T(31,2)
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Edit T(31,2) Further Notes and Views
Knot presentations
| Planar diagram presentation | X17,49,18,48 X49,19,50,18 X19,51,20,50 X51,21,52,20 X21,53,22,52 X53,23,54,22 X23,55,24,54 X55,25,56,24 X25,57,26,56 X57,27,58,26 X27,59,28,58 X59,29,60,28 X29,61,30,60 X61,31,62,30 X31,1,32,62 X1,33,2,32 X33,3,34,2 X3,35,4,34 X35,5,36,4 X5,37,6,36 X37,7,38,6 X7,39,8,38 X39,9,40,8 X9,41,10,40 X41,11,42,10 X11,43,12,42 X43,13,44,12 X13,45,14,44 X45,15,46,14 X15,47,16,46 X47,17,48,16 |
| Gauss code | -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -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, 28, -29, 30, -31, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15 |
| Dowker-Thistlethwaite code | 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ t^{15}-t^{14}+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} - t^{-14} + t^{-15} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{30}+29 z^{28}+378 z^{26}+2925 z^{24}+14950 z^{22}+53130 z^{20}+134596 z^{18}+245157 z^{16}+319770 z^{14}+293930 z^{12}+184756 z^{10}+75582 z^8+18564 z^6+2380 z^4+120 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 31, 30 } |
| Jones polynomial | [math]\displaystyle{ -q^{46}+q^{45}-q^{44}+q^{43}-q^{42}+q^{41}-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^{15} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^{30}a^{-30}-30z^{28}a^{-30}-z^{28}a^{-32}+406z^{26}a^{-30}+28z^{26}a^{-32}-3276z^{24}a^{-30}-351z^{24}a^{-32}+17550z^{22}a^{-30}+2600z^{22}a^{-32}-65780z^{20}a^{-30}-12650z^{20}a^{-32}+177100z^{18}a^{-30}+42504z^{18}a^{-32}-346104z^{16}a^{-30}-100947z^{16}a^{-32}+490314z^{14}a^{-30}+170544z^{14}a^{-32}-497420z^{12}a^{-30}-203490z^{12}a^{-32}+352716z^{10}a^{-30}+167960z^{10}a^{-32}-167960z^8a^{-30}-92378z^8a^{-32}+50388z^6a^{-30}+31824z^6a^{-32}-8568z^4a^{-30}-6188z^4a^{-32}+680z^2a^{-30}+560z^2a^{-32}-16a^{-30}-15a^{-32} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{30}a^{-30}+z^{30}a^{-32}+z^{29}a^{-31}+z^{29}a^{-33}-30z^{28}a^{-30}-29z^{28}a^{-32}+z^{28}a^{-34}-28z^{27}a^{-31}-27z^{27}a^{-33}+z^{27}a^{-35}+406z^{26}a^{-30}+379z^{26}a^{-32}-26z^{26}a^{-34}+z^{26}a^{-36}+351z^{25}a^{-31}+325z^{25}a^{-33}-25z^{25}a^{-35}+z^{25}a^{-37}-3276z^{24}a^{-30}-2951z^{24}a^{-32}+300z^{24}a^{-34}-24z^{24}a^{-36}+z^{24}a^{-38}-2600z^{23}a^{-31}-2300z^{23}a^{-33}+276z^{23}a^{-35}-23z^{23}a^{-37}+z^{23}a^{-39}+17550z^{22}a^{-30}+15250z^{22}a^{-32}-2024z^{22}a^{