T(33,2)

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[[Image:T(11,4).{{{ext}}}|80px|link=T(11,4)]]

T(11,4)

[[Image:T(17,3).{{{ext}}}|80px|link=T(17,3)]]

T(17,3)

Visit T(33,2)'s page at Knotilus!

Visit T(33,2)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation X31,65,32,64 X65,33,66,32 X33,1,34,66 X1,35,2,34 X35,3,36,2 X3,37,4,36 X37,5,38,4 X5,39,6,38 X39,7,40,6 X7,41,8,40 X41,9,42,8 X9,43,10,42 X43,11,44,10 X11,45,12,44 X45,13,46,12 X13,47,14,46 X47,15,48,14 X15,49,16,48 X49,17,50,16 X17,51,18,50 X51,19,52,18 X19,53,20,52 X53,21,54,20 X21,55,22,54 X55,23,56,22 X23,57,24,56 X57,25,58,24 X25,59,26,58 X59,27,60,26 X27,61,28,60 X61,29,62,28 X29,63,30,62 X63,31,64,30
Gauss code {-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, -32, 33, -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, 32, -33, 1, -2, 3}
Dowker-Thistlethwaite code 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Polynomial invariants

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^{16}-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} + t^{-16} }[/math]
Conway polynomial [math]\displaystyle{ z^{32}+31 z^{30}+435 z^{28}+3654 z^{26}+20475 z^{24}+80730 z^{22}+230230 z^{20}+480700 z^{18}+735471 z^{16}+817190 z^{14}+646646 z^{12}+352716 z^{10}+125970 z^8+27132 z^6+3060 z^4+136 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 33, 32 }
Jones polynomial [math]\displaystyle{ -q^{49}+q^{48}-q^{47}+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^{16} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^{32}a^{-32}-32z^{30}a^{-32}-z^{30}a^{-34}+465z^{28}a^{-32}+30z^{28}a^{-34}-4060z^{26}a^{-32}-406z^{26}a^{-34}+23751z^{24}a^{-32}+3276z^{24}a^{-34}-98280z^{22}a^{-32}-17550z^{22}a^{-34}+296010z^{20}a^{-32}+65780z^{20}a^{-34}-657800z^{18}a^{-32}-177100z^{18}a^{-34}+1081575z^{16}a^{-32}+346104z^{16}a^{-34}-1307504z^{14}a^{-32}-490314z^{14}a^{-34}+1144066z^{12}a^{-32}+497420z^{12}a^{-34}-705432z^{10}a^{-32}-352716z^{10}a^{-34}+293930z^8a^{-32}+167960z^8a^{-34}-77520z^6a^{-32}-50388z^6a^{-34}+11628z^4a^{-32}+8568z^4a^{-34}-816z^2a^{-32}-680z^2a^{-34}+17a^{-32}+16a^{-34} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^{32}a^{-32}+z^{32}a^{-34