T(27,2)
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See other torus knots |
| Edit T(27,2) Quick Notes
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Edit T(27,2) Further Notes and Views
Knot presentations
| Planar diagram presentation | X21,49,22,48 X49,23,50,22 X23,51,24,50 X51,25,52,24 X25,53,26,52 X53,27,54,26 X27,1,28,54 X1,29,2,28 X29,3,30,2 X3,31,4,30 X31,5,32,4 X5,33,6,32 X33,7,34,6 X7,35,8,34 X35,9,36,8 X9,37,10,36 X37,11,38,10 X11,39,12,38 X39,13,40,12 X13,41,14,40 X41,15,42,14 X15,43,16,42 X43,17,44,16 X17,45,18,44 X45,19,46,18 X19,47,20,46 X47,21,48,20 |
| Gauss code | -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 1, -2, 3, -4, 5, -6, 7 |
| Dowker-Thistlethwaite code | 28 30 32 34 36 38 40 42 44 46 48 50 52 54 2 4 6 8 10 12 14 16 18 20 22 24 26 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ t^{13}-t^{12}+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} - t^{-12} + t^{-13} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{26}+25 z^{24}+276 z^{22}+1771 z^{20}+7315 z^{18}+20349 z^{16}+38760 z^{14}+50388 z^{12}+43758 z^{10}+24310 z^8+8008 z^6+1365 z^4+91 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 27, 26 } |
| Jones polynomial | [math]\displaystyle{ -q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-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^{13} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^{26}a^{-26}-26z^{24}a^{-26}-z^{24}a^{-28}+300z^{22}a^{-26}+24z^{22}a^{-28}-2024z^{20}a^{-26}-253z^{20}a^{-28}+8855z^{18}a^{-26}+1540z^{18}a^{-28}-26334z^{16}a^{-26}-5985z^{16}a^{-28}+54264z^{14}a^{-26}+15504z^{14}a^{-28}-77520z^{12}a^{-26}-27132z^{12}a^{-28}+75582z^{10}a^{-26}+31824z^{10}a^{-28}-48620z^8a^{-26}-24310z^8a^{-28}+19448z^6a^{-26}+11440z^6a^{-28}-4368z^4a^{-26}-3003z^4a^{-28}+455z^2a^{-26}+364z^2a^{-28}-14a^{-26}-13a^{-28} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{26}a^{-26}+z^{26}a^{-28}+z^{25}a^{-27}+z^{25}a^{-29}-26z^{24}a^{-26}-25z^{24}a^{-28}+z^{24}a^{-30}-24z^{23}a^{-27}-23z^{23}a^{-29}+z^{23}a^{-31}+300z^{22}a^{-26}+277z^{22}a^{-28}-22z^{22}a^{-30}+z^{22}a^{-32}+253z^{21}a^{-27}+231z^{21}a^{-29}-21z^{21}a^{-31}+z^{21}a^{-33}-2024z^{20}a^{-26}-1793z^{20}a^{-28}+210z^{20}a^{-30}-20z^{20}a^{-32}+z^{20}a^{-34}-1540z^{19}a^{-27}-1330z^{19}a^{-29}+190z^{19}a^{-31}-19z^{19}a^{-33}+z^{19}a^{-35}+8855z^{18}a^{-26}+7525z^{18}a^{-28}-1140z^{18}a^{-30}+171z^{18}a^{-32}-18z^{18}a^{-34}+z^{18}a^{-36}+5985z^{17}a^{-27}+4845z^{17}a^{-29}-969z^{17}a^{-31}+153z^{17}a^