-34}+253z^{22}a^{-36}-22z^{22}a^{-38}+z^{22}a^{-40}+12650z^{21}a^{-31}+10626z^{21}a^{-33}-1771z^{21}a^{-35}+231z^{21}a^{-37}-21z^{21}a^{-39}+z^{21}a^{-41}-65780z^{20}a^{-30}-55154z^{20}a^{-32}+8855z^{20}a^{-34}-1540z^{20}a^{-36}+210z^{20}a^{-38}-20z^{20}a^{-40}+z^{20}a^{-42}-42504z^{19}a^{-31}-33649z^{19}a^{-33}+7315z^{19}a^{-35}-1330z^{19}a^{-37}+190z^{19}a^{-39}-19z^{19}a^{-41}+z^{19}a^{-43}+177100z^{18}a^{-30}+143451z^{18}a^{-32}-26334z^{18}a^{-34}+5985z^{18}a^{-36}-1140z^{18}a^{-38}+171z^{18}a^{-40}-18z^{18}a^{-42}+z^{18}a^{-44}+100947z^{17}a^{-31}+74613z^{17}a^{-33}-20349z^{17}a^{-35}+4845z^{17}a^{-37}-969z^{17}a^{-39}+153z^{17}a^{-41}-17z^{17}a^{-43}+z^{17}a^{-45}-346104z^{16}a^{-30}-271491z^{16}a^{-32}+54264z^{16}a^{-34}-15504z^{16}a^{-36}+3876z^{16}a^{-38}-816z^{16}a^{-40}+136z^{16}a^{-42}-16z^{16}a^{-44}+z^{16}a^{-46}-170544z^{15}a^{-31}-116280z^{15}a^{-33}+38760z^{15}a^{-35}-11628z^{15}a^{-37}+3060z^{15}a^{-39}-680z^{15}a^{-41}+120z^{15}a^{-43}-15z^{15}a^{-45}+z^{15}a^{-47}+490314z^{14}a^{-30}+374034z^{14}a^{-32}-77520z^{14}a^{-34}+27132z^{14}a^{-36}-8568z^{14}a^{-38}+2380z^{14}a^{-40}-560z^{14}a^{-42}+105z^{14}a^{-44}-14z^{14}a^{-46}+z^{14}a^{-48}+203490z^{13}a^{-31}+125970z^{13}a^{-33}-50388z^{13}a^{-35}+18564z^{13}a^{-37}-6188z^{13}a^{-39}+1820z^{13}a^{-41}-455z^{13}a^{-43}+91z^{13}a^{-45}-13z^{13}a^{-47}+z^{13}a^{-49}-497420z^{12}a^{-30}-371450z^{12}a^{-32}+75582z^{12}a^{-34}-31824z^{12}a^{-36}+12376z^{12}a^{-38}-4368z^{12}a^{-40}+1365z^{12}a^{-42}-364z^{12}a^{-44}+78z^{12}a^{-46}-12z^{12}a^{-48}+z^{12}a^{-50}-167960z^{11}a^{-31}-92378z^{11}a^{-33}+43758z^{11}a^{-35}-19448z^{11}a^{-37}+8008z^{11}a^{-39}-3003z^{11}a^{-41}+1001z^{11}a^{-43}-286z^{11}a^{-45}+66z^{11}a^{-47}-11z^{11}a^{-49}+z^{11}a^{-51}+352716z^{10}a^{-30}+260338z^{10}a^{-32}-48620z^{10}a^{-34}+24310z^{10}a^{-36}-11440z^{10}a^{-38}+5005z^{10}a^{-40}-2002z^{10}a^{-42}+715z^{10}a^{-44}-220z^{10}a^{-46}+55z^{10}a^{-48}-10z^{10}a^{-50}+z^{10}a^{-52}+92378z^9a^{-31}+43758z^9a^{-33}-24310z^9a^{-35}+12870z^9a^{-37}-6435z^9a^{-39}+3003z^9a^{-41}-1287z^9a^{-43}+495z^9a^{-45}-165z^9a^{-47}+45z^9a^{-49}-9z^9a^{-51}+z^9a^{-53}-167960z^8a^{-30}-124202z^8a^{-32}+19448z^8a^{-34}-11440z^8a^{-36}+6435z^8a^{-38}-3432z^8a^{-40}+1716z^8a^{-42}-792z^8a^{-44}+330z^8a^{-46}-120z^8a^{-48}+36z^8a^{-50}-8z^8a^{-52}+z^8a^{-54}-31824z^7a^{-31}-12376z^7a^{-33}+8008z^7a^{-35}-5005z^7a^{-37}+3003z^7a^{-39}-1716z^7a^{-41}+924z^7a^{-43}-462z^7a^{-45}+210z^7a^{-47}-84z^7a^{