K11a221
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 X18,5,19,6 X20,7,21,8 X14,10,15,9 X16,12,17,11 X2,13,3,14 X10,16,11,15 X22,17,1,18 X6,19,7,20 X8,21,9,22 |
| Gauss code | 1, -7, 2, -1, 3, -10, 4, -11, 5, -8, 6, -2, 7, -5, 8, -6, 9, -3, 10, -4, 11, -9 |
| Dowker-Thistlethwaite code | 4 12 18 20 14 16 2 10 22 6 8 |
| 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^3-10 t^2+15 t-17+15 t^{-1} -10 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+4 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 | { 79, -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^3-3 q^2+5 q-8+11 q^{-1} -12 q^{-2} +13 q^{-3} -10 q^{-4} +8 q^{-5} -5 q^{-6} +2 q^{-7} - q^{-8} } |
| 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+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+10 a^4 z^4-13 a^2 z^4+4 z^4-4 a^6 z^2+16 a^4 z^2-12 a^2 z^2+4 z^2-4 a^6+8 a^4-4 a^2+1} |
| 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 a^4 z^{10}+a^2 z^{10}+2 a^5 z^9+5 a^3 z^9+3 a z^9+3 a^6 z^8+2 a^4 z^8+3 a^2 z^8+4 z^8+3 a^7 z^7-a^5 z^7-14 a^3 z^7-7 a z^7+3 z^7 a^{-1} +2 a^8 z^6-6 a^6 z^6-11 a^4 z^6-17 a^2 z^6+z^6 a^{-2} -13 z^6+a^9 z^5-6 a^7 z^5-4 a^5 z^5+14 a^3 z^5+a z^5-10 z^5 a^{-1} -4 a^8 z^4+9 a^6 z^4+24 a^4 z^4+25 a^2 z^4-3 z^4 a^{-2} +11 z^4-3 a^9 z^3+4 a^7 z^3+11 a^5 z^3+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +a^8 z^2-10 a^6 z^2-21 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2+2 a^9 z-2 a^7 z-7 a^5 z-5 a^3 z-3 a z-z a^{-1} +4 a^6+8 a^4+4 a^2+1} |
| 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^{24}-q^{22}-q^{20}-2 q^{18}+2 q^{16}+3 q^{12}+3 q^{10}+3 q^6-3 q^4+q^2-1- q^{-2} + q^{-4} - q^{-6} + q^{-8} } |
| 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^{128}-q^{126}+3 q^{124}-4 q^{122}+4 q^{120}-3 q^{118}-q^{116}+8 q^{114}-15 q^{112}+21 q^{110}-22 q^{108}+14 q^{106}-2 q^{104}-18 q^{102}+39 q^{100}-53 q^{98}+53 q^{96}-41 q^{94}+9 q^{92}+23 q^{90}-57 q^{88}+78 q^{86}-84 q^{84}+66 q^{82}-32 q^{80}-20 q^{78}+63 q^{76}-91 q^{74}+91 q^{72}-62 q^{70}+11 q^{68}+39 q^{66}-70 q^{64}+68 q^{62}-27 q^{60}-25 q^{58}+76 q^{56}-83 q^{54}+50 q^{52}+20 q^{50}-89 q^{48}+143 q^{46}-137 q^{44}+89 q^{42}-q^{40}-84 q^{38}+151 q^{36}-164 q^{34}+128 q^{32}-53 q^{30}-32 q^{28}+98 q^{26}-124 q^{24}+104 q^{22}-46 q^{20}-23 q^{18}+69 q^{16}-85 q^{14}+50 q^{12}+10 q^{10}-72 q^8+103 q^6-93 q^4+38 q^2+35-100 q^{-2} +128 q^{-4} -112 q^{-6} +63 q^{-8} +2 q^{-10} -61 q^{-12} +93 q^{-14} -91 q^{-16} +68 q^{-18} -28 q^{-20} -8 q^{-22} +31 q^{-24} -40 q^{-26} +36 q^{-28} -24 q^{-30} +12 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46} } |
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["K11a221"];
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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^3-10 t^2+15 t-17+15 t^{-1} -10 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+4 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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{ 79, -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^3-3 q^2+5 q-8+11 q^{-1} -12 q^{-2} +13 q^{-3} -10 q^{-4} +8 q^{-5} -5 q^{-6} +2 q^{-7} - q^{-8} } |
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+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+10 a^4 z^4-13 a^2 z^4+4 z^4-4 a^6 z^2+16 a^4 z^2-12 a^2 z^2+4 z^2-4 a^6+8 a^4-4 a^2+1} |
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 a^4 z^{10}+a^2 z^{10}+2 a^5 z^9+5 a^3 z^9+3 a z^9+3 a^6 z^8+2 a^4 z^8+3 a^2 z^8+4 z^8+3 a^7 z^7-a^5 z^7-14 a^3 z^7-7 a z^7+3 z^7 a^{-1} +2 a^8 z^6-6 a^6 z^6-11 a^4 z^6-17 a^2 z^6+z^6 a^{-2} -13 z^6+a^9 z^5-6 a^7 z^5-4 a^5 z^5+14 a^3 z^5+a z^5-10 z^5 a^{-1} -4 a^8 z^4+9 a^6 z^4+24 a^4 z^4+25 a^2 z^4-3 z^4 a^{-2} +11 z^4-3 a^9 z^3+4 a^7 z^3+11 a^5 z^3+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +a^8 z^2-10 a^6 z^2-21 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2+2 a^9 z-2 a^7 z-7 a^5 z-5 a^3 z-3 a z-z a^{-1} +4 a^6+8 a^4+4 a^2+1} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a259,}
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["K11a221"];
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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^3-10 t^2+15 t-17+15 t^{-1} -10 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^3-3 q^2+5 q-8+11 q^{-1} -12 q^{-2} +13 q^{-3} -10 q^{-4} +8 q^{-5} -5 q^{-6} +2 q^{-7} - q^{-8} } } |
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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{K11a259,} |
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: | (4, -8) |
| 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 K11a221. 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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