{-33}-17z^{17}a^{-35}+z^{17}a^{-37}-26334z^{16}a^{-26}-21489z^{16}a^{-28}+3876z^{16}a^{-30}-816z^{16}a^{-32}+136z^{16}a^{-34}-16z^{16}a^{-36}+z^{16}a^{-38}-15504z^{15}a^{-27}-11628z^{15}a^{-29}+3060z^{15}a^{-31}-680z^{15}a^{-33}+120z^{15}a^{-35}-15z^{15}a^{-37}+z^{15}a^{-39}+54264z^{14}a^{-26}+42636z^{14}a^{-28}-8568z^{14}a^{-30}+2380z^{14}a^{-32}-560z^{14}a^{-34}+105z^{14}a^{-36}-14z^{14}a^{-38}+z^{14}a^{-40}+27132z^{13}a^{-27}+18564z^{13}a^{-29}-6188z^{13}a^{-31}+1820z^{13}a^{-33}-455z^{13}a^{-35}+91z^{13}a^{-37}-13z^{13}a^{-39}+z^{13}a^{-41}-77520z^{12}a^{-26}-58956z^{12}a^{-28}+12376z^{12}a^{-30}-4368z^{12}a^{-32}+1365z^{12}a^{-34}-364z^{12}a^{-36}+78z^{12}a^{-38}-12z^{12}a^{-40}+z^{12}a^{-42}-31824z^{11}a^{-27}-19448z^{11}a^{-29}+8008z^{11}a^{-31}-3003z^{11}a^{-33}+1001z^{11}a^{-35}-286z^{11}a^{-37}+66z^{11}a^{-39}-11z^{11}a^{-41}+z^{11}a^{-43}+75582z^{10}a^{-26}+56134z^{10}a^{-28}-11440z^{10}a^{-30}+5005z^{10}a^{-32}-2002z^{10}a^{-34}+715z^{10}a^{-36}-220z^{10}a^{-38}+55z^{10}a^{-40}-10z^{10}a^{-42}+z^{10}a^{-44}+24310z^9a^{-27}+12870z^9a^{-29}-6435z^9a^{-31}+3003z^9a^{-33}-1287z^9a^{-35}+495z^9a^{-37}-165z^9a^{-39}+45z^9a^{-41}-9z^9a^{-43}+z^9a^{-45}-48620z^8a^{-26}-35750z^8a^{-28}+6435z^8a^{-30}-3432z^8a^{-32}+1716z^8a^{-34}-792z^8a^{-36}+330z^8a^{-38}-120z^8a^{-40}+36z^8a^{-42}-8z^8a^{-44}+z^8a^{-46}-11440z^7a^{-27}-5005z^7a^{-29}+3003z^7a^{-31}-1716z^7a^{-33}+924z^7a^{-35}-462z^7a^{-37}+210z^7a^{-39}-84z^7a^{-41}+28z^7a^{-43}-7z^7a^{-45}+z^7a^{-47}+19448z^6a^{-26}+14443z^6a^{-28}-2002z^6a^{-30}+1287z^6a^{-32}-792z^6a^{-34}+462z^6a^{-36}-252z^6a^{-38}+126z^6a^{-40}-56z^6a^{-42}+21z^6a^{-44}-6z^6a^{-46}+z^6a^{-48}+3003z^5a^{-27}+1001z^5a^{-29}-715z^5a^{-31}+495z^5a^{-33}-330z^5a^{-35}+210z^5a^{-37}-126z^5a^{-39}+70z^5a^{-41}-35z^5a^{-43}+15z^5a^{-45}-5z^5a^{-47}+z^5a^{-49}-4368z^4a^{-26}-3367z^4a^{-28}+286z^4a^{-30}-220z^4a^{-32}+165z^4a^{-34}-120z^4a^{-36}+84z^4a^{-38}-56z^4a^{-40}+35z^4a^{-42}-20z^4a^{-44}+10z^4a^{-46}-4z^4a^{-48}+z^4a^{-50}-364z^3a^{-27}-78z^3a^{-29}+66z^3a^{-31}-55z^3a^{-33}+45z^3a^{-35}-36z^3a^{-37}+28z^3a^{-39}-21z^3a^{-41}+15z^3a^{-43}-10z^3a^{-45}+6z^3a^{-47}-3z^3a^{-49}+z^3a^{-51}+455z^2a^{-26}+377z^2a^{-28}-12z^2a^{-30}+11z^2a^{-32}-10z^2a^{-34}+9z^2a^{-36}-8z^2a^{-38}+7z^2a^{-40}-6z^2a^{-42}+5z^2a^{-44}-4z^2a^{-46}+3z^2a^{-48}-2z^2a^{-50}+z^2a^{-52}+13za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-14a^{-26}-13a^{-28} }[/math] |
