K11a196
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 X20,7,21,8 X22,10,1,9 X18,11,19,12 X2,13,3,14 X8,16,9,15 X10,17,11,18 X6,19,7,20 X16,21,17,22 |
| Gauss code | 1, -7, 2, -1, 3, -10, 4, -8, 5, -9, 6, -2, 7, -3, 8, -11, 9, -6, 10, -4, 11, -5 |
| Dowker-Thistlethwaite code | 4 12 14 20 22 18 2 8 10 6 16 |
| 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 | |
| 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-z^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 \{7,t+1\}} |
| Determinant and Signature | { 147, -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^3-4 q^2+9 q-15+21 q^{-1} -23 q^{-2} +24 q^{-3} -21 q^{-4} +15 q^{-5} -9 q^{-6} +4 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^6-5 a^2 z^6+z^6-a^6 z^4+7 a^4 z^4-10 a^2 z^4+3 z^4-2 a^6 z^2+8 a^4 z^2-8 a^2 z^2+3 z^2-a^6+2 a^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 2 a^4 z^{10}+2 a^2 z^{10}+7 a^5 z^9+13 a^3 z^9+6 a z^9+10 a^6 z^8+18 a^4 z^8+15 a^2 z^8+7 z^8+8 a^7 z^7-16 a^3 z^7-4 a z^7+4 z^7 a^{-1} +4 a^8 z^6-15 a^6 z^6-47 a^4 z^6-44 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-12 a^7 z^5-18 a^5 z^5-9 a^3 z^5-13 a z^5-9 z^5 a^{-1} -5 a^8 z^4+10 a^6 z^4+41 a^4 z^4+38 a^2 z^4-2 z^4 a^{-2} +10 z^4-a^9 z^3+7 a^7 z^3+18 a^5 z^3+16 a^3 z^3+12 a z^3+6 z^3 a^{-1} +a^8 z^2-4 a^6 z^2-15 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^7 z-4 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^6+2 a^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}-2 q^{18}+4 q^{16}-4 q^{14}+q^{12}+q^{10}-2 q^8+6 q^6-4 q^4+4 q^2-1-2 q^{-2} +3 q^{-4} -2 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}-3 q^{126}+7 q^{124}-13 q^{122}+16 q^{120}-16 q^{118}+7 q^{116}+16 q^{114}-45 q^{112}+83 q^{110}-114 q^{108}+116 q^{106}-81 q^{104}-10 q^{102}+143 q^{100}-285 q^{98}+395 q^{96}-412 q^{94}+288 q^{92}-19 q^{90}-340 q^{88}+676 q^{86}-847 q^{84}+752 q^{82}-393 q^{80}-154 q^{78}+682 q^{76}-983 q^{74}+934 q^{72}-512 q^{70}-90 q^{68}+619 q^{66}-839 q^{64}+636 q^{62}-113 q^{60}-501 q^{58}+913 q^{56}-917 q^{54}+476 q^{52}+253 q^{50}-968 q^{48}+1385 q^{46}-1320 q^{44}+767 q^{42}+61 q^{40}-878 q^{38}+1394 q^{36}-1426 q^{34}+992 q^{32}-239 q^{30}-517 q^{28}+1002 q^{26}-1043 q^{24}+647 q^{22}-12 q^{20}-573 q^{18}+842 q^{16}-688 q^{14}+185 q^{12}+451 q^{10}-920 q^8+1023 q^6-718 q^4+123 q^2+493-913 q^{-2} +997 q^{-4} -744 q^{-6} +304 q^{-8} +158 q^{-10} -480 q^{-12} +587 q^{-14} -495 q^{-16} +292 q^{-18} -70 q^{-20} -97 q^{-22} +174 q^{-24} -178 q^{-26} +130 q^{-28} -66 q^{-30} +18 q^{-32} +14 q^{-34} -25 q^{-36} +22 q^{-38} -16 q^{-40} +8 q^{-42} -3 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["K11a196"];
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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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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-z^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 \{7,t+1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 147, -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^3-4 q^2+9 q-15+21 q^{-1} -23 q^{-2} +24 q^{-3} -21 q^{-4} +15 q^{-5} -9 q^{-6} +4 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^6-5 a^2 z^6+z^6-a^6 z^4+7 a^4 z^4-10 a^2 z^4+3 z^4-2 a^6 z^2+8 a^4 z^2-8 a^2 z^2+3 z^2-a^6+2 a^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 2 a^4 z^{10}+2 a^2 z^{10}+7 a^5 z^9+13 a^3 z^9+6 a z^9+10 a^6 z^8+18 a^4 z^8+15 a^2 z^8+7 z^8+8 a^7 z^7-16 a^3 z^7-4 a z^7+4 z^7 a^{-1} +4 a^8 z^6-15 a^6 z^6-47 a^4 z^6-44 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-12 a^7 z^5-18 a^5 z^5-9 a^3 z^5-13 a z^5-9 z^5 a^{-1} -5 a^8 z^4+10 a^6 z^4+41 a^4 z^4+38 a^2 z^4-2 z^4 a^{-2} +10 z^4-a^9 z^3+7 a^7 z^3+18 a^5 z^3+16 a^3 z^3+12 a z^3+6 z^3 a^{-1} +a^8 z^2-4 a^6 z^2-15 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^7 z-4 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^6+2 a^4+a^2+1} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a216, K11a286,}
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}} ): {K11a216,}
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["K11a196"];
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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^3-17 t^2+31 t-37+31 t^{-1} -17 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^3-4 q^2+9 q-15+21 q^{-1} -23 q^{-2} +24 q^{-3} -21 q^{-4} +15 q^{-5} -9 q^{-6} +4 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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{K11a216, K11a286,} |
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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{K11a216,} |
Vassiliev invariants
| V2 and V3: | (1, -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 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 K11a196. 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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