}+z^{31}a^{-33}+z^{31}a^{-35}-32z^{30}a^{-32}-31z^{30}a^{-34}+z^{30}a^{-36}-30z^{29}a^{-33}-29z^{29}a^{-35}+z^{29}a^{-37}+465z^{28}a^{-32}+436z^{28}a^{-34}-28z^{28}a^{-36}+z^{28}a^{-38}+406z^{27}a^{-33}+378z^{27}a^{-35}-27z^{27}a^{-37}+z^{27}a^{-39}-4060z^{26}a^{-32}-3682z^{26}a^{-34}+351z^{26}a^{-36}-26z^{26}a^{-38}+z^{26}a^{-40}-3276z^{25}a^{-33}-2925z^{25}a^{-35}+325z^{25}a^{-37}-25z^{25}a^{-39}+z^{25}a^{-41}+23751z^{24}a^{-32}+20826z^{24}a^{-34}-2600z^{24}a^{-36}+300z^{24}a^{-38}-24z^{24}a^{-40}+z^{24}a^{-42}+17550z^{23}a^{-33}+14950z^{23}a^{-35}-2300z^{23}a^{-37}+276z^{23}a^{-39}-23z^{23}a^{-41}+z^{23}a^{-43}-98280z^{22}a^{-32}-83330z^{22}a^{-34}+12650z^{22}a^{-36}-2024z^{22}a^{-38}+253z^{22}a^{-40}-22z^{22}a^{-42}+z^{22}a^{-44}-65780z^{21}a^{-33}-53130z^{21}a^{-35}+10626z^{21}a^{-37}-1771z^{21}a^{-39}+231z^{21}a^{-41}-21z^{21}a^{-43}+z^{21}a^{-45}+296010z^{20}a^{-32}+242880z^{20}a^{-34}-42504z^{20}a^{-36}+8855z^{20}a^{-38}-1540z^{20}a^{-40}+210z^{20}a^{-42}-20z^{20}a^{-44}+z^{20}a^{-46}+177100z^{19}a^{-33}+134596z^{19}a^{-35}-33649z^{19}a^{-37}+7315z^{19}a^{-39}-1330z^{19}a^{-41}+190z^{19}a^{-43}-19z^{19}a^{-45}+z^{19}a^{-47}-657800z^{18}a^{-32}-523204z^{18}a^{-34}+100947z^{18}a^{-36}-26334z^{18}a^{-38}+5985z^{18}a^{-40}-1140z^{18}a^{-42}+171z^{18}a^{-44}-18z^{18}a^{-46}+z^{18}a^{-48}-346104z^{17}a^{-33}-245157z^{17}a^{-35}+74613z^{17}a^{-37}-20349z^{17}a^{-39}+4845z^{17}a^{-41}-969z^{17}a^{-43}+153z^{17}a^{-45}-17z^{17}a^{-47}+z^{17}a^{-49}+1081575z^{16}a^{-32}+836418z^{16}a^{-34}-170544z^{16}a^{-36}+54264z^{16}a^{-38}-15504z^{16}a^{-40}+3876z^{16}a^{-42}-816z^{16}a^{-44}+136z^{16}a^{-46}-16z^{16}a^{-48}+z^{16}a^{-50}+490314z^{15}a^{-33}+319770z^{15}a^{-35}-116280z^{15}a^{-37}+38760z^{15}a^{-39}-11628z^{15}a^{-41}+3060z^{15}a^{-43}-680z^{15}a^{-45}+120z^{15}a^{-47}-15z^{15}a^{-49}+z^{15}a^{-51}-1307504z^{14}a^{-32}-987734z^{14}a^{-34}+203490z^{14}a^{-36}-77520z^{14}a^{-38}+27132z^{14}a^{-40}-8568z^{14}a^{-42}+2380z^{14}a^{-44}-560z^{14}a^{-46}+105z^{14}a^{-48}-14z^{14}a^{-50}+z^{14}a^{-52}-497420z^{13}a^{-33}-293930z^{13}a^{-35}+125970z^{13}a^{-37}-50388z^{13}a^{-39}+18564z^{13}a^{-41}-6188z^{13}a^{-43}+1820z^{13}a^{-45}-455z^{13}a^{-47}+91