| The A2 invariant | Data:T(27,2)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(27,2)/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(27,2)"];
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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^{13}-t^{12}+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} - t^{-12} + t^{-13} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{26}+25 z^{24}+276 z^{22}+1771 z^{20}+7315 z^{18}+20349 z^{16}+38760 z^{14}+50388 z^{12}+43758 z^{10}+24310 z^8+8008 z^6+1365 z^4+91 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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{ 27, 26 } |
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^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-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^{13} }[/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^{26}a^{-26}-26z^{24}a^{-26}-z^{24}a^{-28}+300z^{22}a^{-26}+24z^{22}a^{-28}-2024z^{20}a^{-26}-253z^{20}a^{-28}+8855z^{18}a^{-26}+1540z^{18}a^{-28}-26334z^{16}a^{-26}-5985z^{16}a^{-28}+54264z^{14}a^{-26}+15504z^{14}a^{-28}-77520z^{12}a^{-26}-27132z^{12}a^{-28}+75582z^{10}a^{-26}+31824z^{10}a^{-28}-48620z^8a^{-26}-24310z^8a^{-28}+19448z^6a^{-26}+11440z^6a^{-28}-4368z^4a^{-26}-3003z^4a^{-28}+455z^2a^{-26}+364z^2a^{-28}-14a^{-26}-13a^{-28} }[/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^{26}a^{-26}+z^{26}a^{-28}+z^{25}a^{-27}+z^{25}a^{-29}-26z^{24}a^{-26}-25z^{24}a^{-28}+z^{24}a^{-30}-24z^{23}a^{-27}-23z^{23}a^{-29}+z^{23}a^{-31}+300z^{22}a^{-26}+277z^{22}a^{-28}-22z^{22}a^{-30}+z^{22}a^{-32}+253z^{21}a^{-27}+231z^{21}a^{-29}-21z^{21}a^{-31}+z^{21}a^{-33}-2024z^{20}a^{-26}-1793z^{20}a^{-28}+210z^{20}a^{-30}-20z^{20}a^{-32}+z^{20}a^{-34}-1540z^{19}a^{-27}-1330z^{19}a^{-29}+190z^{19}a^{-31}-19z^{19}a^{-33}+z^{19}a^{-35}+8855z^{18}a^{-26}+7525z^{18}a^{-28}-1140z^{18}a^{-30}+171z^{18}a^{-32}-18z^{18}a^{-34}+z^{18}a^{-36}+5985z^{17}a^{-27}+4845z^{17}a^{-29}-969z^{17}a^{-31}+153z^{17}a^{-33}-17z^{17}a^{-35}+z^{17}a^{-37}-26334z^{16}a^{-26}-21489z^{16}a^{-28}+3876z^{16}a^{-30}-816z^{16}a^{-32}+136z^{16}a^{-34}-16z^{16}a^{-36}+z^{16}a^{-38}-15504z^{15}a^{-27}-11628z^{15}a^{-29}+3060z^{15}a^{-31}-680z^{15}a^{-33}+120z^{15}a^{-35}-15z^{15}a^{-37}+z^{15}a^{-39}+54264z^{14}a^{-26}+42636z^{14}a^{-28}-8568z^{14}a^{-30}+2380z^{14}a^{-32}-560z^{14}a^{-34}+105z^{14}a^{-36}-14z^{14}a^{-38}+z^{14}a^{-40}+27132z^{13}a^{-27}+18564z^{13}a^{-29}-6188z^{13}a^{-31}+1820z^{13}a^{-33}-455z^{13}a^{-35}+91z^{13}a^{-37}-13z^{13}a^{-39}+z^{13}a^{-41}-77520z^{12}a^{-26}-58956z^{12}a^{-28