T(25,2)
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See other torus knots |
| Edit T(25,2) Quick Notes
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Edit T(25,2) Further Notes and Views
Knot presentations
| Planar diagram presentation | X23,49,24,48 X49,25,50,24 X25,1,26,50 X1,27,2,26 X27,3,28,2 X3,29,4,28 X29,5,30,4 X5,31,6,30 X31,7,32,6 X7,33,8,32 X33,9,34,8 X9,35,10,34 X35,11,36,10 X11,37,12,36 X37,13,38,12 X13,39,14,38 X39,15,40,14 X15,41,16,40 X41,17,42,16 X17,43,18,42 X43,19,44,18 X19,45,20,44 X45,21,46,20 X21,47,22,46 X47,23,48,22 |
| Gauss code | -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -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, 1, -2, 3 |
| Dowker-Thistlethwaite code | 26 28 30 32 34 36 38 40 42 44 46 48 50 2 4 6 8 10 12 14 16 18 20 22 24 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ 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} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{24}+23 z^{22}+231 z^{20}+1330 z^{18}+4845 z^{16}+11628 z^{14}+18564 z^{12}+19448 z^{10}+12870 z^8+5005 z^6+1001 z^4+78 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 25, 24 } |
| Jones polynomial | [math]\displaystyle{ -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^{14}+q^{12} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^{24}a^{-24}-24z^{22}a^{-24}-z^{22}a^{-26}+253z^{20}a^{-24}+22z^{20}a^{-26}-1540z^{18}a^{-24}-210z^{18}a^{-26}+5985z^{16}a^{-24}+1140z^{16}a^{-26}-15504z^{14}a^{-24}-3876z^{14}a^{-26}+27132z^{12}a^{-24}+8568z^{12}a^{-26}-31824z^{10}a^{-24}-12376z^{10}a^{-26}+24310z^8a^{-24}+11440z^8a^{-26}-11440z^6a^{-24}-6435z^6a^{-26}+3003z^4a^{-24}+2002z^4a^{-26}-364z^2a^{-24}-286z^2a^{-26}+13a^{-24}+12a^{-26} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{24}a^{-24}+z^{24}a^{-26}+z^{23}a^{-25}+z^{23}a^{-27}-24z^{22}a^{-24}-23z^{22}a^{-26}+z^{22}a^{-28}-22z^{21}a^{-25}-21z^{21}a^{-27}+z^{21}a^{-29}+253z^{20}a^{-24}+232z^{20}a^{-26}-20z^{20}a^{-28}+z^{20}a^{-30}+210z^{19}a^{-25}+190z^{19}a^{-27}-19z^{19}a^{-29}+z^{19}a^{-31}-1540z^{18}a^{-24}-1350z^{18}a^{-26}+171z^{18}a^{-28}-18z^{18}a^{-30}+z^{18}a^{-32}-1140z^{17}a^{-25}-969z^{17}a^{-27}+153z^{17}a^{-29}-17z^{17}a^{-31}+z^{17}a^{-33}+5985z^{16}a^{-24}+5016z^{16}a^{-26}-816z^{16}a^{-28}+136z^{16}a^{-30}-16z^{16}a^{-32}+z^{16}a^{-34}+3876z^{15}a^{-25}+3060z^{15}a^{-27}-680z^{15}a^{-29}+120z^{15}a^{-31}-15z^{15}a^{-33}+z^{15}a^{-35}-15504z^{14}a^{-24}-12444z^{14}a^{-26}+2380z^{14}a^{-28}-560z^{14}a^{-30}+105z^{14}a^{-32}-14z^{14}a^{-34}+z^{14}a^{-36}-8568z^{13}a^{-25}-6188z^{13}a^{-27}+1820z^{13}a^{-29}-455z^{13}a^{-31}+91z^{13}a^{-33}-13z^{13}a^{-35}+z^{13}a^{-37}+27132z^{12}a^{-24}+20944z^{12}a^{-26}-4368z^{12}a^{-28}+1365z^{12}a^{-30}-364z^{12}a^{-32}+78z^{12}a^{-34}-12z^{12}a^{-36}+z^{12}a^{-38}+12376z^{11}a^{-25}+8008z^{11}a^{-27}-3003z^{11}a^{-29}+1001z^{11}a^{-31}-286z^{11}a^{-33}+66z^{11}a^{-35}-11z^{11}a^{-37}+z^{11}a^{-39