z^{13}a^{-49}-13z^{13}a^{-51}+z^{13}a^{-53}+1144066z^{12}a^{-32}+850136z^{12}a^{-34}-167960z^{12}a^{-36}+75582z^{12}a^{-38}-31824z^{12}a^{-40}+12376z^{12}a^{-42}-4368z^{12}a^{-44}+1365z^{12}a^{-46}-364z^{12}a^{-48}+78z^{12}a^{-50}-12z^{12}a^{-52}+z^{12}a^{-54}+352716z^{11}a^{-33}+184756z^{11}a^{-35}-92378z^{11}a^{-37}+43758z^{11}a^{-39}-19448z^{11}a^{-41}+8008z^{11}a^{-43}-3003z^{11}a^{-45}+1001z^{11}a^{-47}-286z^{11}a^{-49}+66z^{11}a^{-51}-11z^{11}a^{-53}+z^{11}a^{-55}-705432z^{10}a^{-32}-520676z^{10}a^{-34}+92378z^{10}a^{-36}-48620z^{10}a^{-38}+24310z^{10}a^{-40}-11440z^{10}a^{-42}+5005z^{10}a^{-44}-2002z^{10}a^{-46}+715z^{10}a^{-48}-220z^{10}a^{-50}+55z^{10}a^{-52}-10z^{10}a^{-54}+z^{10}a^{-56}-167960z^9a^{-33}-75582z^9a^{-35}+43758z^9a^{-37}-24310z^9a^{-39}+12870z^9a^{-41}-6435z^9a^{-43}+3003z^9a^{-45}-1287z^9a^{-47}+495z^9a^{-49}-165z^9a^{-51}+45z^9a^{-53}-9z^9a^{-55}+z^9a^{-57}+293930z^8a^{-32}+218348z^8a^{-34}-31824z^8a^{-36}+19448z^8a^{-38}-11440z^8a^{-40}+6435z^8a^{-42}-3432z^8a^{-44}+1716z^8a^{-46}-792z^8a^{-48}+330z^8a^{-50}-120z^8a^{-52}+36z^8a^{-54}-8z^8a^{-56}+z^8a^{-58}+50388z^7a^{-33}+18564z^7a^{-35}-12376z^7a^{-37}+8008z^7a^{-39}-5005z^7a^{-41}+3003z^7a^{-43}-1716z^7a^{-45}+924z^7a^{-47}-462z^7a^{-49}+210z^7a^{-51}-84z^7a^{-53}+28z^7a^{-55}-7z^7a^{-57}+z^7a^{-59}-77520z^6a^{-32}-58956z^6a^{-34}+6188z^6a^{-36}-4368z^6a^{-38}+3003z^6a^{-40}-2002z^6a^{-42}+1287z^6a^{-44}-792z^6a^{-46}+462z^6a^{-48}-252z^6a^{-50}+126z^6a^{-52}-56z^6a^{-54}+21z^6a^{-56}-6z^6a^{-58}+z^6a^{-60}-8568z^5a^{-33}-2380z^5a^{-35}+1820z^5a^{-37}-1365z^5a^{-39}+1001z^5a^{-41}-715z^5a^{-43}+495z^5a^{-45}-330z^5a^{-47}+210z^5a^{-49}-126z^5a^{-51}+70z^5a^{-53}-35z^5a^{-55}+15z^5a^{-57}-5z^5a^{-59}+z^5a^{-61}+11628z^4a^{-32}+9248z^4a^{-34}-560z^4a^{-36}+455z^4a^{-38}-364z^4a^{-40}+286z^4a^{-42}-220z^4a^{-44}+165z^4a^{-46}-120z^4a^{-48}+84z^4a^{-50}-56z^4a^{-52}+35z^4a^{-54}-20z^4a^{-56}+10z^4a^{-58}-4z^4a^{-60}+z^4a^{-62}+680z^3a^{-33}+120z^3a^{-35}-105z^3a^{-37}+91z^3a^{-39}-78z^3a^{-41}+66z^3a^{-43}-55z^3a^{-45}+45z^3a^{-47}-36z^3a^{-49}+28z^3a^{-51}-21z^3a^{-53}+15z^3a^{-55}-10z^3a^{-57}+6z^3a^{-59}-3z^3a^{-61}+z^3a^{-63}-816z^2a^{-32}-696z^2a^{-34}+15z^2a^{-36}-14z^2a^{-38}+13z^2a^{-40}-12z^2a^{-42}+11z^2a^{-44}-10z^2a^{-46}+9z^2a^{-48}-8z^2a^{-50}+7z^2a^{-52}-6z^2a^{-54}+5z^2a^{-56}-4z^2a^{-58}+3z^2a^{-60}-2z^2a^{-62}+z^2a^{-64}-16za^{-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}-za^{-63}+za^{-65}+17a^{-32}+16a^{-34} }[/math]
