T(23,2)
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Edit T(23,2) Further Notes and Views
Knot presentations
| Planar diagram presentation | X9,33,10,32 X33,11,34,10 X11,35,12,34 X35,13,36,12 X13,37,14,36 X37,15,38,14 X15,39,16,38 X39,17,40,16 X17,41,18,40 X41,19,42,18 X19,43,20,42 X43,21,44,20 X21,45,22,44 X45,23,46,22 X23,1,24,46 X1,25,2,24 X25,3,26,2 X3,27,4,26 X27,5,28,4 X5,29,6,28 X29,7,30,6 X7,31,8,30 X31,9,32,8 |
| Gauss code | -16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15 |
| Dowker-Thistlethwaite code | 24 26 28 30 32 34 36 38 40 42 44 46 2 4 6 8 10 12 14 16 18 20 22 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ t^{11}-t^{10}+t^9-t^8+t^7-t^6+t^5-t^4+t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5} - t^{-6} + t^{-7} - t^{-8} + t^{-9} - t^{-10} + t^{-11} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{22}+21 z^{20}+190 z^{18}+969 z^{16}+3060 z^{14}+6188 z^{12}+8008 z^{10}+6435 z^8+3003 z^6+715 z^4+66 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 23, 22 } |
| Jones polynomial | [math]\displaystyle{ -q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^{22} a^{-22} +22 z^{20} a^{-22} -z^{20} a^{-24} +210 z^{18} a^{-22} -20 z^{18} a^{-24} +1140 z^{16} a^{-22} -171 z^{16} a^{-24} +3876 z^{14} a^{-22} -816 z^{14} a^{-24} +8568 z^{12} a^{-22} -2380 z^{12} a^{-24} +12376 z^{10} a^{-22} -4368 z^{10} a^{-24} +11440 z^8 a^{-22} -5005 z^8 a^{-24} +6435 z^6 a^{-22} -3432 z^6 a^{-24} +2002 z^4 a^{-22} -1287 z^4 a^{-24} +286 z^2 a^{-22} -220 z^2 a^{-24} +12 a^{-22} -11 a^{-24} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{22}a^{-22}+z^{22}a^{-24}+z^{21}a^{-23}+z^{21}a^{-25}-22z^{20}a^{-22}-21z^{20}a^{-24}+z^{20}a^{-26}-20z^{19}a^{-23}-19z^{19}a^{-25}+z^{19}a^{-27}+210z^{18}a^{-22}+191z^{18}a^{-24}-18z^{18}a^{-26}+z^{18}a^{-28}+171z^{17}a^{-23}+153z^{17}a^{-25}-17z^{17}a^{-27}+z^{17}a^{-29}-1140z^{16}a^{-22}-987z^{16}a^{-24}+136z^{16}a^{-26}-16z^{16}a^{-28}+z^{16}a^{-30}-816z^{15}a^{-23}-680z^{15}a^{-25}+120z^{15}a^{-27}-15z^{15}a^{-29}+z^{15}a^{-31}+3876z^{14}a^{-22}+3196z^{14}a^{-24}-560z^{14}a^{-26}+105z^{14}a^{-28}-14z^{14}a^{-30}+z^{14}a^{-32}+2380z^{13}a^{-23}+1820z^{13}a^{-25}-455z^{13}a^{-27}+91z^{13}a^{-29}-13z^{13}a^{-31}+z^{13}a^{-33}-8568z^{12}a^{-22}-6748z^{12}a^{-24}+1365z^{12}a^{-26}-364z^{12}a^{-28}+78z^{12}a^{-30}-12z^{12}a^{-32}+z^{12}a^{-34}-4368z^{11}a^{-23}-3003z^{11}a^{-25}+1001z^{11}a^{-27}-286z^{11}a^{-29}+66z^{11}a^{-31}-11z^{11}a^{-33}+z^{11}a^{-35}+12376z^{10}a^{-22}+9373z^{10}a^{-24}-2002z^{10}a^{-26}+715z^{10}a^{-28}-220z^{10}a^{-30}+55z^{10}a^{-32}-10z^{10}a^{-34}+z^{10}a^{-36}+5005z^9a^{-23}+3003z^9a^{-25}-1287z^9a^{-27}+495z^9a^{-29}-165z^9a^{-31}+45z^9a^{-33}-9z^9a^{-35}+z^9a^{-37}-11440z^8a^{-22}-8437z^8a^{-24}+1716z^8a^{-26}-792z^8a^{-28}+330z^8a^{-30}-120z^8a^{-32}+36z^8a^{-34}-8z^8a^{-36}+z^8