}-31824z^{10}a^{-24}-23816z^{10}a^{-26}+5005z^{10}a^{-28}-2002z^{10}a^{-30}+715z^{10}a^{-32}-220z^{10}a^{-34}+55z^{10}a^{-36}-10z^{10}a^{-38}+z^{10}a^{-40}-11440z^9a^{-25}-6435z^9a^{-27}+3003z^9a^{-29}-1287z^9a^{-31}+495z^9a^{-33}-165z^9a^{-35}+45z^9a^{-37}-9z^9a^{-39}+z^9a^{-41}+24310z^8a^{-24}+17875z^8a^{-26}-3432z^8a^{-28}+1716z^8a^{-30}-792z^8a^{-32}+330z^8a^{-34}-120z^8a^{-36}+36z^8a^{-38}-8z^8a^{-40}+z^8a^{-42}+6435z^7a^{-25}+3003z^7a^{-27}-1716z^7a^{-29}+924z^7a^{-31}-462z^7a^{-33}+210z^7a^{-35}-84z^7a^{-37}+28z^7a^{-39}-7z^7a^{-41}+z^7a^{-43}-11440z^6a^{-24}-8437z^6a^{-26}+1287z^6a^{-28}-792z^6a^{-30}+462z^6a^{-32}-252z^6a^{-34}+126z^6a^{-36}-56z^6a^{-38}+21z^6a^{-40}-6z^6a^{-42}+z^6a^{-44}-2002z^5a^{-25}-715z^5a^{-27}+495z^5a^{-29}-330z^5a^{-31}+210z^5a^{-33}-126z^5a^{-35}+70z^5a^{-37}-35z^5a^{-39}+15z^5a^{-41}-5z^5a^{-43}+z^5a^{-45}+3003z^4a^{-24}+2288z^4a^{-26}-220z^4a^{-28}+165z^4a^{-30}-120z^4a^{-32}+84z^4a^{-34}-56z^4a^{-36}+35z^4a^{-38}-20z^4a^{-40}+10z^4a^{-42}-4z^4a^{-44}+z^4a^{-46}+286z^3a^{-25}+66z^3a^{-27}-55z^3a^{-29}+45z^3a^{-31}-36z^3a^{-33}+28z^3a^{-35}-21z^3a^{-37}+15z^3a^{-39}-10z^3a^{-41}+6z^3a^{-43}-3z^3a^{-45}+z^3a^{-47}-364z^2a^{-24}-298z^2a^{-26}+11z^2a^{-28}-10z^2a^{-30}+9z^2a^{-32}-8z^2a^{-34}+7z^2a^{-36}-6z^2a^{-38}+5z^2a^{-40}-4z^2a^{-42}+3z^2a^{-44}-2z^2a^{-46}+z^2a^{-48}-12za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}+13a^{-24}+12a^{-26} }[/math] |
| The A2 invariant | Data:T(25,2)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(25,2)/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(25,2)"];
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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^{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} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{24}+23 z^{22}+231 z^{20}+1330 z^{18}+4845 z^{16}+11628 z^{14}+18564 z^{12}+19448 z^{10}+12870 z^8+5005 z^6+1001 z^4+78 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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{ 25, 24 } |
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^{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^{14}+q^{12} }[/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^{24}a^{-24}-24z^{22}a^{-24}-z^{22}a^{-26}+253z^{20}a^{-24}+22z^{20}a^{-26}-1540z^{18}a^{-24}-210z^{18}a^{-26}+5985z^{16}a^{-24}+1140z^{16}a^{-26}-15504z^{14}a^{-24}-3876z^{14}a^{-26}+27132z^{12}a^{-24}+8568z^{12}a^{-26}-31824z^{10}a^{-24}-12376z^{10}a^{-26}+24310z^8a^{-24}+11440z^8a^{-26}-11440z^6a^{-24}-6435z^6a^{-26}+3003z^4a^{-24}+2002z^4a^{-26}-364z^2a^{-24}-286z^2a^{-26}+13a^{-24}+12a^{-26} }[/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^{24}a^{-24}+z^{24}a^{-26}+z^{23}a^{-25}+z^{23}a^{-27}-24z^{22}a^{-24}-23z^{22}a^{-26}+z^{22}a^{-28}-22z^{21}a^{-25}-21z^{21}a^{-27}+z^{21}a^{-29}+253z^{20}a^{-24}+232z^{20}a^{-26}-20z^{20}a^{-28}+z^{20}a^{-30}+210z^{19}a^{-25}+190z^{19}a^{-27