The A2 invariant Data:T(33,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(33,2)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(33,2) 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]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["T(33,2)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^{16}-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} + t^{-16} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^{32}+31 z^{30}+435 z^{28}+3654 z^{26}+20475 z^{24}+80730 z^{22}+230230 z^{20}+480700 z^{18}+735471 z^{16}+817190 z^{14}+646646 z^{12}+352716 z^{10}+125970 z^8+27132 z^6+3060 z^4+136 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
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.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 33, 32 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^{49}+q^{48}-q^{47}+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^{16} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^{32}a^{-32}-32z^{30}a^{-32}-z^{30}a^{-34}+465z^{28}a^{-32}+30z^{28}a^{-34}-4060z^{26}a^{-32}-406z^{26}a^{-34}+23751z^{24}a^{-32}+3276z^{24}a^{-34}-98280z^{22}a^{-32}-17550z^{22}a^{-34}+296010z^{20}a^{-32}+65780z^{20}a^{-34}-657800z^{18}a^{-32}-177100z^{18}a^{-34}+1081575z^{16}a^{-32}+346104z^{16}a^{-34}-1307504z^{14}a^{-32}-490314z^{14}a^{-34}+1144066z^{12}a^{-32}+497420z^{12}a^{-34}-705432z^{10}a^{-32}-352716z^{10}a^{-34}+293930z^8a^{-32}+167960z^8a^{-34}-77520z^6a^{-32}-50388z^6a^{-34}+11628z^4a^{-32}+8568z^4a^{-34}-816z^2a^{-32}-680z^2a^{-34}+17a^{-32}+16a^{-34} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^{32}a^{-32}+z^{32}a^{-34}+z^{31}a^{-33}+z^{31}a^{-35}-32z^{30}a^{-32}-31z^{30}a^{-34}+z^{30}a^{-36}-30z^{29}a^{-33}-29z^{29}a^{-35}+z^{29}a^{-37}+465z^{28}a^{-32}+436z^{28}a^{-34}-28z^{28}a^{-36}+z^{28}a^{-38}+406z^{27}a^{-33}+378z^{27}a^{-35}-27z^{27}a^{-37}+z^{27}a^{-39}-4060z^{26}a^{-32}-3682z^{26}a^{