-49}+28z^7a^{-51}-7z^7a^{-53}+z^7a^{-55}+50388z^6a^{-30}+38012z^6a^{-32}-4368z^6a^{-34}+3003z^6a^{-36}-2002z^6a^{-38}+1287z^6a^{-40}-792z^6a^{-42}+462z^6a^{-44}-252z^6a^{-46}+126z^6a^{-48}-56z^6a^{-50}+21z^6a^{-52}-6z^6a^{-54}+z^6a^{-56}+6188z^5a^{-31}+1820z^5a^{-33}-1365z^5a^{-35}+1001z^5a^{-37}-715z^5a^{-39}+495z^5a^{-41}-330z^5a^{-43}+210z^5a^{-45}-126z^5a^{-47}+70z^5a^{-49}-35z^5a^{-51}+15z^5a^{-53}-5z^5a^{-55}+z^5a^{-57}-8568z^4a^{-30}-6748z^4a^{-32}+455z^4a^{-34}-364z^4a^{-36}+286z^4a^{-38}-220z^4a^{-40}+165z^4a^{-42}-120z^4a^{-44}+84z^4a^{-46}-56z^4a^{-48}+35z^4a^{-50}-20z^4a^{-52}+10z^4a^{-54}-4z^4a^{-56}+z^4a^{-58}-560z^3a^{-31}-105z^3a^{-33}+91z^3a^{-35}-78z^3a^{-37}+66z^3a^{-39}-55z^3a^{-41}+45z^3a^{-43}-36z^3a^{-45}+28z^3a^{-47}-21z^3a^{-49}+15z^3a^{-51}-10z^3a^{-53}+6z^3a^{-55}-3z^3a^{-57}+z^3a^{-59}+680z^2a^{-30}+575z^2a^{-32}-14z^2a^{-34}+13z^2a^{-36}-12z^2a^{-38}+11z^2a^{-40}-10z^2a^{-42}+9z^2a^{-44}-8z^2a^{-46}+7z^2a^{-48}-6z^2a^{-50}+5z^2a^{-52}-4z^2a^{-54}+3z^2a^{-56}-2z^2a^{-58}+z^2a^{-60}+15za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-za^{-55}+za^{-57}-za^{-59}+za^{-61}-16a^{-30}-15a^{-32} }[/math] |
| The A2 invariant | Data:T(31,2)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(31,2)/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(31,2)"];
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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^{15}-t^{14}+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} - t^{-14} + t^{-15} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{30}+29 z^{28}+378 z^{26}+2925 z^{24}+14950 z^{22}+53130 z^{20}+134596 z^{18}+245157 z^{16}+319770 z^{14}+293930 z^{12}+184756 z^{10}+75582 z^8+18564 z^6+2380 z^4+120 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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{ 31, 30 } |
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^{46}+q^{45}-q^{44}+q^{43}-q^{42}+q^{41}-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^{15} }[/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^{30}a^{-30}-30z^{28}a^{-30}-z^{28}a^{-32}+406z^{26}a^{-30}+28z^{26}a^{-32}-3276z^{24}a^{-30}-351z^{24}a^{-32}+17550z^{22}a^{-30}+2600z^{22}a^{-32}-65780z^{20}a^{-30}-12650z^{20}a^{-32}+177100z^{18}a^{-30}+42504z^{18}a^{-32}-346104z^{16}a^{-30}-100947z^{16}a^{-32}+490314z^{14}a^{-30}+170544z^{14}a^{-32}-497420z^{12}a^{-30}-203490z^{12}a^{-32}+352716z^{10}a^{-30}+167960z^{10}a^{-32}-167960z^8a^{-30}-92378z^8a^{-32