T(19,2)
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See other torus knots |
| Edit T(19,2) Quick Notes
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Edit T(19,2) Further Notes and Views
Knot presentations
| Planar diagram presentation | X13,33,14,32 X33,15,34,14 X15,35,16,34 X35,17,36,16 X17,37,18,36 X37,19,38,18 X19,1,20,38 X1,21,2,20 X21,3,22,2 X3,23,4,22 X23,5,24,4 X5,25,6,24 X25,7,26,6 X7,27,8,26 X27,9,28,8 X9,29,10,28 X29,11,30,10 X11,31,12,30 X31,13,32,12 |
| Gauss code | -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 1, -2, 3, -4, 5, -6, 7 |
| Dowker-Thistlethwaite code | 20 22 24 26 28 30 32 34 36 38 2 4 6 8 10 12 14 16 18 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ 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} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{18}+17 z^{16}+120 z^{14}+455 z^{12}+1001 z^{10}+1287 z^8+924 z^6+330 z^4+45 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 19, 18 } |
| Jones polynomial | [math]\displaystyle{ -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^{12}+q^{11}+q^9 }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^{18} a^{-18} +18 z^{16} a^{-18} -z^{16} a^{-20} +136 z^{14} a^{-18} -16 z^{14} a^{-20} +560 z^{12} a^{-18} -105 z^{12} a^{-20} +1365 z^{10} a^{-18} -364 z^{10} a^{-20} +2002 z^8 a^{-18} -715 z^8 a^{-20} +1716 z^6 a^{-18} -792 z^6 a^{-20} +792 z^4 a^{-18} -462 z^4 a^{-20} +165 z^2 a^{-18} -120 z^2 a^{-20} +10 a^{-18} -9 a^{-20} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{18}a^{-18}+z^{18}a^{-20}+z^{17}a^{-19}+z^{17}a^{-21}-18z^{16}a^{-18}-17z^{16}a^{-20}+z^{16}a^{-22}-16z^{15}a^{-19}-15z^{15}a^{-21}+z^{15}a^{-23}+136z^{14}a^{-18}+121z^{14}a^{-20}-14z^{14}a^{-22}+z^{14}a^{-24}+105z^{13}a^{-19}+91z^{13}a^{-21}-13z^{13}a^{-23}+z^{13}a^{-25}-560z^{12}a^{-18}-469z^{12}a^{-20}+78z^{12}a^{-22}-12z^{12}a^{-24}+z^{12}a^{-26}-364z^{11}a^{-19}-286z^{11}a^{-21}+66z^{11}a^{-23}-11z^{11}a^{-25}+z^{11}a^{-27}+1365z^{10}a^{-18}+1079z^{10}a^{-20}-220z^{10}a^{-22}+55z^{10}a^{-24}-10z^{10}a^{-26}+z^{10}a^{-28}+715z^9a^{-19}+495z^9a^{-21}-165z^9a^{-23}+45z^9a^{-25}-9z^9a^{-27}+z^9a^{-29}-2002z^8a^{-18}-1507z^8a^{-20}+330z^8a^{-22}-120z^8a^{-24}+36z^8a^{-26}-8z^8a^{-28}+z^8a^{-30}-792z^7a^{-19}-462z^7a^{-21}+210z^7a^{-23}-84z^7a^{-25}+28z^7a^{-27}-7z^7a^{-29}+z^7a^{-31}+1716z^6a^{-18}+1254z^6a^{-20}-252z^6a^{-22}+126z^6a^{-24}-56z^6a^{-26}+21z^6a^{-28}-6z^6a^{-30}+z^6a^{-32}+462z^5a^{-19}+210z^5a^{-21}-126z^5a^{-23}+70z^5a^{-25}-35z^5a^{-27}+15z^5a^{-29}-5z^5a^{-31}+z^5a^{-33}-792z^4a^{-18}-582z^4a^{-20}+84z^4a^{-22}-56z^4a^{-24}+35z^4a^{-26}-20z^4a^{-28}+10z^4a^{-30}-4z^4a^{-32}+z^4a^{-34}-120z^3a^{-19}-36z^3a^{-21}+28z^3a^{-23}-21z^3a^{-25}+15z^3a^{-27}-10z^3a^{-29}+6z^3a^{-31}-3z^3a^{-33}+z^3a^{-35}+165z^2a^{-18}+129z^2a^{-20}-8z^2a^{-22}+7z^2a^{-24}-6z^2a^{-26}+5z^2a^{-28}-4z^2a^{-30}+3z^2a^{-32}-2z^2a^{-34}+z^2a^{-36}+9za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-10a^{-18}-9a^{-20} }[/math] |
| The A2 invariant | Data:T(19,2)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(19,2)/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(19,2)"];