-34}+351z^{26}a^{-36}-26z^{26}a^{-38}+z^{26}a^{-40}-3276z^{25}a^{-33}-2925z^{25}a^{-35}+325z^{25}a^{-37}-25z^{25}a^{-39}+z^{25}a^{-41}+23751z^{24}a^{-32}+20826z^{24}a^{-34}-2600z^{24}a^{-36}+300z^{24}a^{-38}-24z^{24}a^{-40}+z^{24}a^{-42}+17550z^{23}a^{-33}+14950z^{23}a^{-35}-2300z^{23}a^{-37}+276z^{23}a^{-39}-23z^{23}a^{-41}+z^{23}a^{-43}-98280z^{22}a^{-32}-83330z^{22}a^{-34}+12650z^{22}a^{-36}-2024z^{22}a^{-38}+253z^{22}a^{-40}-22z^{22}a^{-42}+z^{22}a^{-44}-65780z^{21}a^{-33}-53130z^{21}a^{-35}+10626z^{21}a^{-37}-1771z^{21}a^{-39}+231z^{21}a^{-41}-21z^{21}a^{-43}+z^{21}a^{-45}+296010z^{20}a^{-32}+242880z^{20}a^{-34}-42504z^{20}a^{-36}+8855z^{20}a^{-38}-1540z^{20}a^{-40}+210z^{20}a^{-42}-20z^{20}a^{-44}+z^{20}a^{-46}+177100z^{19}a^{-33}+134596z^{19}a^{-35}-33649z^{19}a^{-37}+7315z^{19}a^{-39}-1330z^{19}a^{-41}+190z^{19}a^{-43}-19z^{19}a^{-45}+z^{19}a^{-47}-657800z^{18}a^{-32}-523204z^{18}a^{-34}+100947z^{18}a^{-36}-26334z^{18}a^{-38}+5985z^{18}a^{-40}-1140z^{18}a^{-42}+171z^{18}a^{-44}-18z^{18}a^{-46}+z^{18}a^{-48}-346104z^{17}a^{-33}-245157z^{17}a^{-35}+74613z^{17}a^{-37}-20349z^{17}a^{-39}+4845z^{17}a^{-41}-969z^{17}a^{-43}+153z^{17}a^{-45}-17z^{17}a^{-47}+z^{17}a^{-49}+1081575z^{16}a^{-32}+836418z^{16}a^{-34}-170544z^{16}a^{-36}+54264z^{16}a^{-38}-15504z^{16}a^{-40}+3876z^{16}a^{-42}-816z^{16}a^{-44}+136z^{16}a^{-46}-16z^{16}a^{-48}+z^{16}a^{-50}+490314z^{15}a^{-33}+319770z^{15}a^{-35}-116280z^{15}a^{-37}+38760z^{15}a^{-39}-11628z^{15}a^{-41}+3060z^{15}a^{-43}-680z^{15}a^{-45}+120z^{15}a^{-47}-15z^{15}a^{-49}+z^{15}a^{-51}-1307504z^{14}a^{-32}-987734z^{14}a^{-34}+203490z^{14}a^{-36}-77520z^{14}a^{-38}+27132z^{14}a^{-40}-8568z^{14}a^{-42}+2380z^{14}a^{-44}-560z^{14}a^{-46}+105z^{14}a^{-48}-14z^{14}a^{-50}+z^{14}a^{-52}-497420z^{13}a^{-33}-293930z^{13}a^{-35}+125970z^{13}a^{-37}-50388z^{13}a^{-39}+18564z^{13}a^{-41}-6188z^{13}a^{-43}+1820z^{13}a^{-45}-455z^{13}a^{-47}+91z^{13}a^{-49}-13z^{13}a^{-51}+z^{13}a^{-53}+1144066z^{12}a^{-32}+850136z^{12}a^{-34}-167960z^{12}a^{-36}+75582z^{12}a^{-38}-31824z^{12}a^{-40}+12376z^{12}a^{-42}-4368z^{12}a^{-44}+1365z^{12}a^{-46}-364z^{12}a^{-48}+78z^{12}a^{-50}-12z^{12}a^{-52}+z^{12}a^{-54}+352716z^{11}a^{-33}+