}+50388z^6a^{-30}+31824z^6a^{-32}-8568z^4a^{-30}-6188z^4a^{-32}+680z^2a^{-30}+560z^2a^{-32}-16a^{-30}-15a^{-32} }[/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^{30}a^{-30}+z^{30}a^{-32}+z^{29}a^{-31}+z^{29}a^{-33}-30z^{28}a^{-30}-29z^{28}a^{-32}+z^{28}a^{-34}-28z^{27}a^{-31}-27z^{27}a^{-33}+z^{27}a^{-35}+406z^{26}a^{-30}+379z^{26}a^{-32}-26z^{26}a^{-34}+z^{26}a^{-36}+351z^{25}a^{-31}+325z^{25}a^{-33}-25z^{25}a^{-35}+z^{25}a^{-37}-3276z^{24}a^{-30}-2951z^{24}a^{-32}+300z^{24}a^{-34}-24z^{24}a^{-36}+z^{24}a^{-38}-2600z^{23}a^{-31}-2300z^{23}a^{-33}+276z^{23}a^{-35}-23z^{23}a^{-37}+z^{23}a^{-39}+17550z^{22}a^{-30}+15250z^{22}a^{-32}-2024z^{22}a^{-34}+253z^{22}a^{-36}-22z^{22}a^{-38}+z^{22}a^{-40}+12650z^{21}a^{-31}+10626z^{21}a^{-33}-1771z^{21}a^{-35}+231z^{21}a^{-37}-21z^{21}a^{-39}+z^{21}a^{-41}-65780z^{20}a^{-30}-55154z^{20}a^{-32}+8855z^{20}a^{-34}-1540z^{20}a^{-36}+210z^{20}a^{-38}-20z^{20}a^{-40}+z^{20}a^{-42}-42504z^{19}a^{-31}-33649z^{19}a^{-33}+7315z^{19}a^{-35}-1330z^{19}a^{-37}+190z^{19}a^{-39}-19z^{19}a^{-41}+z^{19}a^{-43}+177100z^{18}a^{-30}+143451z^{18}a^{-32}-26334z^{18}a^{-34}+5985z^{18}a^{-36}-1140z^{18}a^{-38}+171z^{18}a^{-40}-18z^{18}a^{-42}+z^{18}a^{-44}+100947z^{17}a^{-31}+74613z^{17}a^{-33}-20349z^{17}a^{-35}+4845z^{17}a^{-37}-969z^{17}a^{-39}+153z^{17}a^{-41}-17z^{17}a^{-43}+z^{17}a^{-45}-346104z^{16}a^{-30}-271491z^{16}a^{-32}+54264z^{16}a^{-34}-15504z^{16}a^{-36}+3876z^{16}a^{-38}-816z^{16}a^{-40}+136z^{16}a^{-42}-16z^{16}a^{-44}+z^{16}a^{-46}-170544z^{15}a^{-31}-116280z^{15}a^{-33}+38760z^{15}a^{-35}-11628z^{15}a^{-37}+3060z^{15}a^{-39}-680z^{15}a^{-41}+120z^{15}a^{-43}-15z^{15}a^{-45}+z^{15}a^{-47}+490314z^{14}a^{-30}+374034z^{14}a^{-32}-77520z^{14}a^{-34}+27132z^{14}a^{-36}-8568z^{14}a^{-38}+2380z^{14}a^{-40}-560z^{14}a^{-42}+105z^{14}a^{-44}-14z^{14}a^{-46}+z^{14}a^{-48}+203490z^{13}a^{-31}+125970z^{13}a^{-33}-50388z^{13}a^{-35}+18564z^{13}a^{-37}-6188z^{13}a^{-39}+1820z^{13}a^{-41}-455z^{13}a^{-43}+91z^{13}a^{-45}-13z^{13}a^{-47}+z^{13}a^{-49}-497420z^{12}a^{-30}-371450z^{12}a^{-32}+75582z^{12}a^{-34}-31824z^{12}a^{-36}+12376z^{12}a^{-38}-4368z^{12}a^{-40}+1365z^{12}a^{-42}-364z^{12}a^{-44}+78z^{12}a^{-46}-12z^{12}a^{-48}+z^{12}a^{-50}-167960z^{11}a^{-31}-92378z^{11}a^{-33}+43758z^{11}a^{-35}-19448z^{11}a^{-37}+8008z^{11}a^{-39}-3003z^{11}a^{-41}+1001z^{11}a^{-43}-286z^{11}a^{-45}+66z^{11}a^{-47}-11z^{11}a^{-49}+z^{11}a^{-51}+352716z^{10}a^{-30}+260338z^{10}a^{-32}-48620z^{10}a^{-34