q^{-58} +191 q^{-60} +17 q^{-62} -189 q^{-64} +243 q^{-66} -167 q^{-68} -6 q^{-70} +197 q^{-72} -313 q^{-74} +315 q^{-76} -192 q^{-78} -7 q^{-80} +203 q^{-82} -339 q^{-84} +362 q^{-86} -274 q^{-88} +118 q^{-90} +50 q^{-92} -179 q^{-94} +237 q^{-96} -219 q^{-98} +152 q^{-100} -60 q^{-102} -23 q^{-104} +72 q^{-106} -92 q^{-108} +77 q^{-110} -48 q^{-112} +22 q^{-114} +2 q^{-116} -13 q^{-118} +15 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} } |
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["K11a176"];
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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^3-13 t^2+23 t-27+23 t^{-1} -13 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^6-3 z^4+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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{ 111, 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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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^8+4 q^7-8 q^6+12 q^5-16 q^4+18 q^3-17 q^2+15 q-10+6 q^{-1} -3 q^{-2} + q^{-3} } |
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^{-2} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-14 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} +4 z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -2 z^2 a^{-6} +5 z^2-3 a^{-2} +3 a^{-4} - a^{-6} +2} |
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} +3 z^9 a^{-1} +7 z^9 a^{-3} +4 z^9 a^{-5} +8 z^8 a^{-2} +11 z^8 a^{-4} +7 z^8 a^{-6} +4 z^8+3 a z^7-2 z^7 a^{-1} -10 z^7 a^{-3} +2 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-26 z^6 a^{-2} -29 z^6 a^{-4} -10 z^6 a^{-6} +4 z^6 a^{-8} -10 z^6-9 a z^5-7 z^5 a^{-1} +z^5 a^{-3} -14 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+32 z^4 a^{-2} +31 z^4 a^{-4} +4 z^4 a^{-6} -6 z^4 a^{-8} +8 z^4+7 a z^3+7 z^3 a^{-1} +6 z^3 a^{-3} +12 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +2 a^2 z^2-19 z^2 a^{-2} -15 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} -5 z^2-a z-2 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2} |
"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["K11a176"];
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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^3-13 t^2+23 t-27+23 t^{-1} -13 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^8+4 q^7-8 q^6+12 q^5-16 q^4+18 q^3-17 q^2+15 q-10+6 q^{-1} -3 q^{-2} + q^{-3} } } |
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: | (0, 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=} 2 is the signature of K11a176. 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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