184756z^{11}a^{-35}-92378z^{11}a^{-37}+43758z^{11}a^{-39}-19448z^{11}a^{-41}+8008z^{11}a^{-43}-3003z^{11}a^{-45}+1001z^{11}a^{-47}-286z^{11}a^{-49}+66z^{11}a^{-51}-11z^{11}a^{-53}+z^{11}a^{-55}-705432z^{10}a^{-32}-520676z^{10}a^{-34}+92378z^{10}a^{-36}-48620z^{10}a^{-38}+24310z^{10}a^{-40}-11440z^{10}a^{-42}+5005z^{10}a^{-44}-2002z^{10}a^{-46}+715z^{10}a^{-48}-220z^{10}a^{-50}+55z^{10}a^{-52}-10z^{10}a^{-54}+z^{10}a^{-56}-167960z^9a^{-33}-75582z^9a^{-35}+43758z^9a^{-37}-24310z^9a^{-39}+12870z^9a^{-41}-6435z^9a^{-43}+3003z^9a^{-45}-1287z^9a^{-47}+495z^9a^{-49}-165z^9a^{-51}+45z^9a^{-53}-9z^9a^{-55}+z^9a^{-57}+293930z^8a^{-32}+218348z^8a^{-34}-31824z^8a^{-36}+19448z^8a^{-38}-11440z^8a^{-40}+6435z^8a^{-42}-3432z^8a^{-44}+1716z^8a^{-46}-792z^8a^{-48}+330z^8a^{-50}-120z^8a^{-52}+36z^8a^{-54}-8z^8a^{-56}+z^8a^{-58}+50388z^7a^{-33}+18564z^7a^{-35}-12376z^7a^{-37}+8008z^7a^{-39}-5005z^7a^{-41}+3003z^7a^{-43}-1716z^7a^{-45}+924z^7a^{-47}-462z^7a^{-49}+210z^7a^{-51}-84z^7a^{-53}+28z^7a^{-55}-7z^7a^{-57}+z^7a^{-59}-77520z^6a^{-32}-58956z^6a^{-34}+6188z^6a^{-36}-4368z^6a^{-38}+3003z^6a^{-40}-2002z^6a^{-42}+1287z^6a^{-44}-792z^6a^{-46}+462z^6a^{-48}-252z^6a^{-50}+126z^6a^{-52}-56z^6a^{-54}+21z^6a^{-56}-6z^6a^{-58}+z^6a^{-60}-8568z^5a^{-33}-2380z^5a^{-35}+1820z^5a^{-37}-1365z^5a^{-39}+1001z^5a^{-41}-715z^5a^{-43}+495z^5a^{-45}-330z^5a^{-47}+210z^5a^{-49}-126z^5a^{-51}+70z^5a^{-53}-35z^5a^{-55}+15z^5a^{-57}-5z^5a^{-59}+z^5a^{-61}+11628z^4a^{-32}+9248z^4a^{-34}-560z^4a^{-36}+455z^4a^{-38}-364z^4a^{-40}+286z^4a^{-42}-220z^4a^{-44}+165z^4a^{-46}-120z^4a^{-48}+84z^4a^{-50}-56z^4a^{-52}+35z^4a^{-54}-20z^4a^{-56}+10z^4a^{-58}-4z^4a^{-60}+z^4a^{-62}+680z^3a^{-33}+120z^3a^{-35}-105z^3a^{-37}+91z^3a^{-39}-78z^3a^{-41}+66z^3a^{-43}-55z^3a^{-45}+45z^3a^{-47}-36z^3a^{-49}+28z^3a^{-51}-21z^3a^{-53}+15z^3a^{-55}-10z^3a^{-57}+6z^3a^{-59}-3z^3a^{-61}+z^3a^{-63}-816z^2a^{-32}-696z^2a^{-34}+15z^2a^{-36}-14z^2a^{-38}+13z^2a^{-40}-12z^2a^{-42}+11z^2a^{-44}-10z^2a^{-46}+9z^2a^{-48}-8z^2a^{-50}+7z^2a^{-52}-6z^2a^{-54}+5z^2a^{-56}-4z^2a^{-58}+3z^2a^{-60}-2z^2a^{-62}+z^2a^{-64}-16za^{-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}-za^{-63}+za^{-65}+17a^{-32}+16a^{-34} }[/math]