}+24310z^{10}a^{-36}-11440z^{10}a^{-38}+5005z^{10}a^{-40}-2002z^{10}a^{-42}+715z^{10}a^{-44}-220z^{10}a^{-46}+55z^{10}a^{-48}-10z^{10}a^{-50}+z^{10}a^{-52}+92378z^9a^{-31}+43758z^9a^{-33}-24310z^9a^{-35}+12870z^9a^{-37}-6435z^9a^{-39}+3003z^9a^{-41}-1287z^9a^{-43}+495z^9a^{-45}-165z^9a^{-47}+45z^9a^{-49}-9z^9a^{-51}+z^9a^{-53}-167960z^8a^{-30}-124202z^8a^{-32}+19448z^8a^{-34}-11440z^8a^{-36}+6435z^8a^{-38}-3432z^8a^{-40}+1716z^8a^{-42}-792z^8a^{-44}+330z^8a^{-46}-120z^8a^{-48}+36z^8a^{-50}-8z^8a^{-52}+z^8a^{-54}-31824z^7a^{-31}-12376z^7a^{-33}+8008z^7a^{-35}-5005z^7a^{-37}+3003z^7a^{-39}-1716z^7a^{-41}+924z^7a^{-43}-462z^7a^{-45}+210z^7a^{-47}-84z^7a^{-49}+28z^7a^{-51}-7z^7a^{-53}+z^7a^{-55}+50388z^6a^{-30}+38012z^6a^{-32}-4368z^6a^{-34}+3003z^6a^{-36}-2002z^6a^{-38}+1287z^6a^{-40}-792z^6a^{-42}+462z^6a^{-44}-252z^6a^{-46}+126z^6a^{-48}-56z^6a^{-50}+21z^6a^{-52}-6z^6a^{-54}+z^6a^{-56}+6188z^5a^{-31}+1820z^5a^{-33}-1365z^5a^{-35}+1001z^5a^{-37}-715z^5a^{-39}+495z^5a^{-41}-330z^5a^{-43}+210z^5a^{-45}-126z^5a^{-47}+70z^5a^{-49}-35z^5a^{-51}+15z^5a^{-53}-5z^5a^{-55}+z^5a^{-57}-8568z^4a^{-30}-6748z^4a^{-32}+455z^4a^{-34}-364z^4a^{-36}+286z^4a^{-38}-220z^4a^{-40}+165z^4a^{-42}-120z^4a^{-44}+84z^4a^{-46}-56z^4a^{-48}+35z^4a^{-50}-20z^4a^{-52}+10z^4a^{-54}-4z^4a^{-56}+z^4a^{-58}-560z^3a^{-31}-105z^3a^{-33}+91z^3a^{-35}-78z^3a^{-37}+66z^3a^{-39}-55z^3a^{-41}+45z^3a^{-43}-36z^3a^{-45}+28z^3a^{-47}-21z^3a^{-49}+15z^3a^{-51}-10z^3a^{-53}+6z^3a^{-55}-3z^3a^{-57}+z^3a^{-59}+680z^2a^{-30}+575z^2a^{-32}-14z^2a^{-34}+13z^2a^{-36}-12z^2a^{-38}+11z^2a^{-40}-10z^2a^{-42}+9z^2a^{-44}-8z^2a^{-46}+7z^2a^{-48}-6z^2a^{-50}+5z^2a^{-52}-4z^2a^{-54}+3z^2a^{-56}-2z^2a^{-58}+z^2a^{-60}+15za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-za^{-55}+za^{-57}-za^{-59}+za^{-61}-16a^{-30}-15a^{-32} }[/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(31,2)"];
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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^{15}-t^{14}+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} - t^{-14} + t^{-15} }[/math], [math]\displaystyle{ -q^{46}+q^{45}-q^{44}+q^{43}-q^{42}+q^{41}-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^{15} }[/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: | (120, 1240) |
| 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]30 is the signature of T(31,2). 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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