K11a182
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 X16,7,17,8 X18,9,19,10 X20,11,21,12 X2,13,3,14 X22,16,1,15 X8,17,9,18 X10,19,11,20 X6,21,7,22 |
| Gauss code | 1, -7, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -8 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 20 2 22 8 10 6 |
| 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+11 t^2-13 t+13-13 t^{-1} +11 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+z^4+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 | { 73, -4 } |
| 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^{-1} +7 q^{-2} -9 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} } |
| 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^4 a^8+3 z^2 a^8+a^8-2 z^6 a^6-9 z^4 a^6-11 z^2 a^6-4 a^6+z^8 a^4+6 z^6 a^4+13 z^4 a^4+13 z^2 a^4+4 a^4-z^6 a^2-4 z^4 a^2-3 z^2 a^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^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-4 z^4 a^{10}+4 z^7 a^9-3 z^5 a^9-2 z^3 a^9+2 z a^9+4 z^8 a^8-7 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+3 z^9 a^7-6 z^7 a^7+5 z^5 a^7-2 z^3 a^7+z^{10} a^6+4 z^8 a^6-24 z^6 a^6+36 z^4 a^6-21 z^2 a^6+4 a^6+6 z^9 a^5-23 z^7 a^5+27 z^5 a^5-11 z^3 a^5+z^{10} a^4+3 z^8 a^4-27 z^6 a^4+42 z^4 a^4-23 z^2 a^4+4 a^4+3 z^9 a^3-12 z^7 a^3+12 z^5 a^3-4 z^3 a^3+z a^3+3 z^8 a^2-14 z^6 a^2+18 z^4 a^2-7 z^2 a^2+z^7 a-4 z^5 a+3 z^3 a} |
| 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^{30}-q^{26}-2 q^{22}+q^{20}-q^{18}+q^{14}-2 q^{12}+3 q^{10}+2 q^6+q^4+1- q^{-2} } |
| 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^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+12 q^{148}-18 q^{146}+22 q^{144}-19 q^{142}+8 q^{140}+7 q^{138}-23 q^{136}+37 q^{134}-40 q^{132}+34 q^{130}-20 q^{128}-2 q^{126}+25 q^{124}-39 q^{122}+48 q^{120}-41 q^{118}+29 q^{116}-11 q^{114}-11 q^{112}+28 q^{110}-40 q^{108}+40 q^{106}-30 q^{104}+11 q^{102}+8 q^{100}-24 q^{98}+27 q^{96}-18 q^{94}-2 q^{92}+21 q^{90}-36 q^{88}+25 q^{86}+q^{84}-35 q^{82}+64 q^{80}-71 q^{78}+51 q^{76}-12 q^{74}-39 q^{72}+78 q^{70}-98 q^{68}+84 q^{66}-43 q^{64}-8 q^{62}+55 q^{60}-76 q^{58}+75 q^{56}-48 q^{54}+8 q^{52}+27 q^{50}-49 q^{48}+44 q^{46}-13 q^{44}-19 q^{42}+51 q^{40}-54 q^{38}+32 q^{36}+8 q^{34}-49 q^{32}+79 q^{30}-81 q^{28}+56 q^{26}-9 q^{24}-36 q^{22}+70 q^{20}-76 q^{18}+61 q^{16}-29 q^{14}-3 q^{12}+26 q^{10}-37 q^8+35 q^6-23 q^4+11 q^2+1-7 q^{-2} +6 q^{-4} -7 q^{-6} +4 q^{-8} -2 q^{-10} + q^{-12} } |
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["K11a182"];
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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+11 t^2-13 t+13-13 t^{-1} +11 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+z^4+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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{ 73, -4 } |
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^{-1} +7 q^{-2} -9 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} } |
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^4 a^8+3 z^2 a^8+a^8-2 z^6 a^6-9 z^4 a^6-11 z^2 a^6-4 a^6+z^8 a^4+6 z^6 a^4+13 z^4 a^4+13 z^2 a^4+4 a^4-z^6 a^2-4 z^4 a^2-3 z^2 a^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^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-4 z^4 a^{10}+4 z^7 a^9-3 z^5 a^9-2 z^3 a^9+2 z a^9+4 z^8 a^8-7 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+3 z^9 a^7-6 z^7 a^7+5 z^5 a^7-2 z^3 a^7+z^{10} a^6+4 z^8 a^6-24 z^6 a^6+36 z^4 a^6-21 z^2 a^6+4 a^6+6 z^9 a^5-23 z^7 a^5+27 z^5 a^5-11 z^3 a^5+z^{10} a^4+3 z^8 a^4-27 z^6 a^4+42 z^4 a^4-23 z^2 a^4+4 a^4+3 z^9 a^3-12 z^7 a^3+12 z^5 a^3-4 z^3 a^3+z a^3+3 z^8 a^2-14 z^6 a^2+18 z^4 a^2-7 z^2 a^2+z^7 a-4 z^5 a+3 z^3 a} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, ): {}
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["K11a182"];
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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+11 t^2-13 t+13-13 t^{-1} +11 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+3-4 q^{-1} +7 q^{-2} -9 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} } } |
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: | (2, -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 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=} -4 is the signature of K11a182. 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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