Vassiliev invariants

V2 and V3 {0, 1496})

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]32 is the signature of T(33,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123456789101112131415161718192021222324252627282930313233χ
99                                 1-1
97                                  0
95                               11 0
93                                  0
91                             11   0
89                                  0
87                           11     0
85                                  0
83                         11       0
81                                  0
79                       11         0
77                                  0
75                     11           0
73                                  0
71                   11             0
69                                  0
67                 11               0
65                                  0
63               11                 0
61                                  0
59             11                   0
57                                  0
55           11                     0
53                                  0
51         11                       0
49                                  0
47       11                         0
45                                  0
43     11                           0
41                                  0
39   11                             0
37                                  0
35  1                               1
331                                 1
311                                 1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session. 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 19, 2005, 13:11:25)...
In[2]:=
Crossings[TorusKnot[33, 2]]
Out[2]=   
33
In[3]:=
PD[TorusKnot[33, 2]]
Out[3]=   
PD[X[31, 65, 32, 64], X[65, 33, 66, 32], X[33, 1, 34, 66], 

 X[1, 35, 2, 34], X[35, 3, 36, 2], X[3, 37, 4, 36], X[37, 5, 38, 4], 

 X[5, 39, 6, 38], X[39, 7, 40, 6], X[7, 41, 8, 40], X[41, 9, 42, 8], 

 X[9, 43, 10, 42], X[43, 11, 44, 10], X[11, 45, 12, 44], 

 X[45, 13, 46, 12], X[13, 47, 14, 46], X[47, 15, 48, 14], 

 X[15, 49, 16, 48], X[49, 17, 50, 16], X[17, 51, 18, 50], 

 X[51, 19, 52, 18], X[19, 53, 20, 52], X[53, 21, 54, 20], 

 X[21, 55, 22, 54], X[55, 23, 56, 22], X[23, 57, 24, 56], 

 X[57, 25, 58, 24], X[25, 59, 26, 58], X[59, 27, 60, 26], 

 X[27, 61, 28, 60], X[61, 29, 62, 28], X[29, 63, 30, 62], 



  X[63, 31, 64, 30]]
In[4]:=
GaussCode[TorusKnot[33, 2]]
Out[4]=   
GaussCode[-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, -32, 33, 

 -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, 32, 



  -33, 1, -2, 3]
In[5]:=
BR[TorusKnot[33, 2]]
Out[5]=   
BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[6]:=
alex = Alexander[TorusKnot[33, 2]][t]
Out[6]=   
     -16    -15    -14    -13    -12    -11    -10    -9    -8    -7

1 + t    - t    + t    - t    + t    - t    + t    - t   + t   - t   + 



  -6    -5    -4    -3    -2   1        2    3    4    5    6    7
 t   - t   + t   - t   + t   - - - t + t  - t  + t  - t  + t  - t  + 
                               t

  8    9    10    11    12    13    14    15    16


  t  - t  + t   - t   + t   - t   + t   - t   + t
In[7]:=
Conway[TorusKnot[33, 2]][z]
Out[7]=   
         2         4          6           8           10           12

1 + 136 z  + 3060 z  + 27132 z  + 125970 z  + 352716 z   + 646646 z   + 



         14           16           18           20          22
 817190 z   + 735471 z   + 480700 z   + 230230 z   + 80730 z   + 

        24         26        28       30    32


  20475 z   + 3654 z   + 435 z   + 31 z   + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{}
In[9]:=
{KnotDet[TorusKnot[33, 2]], KnotSignature[TorusKnot[33, 2]]}
Out[9]=   
{33, 32}
In[10]:=
J=Jones[TorusKnot[33, 2]][q]
Out[10]=   
 16    18    19    20    21    22    23    24    25    26    27    28

q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - 



  29    30    31    32    33    34    35    36    37    38    39
 q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 

  40    41    42    43    44    45    46    47    48    49


  q   - q   + q   - q   + q   - q   + q   - q   + q   - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{}
In[12]:=
A2Invariant[TorusKnot[33, 2]][q]
Out[12]=   
NotAvailable
In[13]:=
Kauffman[TorusKnot[33, 2]][a, z]
Out[13]=   
NotAvailable
In[14]:=
{Vassiliev[2][TorusKnot[33, 2]], Vassiliev[3][TorusKnot[33, 2]]}
Out[14]=   
{0, 1496}
In[15]:=
Kh[TorusKnot[33, 2]][q, t]
Out[15]=   
 31    33    35  2    39  3    39  4    43  5    43  6    47  7

q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 



  47  8    51  9    51  10    55  11    55  12    59  13    59  14
 q   t  + q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + 

  63  15    63  16    67  17    67  18    71  19    71  20    75  21
 q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 

  75  22    79  23    79  24    83  25    83  26    87  27    87  28
 q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 

  91  29    91  30    95  31    95  32    99  33


  q   t   + q   t   + q   t   + q   t   + q   t
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