a^{-38}-3432z^7a^{-23}-1716z^7a^{-25}+924z^7a^{-27}-462z^7a^{-29}+210z^7a^{-31}-84z^7a^{-33}+28z^7a^{-35}-7z^7a^{-37}+z^7a^{-39}+6435z^6a^{-22}+4719z^6a^{-24}-792z^6a^{-26}+462z^6a^{-28}-252z^6a^{-30}+126z^6a^{-32}-56z^6a^{-34}+21z^6a^{-36}-6z^6a^{-38}+z^6a^{-40}+1287z^5a^{-23}+495z^5a^{-25}-330z^5a^{-27}+210z^5a^{-29}-126z^5a^{-31}+70z^5a^{-33}-35z^5a^{-35}+15z^5a^{-37}-5z^5a^{-39}+z^5a^{-41}-2002z^4a^{-22}-1507z^4a^{-24}+165z^4a^{-26}-120z^4a^{-28}+84z^4a^{-30}-56z^4a^{-32}+35z^4a^{-34}-20z^4a^{-36}+10z^4a^{-38}-4z^4a^{-40}+z^4a^{-42}-220z^3a^{-23}-55z^3a^{-25}+45z^3a^{-27}-36z^3a^{-29}+28z^3a^{-31}-21z^3a^{-33}+15z^3a^{-35}-10z^3a^{-37}+6z^3a^{-39}-3z^3a^{-41}+z^3a^{-43}+286z^2a^{-22}+231z^2a^{-24}-10z^2a^{-26}+9z^2a^{-28}-8z^2a^{-30}+7z^2a^{-32}-6z^2a^{-34}+5z^2a^{-36}-4z^2a^{-38}+3z^2a^{-40}-2z^2a^{-42}+z^2a^{-44}+11za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-12a^{-22}-11a^{-24} }[/math] |
| The A2 invariant | Data:T(23,2)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(23,2)/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["T(23,2)"];
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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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[math]\displaystyle{ t^{11}-t^{10}+t^9-t^8+t^7-t^6+t^5-t^4+t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5} - t^{-6} + t^{-7} - t^{-8} + t^{-9} - t^{-10} + t^{-11} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{22}+21 z^{20}+190 z^{18}+969 z^{16}+3060 z^{14}+6188 z^{12}+8008 z^{10}+6435 z^8+3003 z^6+715 z^4+66 z^2+1 }[/math] |
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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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 23, 22 } |
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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[math]\displaystyle{ -q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11} }[/math] |
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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[math]\displaystyle{ z^{22} a^{-22} +22 z^{20} a^{-22} -z^{20} a^{-24} +210 z^{18} a^{-22} -20 z^{18} a^{-24} +1140 z^{16} a^{-22} -171 z^{16} a^{-24} +3876 z^{14} a^{-22} -816 z^{14} a^{-24} +8568 z^{12} a^{-22} -2380 z^{12} a^{-24} +12376 z^{10} a^{-22} -4368 z^{10} a^{-24} +11440 z^8 a^{-22} -5005 z^8 a^{-24} +6435 z^6 a^{-22} -3432 z^6 a^{-24} +2002 z^4 a^{-22} -1287 z^4 a^{-24} +286 z^2 a^{-22} -220 z^2 a^{-24} +12 a^{-22} -11 a^{-24} }[/math] |