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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^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} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{18}+17 z^{16}+120 z^{14}+455 z^{12}+1001 z^{10}+1287 z^8+924 z^6+330 z^4+45 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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{ 19, 18 } |
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^{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^{12}+q^{11}+q^9 }[/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^{18} a^{-18} +18 z^{16} a^{-18} -z^{16} a^{-20} +136 z^{14} a^{-18} -16 z^{14} a^{-20} +560 z^{12} a^{-18} -105 z^{12} a^{-20} +1365 z^{10} a^{-18} -364 z^{10} a^{-20} +2002 z^8 a^{-18} -715 z^8 a^{-20} +1716 z^6 a^{-18} -792 z^6 a^{-20} +792 z^4 a^{-18} -462 z^4 a^{-20} +165 z^2 a^{-18} -120 z^2 a^{-20} +10 a^{-18} -9 a^{-20} }[/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^{18}a^{-18}+z^{18}a^{-20}+z^{17}a^{-19}+z^{17}a^{-21}-18z^{16}a^{-18}-17z^{16}a^{-20}+z^{16}a^{-22}-16z^{15}a^{-19}-15z^{15}a^{-21}+z^{15}a^{-23}+136z^{14}a^{-18}+121z^{14}a^{-20}-14z^{14}a^{-22}+z^{14}a^{-24}+105z^{13}a^{-19}+91z^{13}a^{-21}-13z^{13}a^{-23}+z^{13}a^{-25}-560z^{12}a^{-18}-469z^{12}a^{-20}+78z^{12}a^{-22}-12z^{12}a^{-24}+z^{12}a^{-26}-364z^{11}a^{-19}-286z^{11}a^{-21}+66z^{11}a^{-23}-11z^{11}a^{-25}+z^{11}a^{-27}+1365z^{10}a^{-18}+1079z^{10}a^{-20}-220z^{10}a^{-22}+55z^{10}a^{-24}-10z^{10}a^{-26}+z^{10}a^{-28}+715z^9a^{-19}+495z^9a^{-21}-165z^9a^{-23}+45z^9a^{-25}-9z^9a^{-27}+z^9a^{-29}-2002z^8a^{-18}-1507z^8a^{-20}+330z^8a^{-22}-120z^8a^{-24}+36z^8a^{-26}-8z^8a^{-28}+z^8a^{-30}-792z^7a^{-19}-462z^7a^{-21}+210z^7a^{-23}-84z^7a^{-25}+28z^7a^{-27}-7z^7a^{-29}+z^7a^{-31}+1716z^6a^{-18}+1254z^6a^{-20}-252z^6a^{-22}+126z^6a^{-24}-56z^6a^{-26}+21z^6a^{-28}-6z^6a^{-30}+z^6a^{-32}+462z^5a^{-19}+210z^5a^{-21}-126z^5a^{-23}+70z^5a^{-25}-35z^5a^{-27}+15z^5a^{-29}-5z^5a^{-31}+z^5a^{-33}-792z^4a^{-18}-582z^4a^{-20}+84z^4a^{-22}-56z^4a^{-24}+35z^4a^{-26}-20z^4a^{-28}+10z^4a^{-30}-4z^4a^{-32}+z^4a^{-34}-120z^3a^{-19}-36z^3a^{-21}+28z^3a^{-23}-21z^3a^{-25}+15z^3a^{-27}-10z^3a^{-29}+6z^3a^{-31}-3z^3a^{-33}+z^3a^{-35}+165z^2a^{-18}+129z^2a^{-20}-8z^2a^{-22}+7z^2a^{-24}-6z^2a^{-26}+5z^2a^{-28}-4z^2a^{-30}+3z^2a^{-32}-2z^2a^{-34}+z^2a^{-36}+9za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-10a^{-18}-9a^{-20} }[/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(19,2)"];
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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^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} }[/math], [math]\displaystyle{ -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^{12}+q^{11}+q^9 }[/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: | (45, 285) |
| 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]18 is the signature of T(19,2). 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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