}-19z^{19}a^{-29}+z^{19}a^{-31}-1540z^{18}a^{-24}-1350z^{18}a^{-26}+171z^{18}a^{-28}-18z^{18}a^{-30}+z^{18}a^{-32}-1140z^{17}a^{-25}-969z^{17}a^{-27}+153z^{17}a^{-29}-17z^{17}a^{-31}+z^{17}a^{-33}+5985z^{16}a^{-24}+5016z^{16}a^{-26}-816z^{16}a^{-28}+136z^{16}a^{-30}-16z^{16}a^{-32}+z^{16}a^{-34}+3876z^{15}a^{-25}+3060z^{15}a^{-27}-680z^{15}a^{-29}+120z^{15}a^{-31}-15z^{15}a^{-33}+z^{15}a^{-35}-15504z^{14}a^{-24}-12444z^{14}a^{-26}+2380z^{14}a^{-28}-560z^{14}a^{-30}+105z^{14}a^{-32}-14z^{14}a^{-34}+z^{14}a^{-36}-8568z^{13}a^{-25}-6188z^{13}a^{-27}+1820z^{13}a^{-29}-455z^{13}a^{-31}+91z^{13}a^{-33}-13z^{13}a^{-35}+z^{13}a^{-37}+27132z^{12}a^{-24}+20944z^{12}a^{-26}-4368z^{12}a^{-28}+1365z^{12}a^{-30}-364z^{12}a^{-32}+78z^{12}a^{-34}-12z^{12}a^{-36}+z^{12}a^{-38}+12376z^{11}a^{-25}+8008z^{11}a^{-27}-3003z^{11}a^{-29}+1001z^{11}a^{-31}-286z^{11}a^{-33}+66z^{11}a^{-35}-11z^{11}a^{-37}+z^{11}a^{-39}-31824z^{10}a^{-24}-23816z^{10}a^{-26}+5005z^{10}a^{-28}-2002z^{10}a^{-30}+715z^{10}a^{-32}-220z^{10}a^{-34}+55z^{10}a^{-36}-10z^{10}a^{-38}+z^{10}a^{-40}-11440z^9a^{-25}-6435z^9a^{-27}+3003z^9a^{-29}-1287z^9a^{-31}+495z^9a^{-33}-165z^9a^{-35}+45z^9a^{-37}-9z^9a^{-39}+z^9a^{-41}+24310z^8a^{-24}+17875z^8a^{-26}-3432z^8a^{-28}+1716z^8a^{-30}-792z^8a^{-32}+330z^8a^{-34}-120z^8a^{-36}+36z^8a^{-38}-8z^8a^{-40}+z^8a^{-42}+6435z^7a^{-25}+3003z^7a^{-27}-1716z^7a^{-29}+924z^7a^{-31}-462z^7a^{-33}+210z^7a^{-35}-84z^7a^{-37}+28z^7a^{-39}-7z^7a^{-41}+z^7a^{-43}-11440z^6a^{-24}-8437z^6a^{-26}+1287z^6a^{-28}-792z^6a^{-30}+462z^6a^{-32}-252z^6a^{-34}+126z^6a^{-36}-56z^6a^{-38}+21z^6a^{-40}-6z^6a^{-42}+z^6a^{-44}-2002z^5a^{-25}-715z^5a^{-27}+495z^5a^{-29}-330z^5a^{-31}+210z^5a^{-33}-126z^5a^{-35}+70z^5a^{-37}-35z^5a^{-39}+15z^5a^{-41}-5z^5a^{-43}+z^5a^{-45}+3003z^4a^{-24}+2288z^4a^{-26}-220z^4a^{-28}+165z^4a^{-30}-120z^4a^{-32}+84z^4a^{-34}-56z^4a^{-36}+35z^4a^{-38}-20z^4a^{-40}+10z^4a^{-42}-4z^4a^{-44}+z^4a^{-46}+286z^3a^{-25}+66z^3a^{-27}-55z^3a^{-29}+45z^3a^{-31}-36z^3a^{-33}+28z^3a^{-35}-21z^3a^{-37}+15z^3a^{-39}-10z^3a^{-41}+6z^3a^{-43}-3z^3a^{-45}+z^3a^{-47}-364z^2a^{-24}-298z^2a^{-26}+11z^2a^{-28}-10z^2a^{-30}+9z^2a^{-32}-8z^2a^{-34}+7z^2a^{-36}-6z^2a^{-38}+5z^2a^{-40}-4z^2a^{-42}+3z^2a^{-44}-2z^2a^{-46}+z^2a^{-48}-12za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}+13a^{-24}+12a^{-26} }[/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(25,2)"];
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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^{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} }[/math], [math]\displaystyle{ -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^{14}+q^{12} }[/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: | (78, 650) |
| 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]24 is the signature of T(25,2). 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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