}+12376z^{12}a^{-30}-4368z^{12}a^{-32}+1365z^{12}a^{-34}-364z^{12}a^{-36}+78z^{12}a^{-38}-12z^{12}a^{-40}+z^{12}a^{-42}-31824z^{11}a^{-27}-19448z^{11}a^{-29}+8008z^{11}a^{-31}-3003z^{11}a^{-33}+1001z^{11}a^{-35}-286z^{11}a^{-37}+66z^{11}a^{-39}-11z^{11}a^{-41}+z^{11}a^{-43}+75582z^{10}a^{-26}+56134z^{10}a^{-28}-11440z^{10}a^{-30}+5005z^{10}a^{-32}-2002z^{10}a^{-34}+715z^{10}a^{-36}-220z^{10}a^{-38}+55z^{10}a^{-40}-10z^{10}a^{-42}+z^{10}a^{-44}+24310z^9a^{-27}+12870z^9a^{-29}-6435z^9a^{-31}+3003z^9a^{-33}-1287z^9a^{-35}+495z^9a^{-37}-165z^9a^{-39}+45z^9a^{-41}-9z^9a^{-43}+z^9a^{-45}-48620z^8a^{-26}-35750z^8a^{-28}+6435z^8a^{-30}-3432z^8a^{-32}+1716z^8a^{-34}-792z^8a^{-36}+330z^8a^{-38}-120z^8a^{-40}+36z^8a^{-42}-8z^8a^{-44}+z^8a^{-46}-11440z^7a^{-27}-5005z^7a^{-29}+3003z^7a^{-31}-1716z^7a^{-33}+924z^7a^{-35}-462z^7a^{-37}+210z^7a^{-39}-84z^7a^{-41}+28z^7a^{-43}-7z^7a^{-45}+z^7a^{-47}+19448z^6a^{-26}+14443z^6a^{-28}-2002z^6a^{-30}+1287z^6a^{-32}-792z^6a^{-34}+462z^6a^{-36}-252z^6a^{-38}+126z^6a^{-40}-56z^6a^{-42}+21z^6a^{-44}-6z^6a^{-46}+z^6a^{-48}+3003z^5a^{-27}+1001z^5a^{-29}-715z^5a^{-31}+495z^5a^{-33}-330z^5a^{-35}+210z^5a^{-37}-126z^5a^{-39}+70z^5a^{-41}-35z^5a^{-43}+15z^5a^{-45}-5z^5a^{-47}+z^5a^{-49}-4368z^4a^{-26}-3367z^4a^{-28}+286z^4a^{-30}-220z^4a^{-32}+165z^4a^{-34}-120z^4a^{-36}+84z^4a^{-38}-56z^4a^{-40}+35z^4a^{-42}-20z^4a^{-44}+10z^4a^{-46}-4z^4a^{-48}+z^4a^{-50}-364z^3a^{-27}-78z^3a^{-29}+66z^3a^{-31}-55z^3a^{-33}+45z^3a^{-35}-36z^3a^{-37}+28z^3a^{-39}-21z^3a^{-41}+15z^3a^{-43}-10z^3a^{-45}+6z^3a^{-47}-3z^3a^{-49}+z^3a^{-51}+455z^2a^{-26}+377z^2a^{-28}-12z^2a^{-30}+11z^2a^{-32}-10z^2a^{-34}+9z^2a^{-36}-8z^2a^{-38}+7z^2a^{-40}-6z^2a^{-42}+5z^2a^{-44}-4z^2a^{-46}+3z^2a^{-48}-2z^2a^{-50}+z^2a^{-52}+13za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-14a^{-26}-13a^{-28} }[/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(27,2)"];
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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^{13}-t^{12}+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} - t^{-12} + t^{-13} }[/math], [math]\displaystyle{ -q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-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^{13} }[/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: | (91, 819) |
| 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]26 is the signature of T(27,2). 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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