K11a15
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 X12,5,13,6 X2837 X14,9,15,10 X18,12,19,11 X6,13,7,14 X20,15,21,16 X22,17,1,18 X10,20,11,19 X16,21,17,22 |
| Gauss code | 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 12 2 14 18 6 20 22 10 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 | 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+22 t-25+22 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-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 | { 107, -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+3 q^3-6 q^2+11 q-14+17 q^{-1} -17 q^{-2} +15 q^{-3} -12 q^{-4} +7 q^{-5} -3 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^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +9 z^4+6 a^4 z^2-18 a^2 z^2-3 z^2 a^{-2} +14 z^2+3 a^4-8 a^2-2 a^{-2} +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 a^2 z^{10}+z^{10}+5 a^3 z^9+8 a z^9+3 z^9 a^{-1} +9 a^4 z^8+15 a^2 z^8+3 z^8 a^{-2} +9 z^8+9 a^5 z^7+a^3 z^7-15 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-16 a^4 z^6-52 a^2 z^6-12 z^6 a^{-2} -42 z^6+3 a^7 z^5-15 a^5 z^5-25 a^3 z^5-9 a z^5-6 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+13 a^4 z^4+58 a^2 z^4+16 z^4 a^{-2} +54 z^4-2 a^7 z^3+14 a^5 z^3+29 a^3 z^3+23 a z^3+15 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-7 a^4 z^2-33 a^2 z^2-9 z^2 a^{-2} -31 z^2-5 a^5 z-11 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +3 a^4+8 a^2+2 a^{-2} +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^{20}-q^{18}+3 q^{16}-q^{14}-q^{12}+q^{10}-5 q^8+2 q^6-3 q^4+2 q^2+3+4 q^{-4} - q^{-6} - q^{-12} } |
| 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^{114}-2 q^{112}+4 q^{110}-6 q^{108}+6 q^{106}-5 q^{104}+9 q^{100}-19 q^{98}+29 q^{96}-37 q^{94}+33 q^{92}-20 q^{90}-5 q^{88}+44 q^{86}-78 q^{84}+105 q^{82}-114 q^{80}+87 q^{78}-31 q^{76}-56 q^{74}+155 q^{72}-222 q^{70}+240 q^{68}-180 q^{66}+52 q^{64}+114 q^{62}-253 q^{60}+318 q^{58}-267 q^{56}+116 q^{54}+78 q^{52}-231 q^{50}+278 q^{48}-189 q^{46}+10 q^{44}+180 q^{42}-295 q^{40}+259 q^{38}-97 q^{36}-144 q^{34}+352 q^{32}-446 q^{30}+366 q^{28}-152 q^{26}-136 q^{24}+378 q^{22}-501 q^{20}+449 q^{18}-254 q^{16}-18 q^{14}+262 q^{12}-389 q^{10}+367 q^8-197 q^6-28 q^4+220 q^2-291+223 q^{-2} -32 q^{-4} -173 q^{-6} +318 q^{-8} -320 q^{-10} +188 q^{-12} +29 q^{-14} -235 q^{-16} +359 q^{-18} -346 q^{-20} +222 q^{-22} -39 q^{-24} -136 q^{-26} +236 q^{-28} -246 q^{-30} +181 q^{-32} -80 q^{-34} -17 q^{-36} +76 q^{-38} -97 q^{-40} +81 q^{-42} -48 q^{-44} +17 q^{-46} +5 q^{-48} -16 q^{-50} +14 q^{-52} -11 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60} } |
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["K11a15"];
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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+22 t-25+22 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-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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{ 107, -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+3 q^3-6 q^2+11 q-14+17 q^{-1} -17 q^{-2} +15 q^{-3} -12 q^{-4} +7 q^{-5} -3 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^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +9 z^4+6 a^4 z^2-18 a^2 z^2-3 z^2 a^{-2} +14 z^2+3 a^4-8 a^2-2 a^{-2} +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 a^2 z^{10}+z^{10}+5 a^3 z^9+8 a z^9+3 z^9 a^{-1} +9 a^4 z^8+15 a^2 z^8+3 z^8 a^{-2} +9 z^8+9 a^5 z^7+a^3 z^7-15 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-16 a^4 z^6-52 a^2 z^6-12 z^6 a^{-2} -42 z^6+3 a^7 z^5-15 a^5 z^5-25 a^3 z^5-9 a z^5-6 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+13 a^4 z^4+58 a^2 z^4+16 z^4 a^{-2} +54 z^4-2 a^7 z^3+14 a^5 z^3+29 a^3 z^3+23 a z^3+15 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-7 a^4 z^2-33 a^2 z^2-9 z^2 a^{-2} -31 z^2-5 a^5 z-11 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +3 a^4+8 a^2+2 a^{-2} +8} |
"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["K11a15"];
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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+22 t-25+22 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^4+3 q^3-6 q^2+11 q-14+17 q^{-1} -17 q^{-2} +15 q^{-3} -12 q^{-4} +7 q^{-5} -3 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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{} |
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: | (-1, 3) |
| 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 K11a15. 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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