K11a305
From Knot Atlas
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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 X10,5,11,6 X22,8,1,7 X16,9,17,10 X18,12,19,11 X20,13,21,14 X8,15,9,16 X4,18,5,17 X2,19,3,20 X14,21,15,22 |
| Gauss code | 1, -10, 2, -9, 3, -1, 4, -8, 5, -3, 6, -2, 7, -11, 8, -5, 9, -6, 10, -7, 11, -4 |
| Dowker-Thistlethwaite code | 6 12 10 22 16 18 20 8 4 2 14 |
| 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+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 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^8-2 z^6+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 | { 135, -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^4+4 q^3-9 q^2+14 q-18+22 q^{-1} -21 q^{-2} +19 q^{-3} -14 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} } |
| 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 -a^2 z^8+a^4 z^6-5 a^2 z^6+2 z^6+3 a^4 z^4-9 a^2 z^4-z^4 a^{-2} +7 z^4+2 a^4 z^2-5 a^2 z^2-2 z^2 a^{-2} +7 z^2-a^4+a^2- a^{-2} +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 3 a^2 z^{10}+3 z^{10}+10 a^3 z^9+16 a z^9+6 z^9 a^{-1} +14 a^4 z^8+15 a^2 z^8+4 z^8 a^{-2} +5 z^8+12 a^5 z^7-14 a^3 z^7-44 a z^7-17 z^7 a^{-1} +z^7 a^{-3} +8 a^6 z^6-27 a^4 z^6-64 a^2 z^6-13 z^6 a^{-2} -42 z^6+4 a^7 z^5-15 a^5 z^5-5 a^3 z^5+26 a z^5+9 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+20 a^4 z^4+64 a^2 z^4+13 z^4 a^{-2} +50 z^4-2 a^7 z^3+4 a^5 z^3+9 a^3 z^3+3 a z^3+3 z^3 a^{-1} +3 z^3 a^{-3} -6 a^4 z^2-19 a^2 z^2-6 z^2 a^{-2} -19 z^2+a^5 z-2 a z-2 z a^{-1} -z a^{-3} -a^4-a^2+ a^{-2} +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^{20}-2 q^{18}+2 q^{16}-3 q^{14}-2 q^{12}+3 q^{10}-3 q^8+6 q^6-q^4+2 q^2+2-3 q^{-2} +3 q^{-4} -2 q^{-6} + q^{-10} - q^{-12} } |
| The G2 invariant | Data:K11a305/QuantumInvariant/G2/1,0 |
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["K11a305"];
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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+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 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^8-2 z^6+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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{ 135, -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^4+4 q^3-9 q^2+14 q-18+22 q^{-1} -21 q^{-2} +19 q^{-3} -14 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} } |
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 -a^2 z^8+a^4 z^6-5 a^2 z^6+2 z^6+3 a^4 z^4-9 a^2 z^4-z^4 a^{-2} +7 z^4+2 a^4 z^2-5 a^2 z^2-2 z^2 a^{-2} +7 z^2-a^4+a^2- a^{-2} +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 3 a^2 z^{10}+3 z^{10}+10 a^3 z^9+16 a z^9+6 z^9 a^{-1} +14 a^4 z^8+15 a^2 z^8+4 z^8 a^{-2} +5 z^8+12 a^5 z^7-14 a^3 z^7-44 a z^7-17 z^7 a^{-1} +z^7 a^{-3} +8 a^6 z^6-27 a^4 z^6-64 a^2 z^6-13 z^6 a^{-2} -42 z^6+4 a^7 z^5-15 a^5 z^5-5 a^3 z^5+26 a z^5+9 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+20 a^4 z^4+64 a^2 z^4+13 z^4 a^{-2} +50 z^4-2 a^7 z^3+4 a^5 z^3+9 a^3 z^3+3 a z^3+3 z^3 a^{-1} +3 z^3 a^{-3} -6 a^4 z^2-19 a^2 z^2-6 z^2 a^{-2} -19 z^2+a^5 z-2 a z-2 z a^{-1} -z a^{-3} -a^4-a^2+ a^{-2} +2} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a157, K11a264,}
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["K11a305"];
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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+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 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^4+4 q^3-9 q^2+14 q-18+22 q^{-1} -21 q^{-2} +19 q^{-3} -14 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} } } |
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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{K11a157, K11a264,} |
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 or , where -2 is the signature of K11a305. 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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