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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[math]\displaystyle{ z^{22}a^{-22}+z^{22}a^{-24}+z^{21}a^{-23}+z^{21}a^{-25}-22z^{20}a^{-22}-21z^{20}a^{-24}+z^{20}a^{-26}-20z^{19}a^{-23}-19z^{19}a^{-25}+z^{19}a^{-27}+210z^{18}a^{-22}+191z^{18}a^{-24}-18z^{18}a^{-26}+z^{18}a^{-28}+171z^{17}a^{-23}+153z^{17}a^{-25}-17z^{17}a^{-27}+z^{17}a^{-29}-1140z^{16}a^{-22}-987z^{16}a^{-24}+136z^{16}a^{-26}-16z^{16}a^{-28}+z^{16}a^{-30}-816z^{15}a^{-23}-680z^{15}a^{-25}+120z^{15}a^{-27}-15z^{15}a^{-29}+z^{15}a^{-31}+3876z^{14}a^{-22}+3196z^{14}a^{-24}-560z^{14}a^{-26}+105z^{14}a^{-28}-14z^{14}a^{-30}+z^{14}a^{-32}+2380z^{13}a^{-23}+1820z^{13}a^{-25}-455z^{13}a^{-27}+91z^{13}a^{-29}-13z^{13}a^{-31}+z^{13}a^{-33}-8568z^{12}a^{-22}-6748z^{12}a^{-24}+1365z^{12}a^{-26}-364z^{12}a^{-28}+78z^{12}a^{-30}-12z^{12}a^{-32}+z^{12}a^{-34}-4368z^{11}a^{-23}-3003z^{11}a^{-25}+1001z^{11}a^{-27}-286z^{11}a^{-29}+66z^{11}a^{-31}-11z^{11}a^{-33}+z^{11}a^{-35}+12376z^{10}a^{-22}+9373z^{10}a^{-24}-2002z^{10}a^{-26}+715z^{10}a^{-28}-220z^{10}a^{-30}+55z^{10}a^{-32}-10z^{10}a^{-34}+z^{10}a^{-36}+5005z^9a^{-23}+3003z^9a^{-25}-1287z^9a^{-27}+495z^9a^{-29}-165z^9a^{-31}+45z^9a^{-33}-9z^9a^{-35}+z^9a^{-37}-11440z^8a^{-22}-8437z^8a^{-24}+1716z^8a^{-26}-792z^8a^{-28}+330z^8a^{-30}-120z^8a^{-32}+36z^8a^{-34}-8z^8a^{-36}+z^8a^{-38}-3432z^7a^{-23}-1716z^7a^{-25}+924z^7a^{-27}-462z^7a^{-29}+210z^7a^{-31}-84z^7a^{-33}+28z^7a^{-35}-7z^7a^{-37}+z^7a^{-39}+6435z^6a^{-22}+4719z^6a^{-24}-792z^6a^{-26}+462z^6a^{-28}-252z^6a^{-30}+126z^6a^{-32}-56z^6a^{-34}+21z^6a^{-36}-6z^6a^{-38}+z^6a^{-40}+1287z^5a^{-23}+495z^5a^{-25}-330z^5a^{-27}+210z^5a^{-29}-126z^5a^{-31}+70z^5a^{-33}-35z^5a^{-35}+15z^5a^{-37}-5z^5a^{-39}+z^5a^{-41}-2002z^4a^{-22}-1507z^4a^{-24}+165z^4a^{-26}-120z^4a^{-28}+84z^4a^{-30}-56z^4a^{-32}+35z^4a^{-34}-20z^4a^{-36}+10z^4a^{-38}-4z^4a^{-40}+z^4a^{-42}-220z^3a^{-23}-55z^3a^{-25}+45z^3a^{-27}-36z^3a^{-29}+28z^3a^{-31}-21z^3a^{-33}+15z^3a^{-35}-10z^3a^{-37}+6z^3a^{-39}-3z^3a^{-41}+z^3a^{-43}+286z^2a^{-22}+231z^2a^{-24}-10z^2a^{-26}+9z^2a^{-28}-8z^2a^{-30}+7z^2a^{-32}-6z^2a^{-34}+5z^2a^{-36}-4z^2a^{-38}+3z^2a^{-40}-2z^2a^{-42}+z^2a^{-44}+11za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-12a^{-22}-11a^{-24} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}
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["T(23,2)"];
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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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{ [math]\displaystyle{ t^{11}-t^{10}+t^9-t^8+t^7-t^6+t^5-t^4+t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5} - t^{-6} + t^{-7} - t^{-8} + t^{-9} - t^{-10} + t^{-11} }[/math], [math]\displaystyle{ -q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11} }[/math] } |
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: | (66, 506) |
| 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]22 is the signature of T(23,2). 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 Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top. |
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