T(33,2)

T(11,4)

T(17,3)

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	T(33,2) Quick Notes

T(33,2) Further Notes and Views

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
Conway Notation	Data:T(33,2)/Conway Notation

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:

(136, 1496)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:T(33,2)/V 2,1 Data:T(33,2)/V 3,1 Data:T(33,2)/V 4,1 Data:T(33,2)/V 4,2 Data:T(33,2)/V 4,3 Data:T(33,2)/V 5,1 Data:T(33,2)/V 5,2 Data:T(33,2)/V 5,3 Data:T(33,2)/V 5,4 Data:T(33,2)/V 6,1 Data:T(33,2)/V 6,2 Data:T(33,2)/V 6,3 Data:T(33,2)/V 6,4 Data:T(33,2)/V 6,5 Data:T(33,2)/V 6,6 Data:T(33,2)/V 6,7 Data:T(33,2)/V 6,8 Data:T(33,2)/V 6,9

V_2,1 through V_6,9 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]32 is the signature of T(33,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 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 χ 99                                                                   1 -1 97                                                                     0 95                                                               1 1   0 93                                                                     0 91                                                           1 1       0 89                                                                     0 87                                                       1 1           0 85                                                                     0 83                                                   1 1               0 81                                                                     0 79                                               1 1                   0 77                                                                     0 75                                           1 1                       0 73                                                                     0 71                                       1 1                           0 69                                                                     0 67                                   1 1                               0 65                                                                     0 63                               1 1                                   0 61                                                                     0 59                           1 1                                       0 57                                                                     0 55                       1 1                                           0 53                                                                     0 51                   1 1                                               0 49                                                                     0 47               1 1                                                   0 45                                                                     0 43           1 1                                                       0 41                                                                     0 39       1 1                                                           0 37                                                                     0 35     1                                                               1 33 1                                                                   1 31 1                                                                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=31 }[/math] [math]\displaystyle{ i=33 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=11 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=12 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=13 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=14 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=15 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=16 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=17 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=18 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=19 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=20 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=21 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=22 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=23 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=24 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=25 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=26 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=27 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=28 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=29 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=30 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=31 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=32 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=33 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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 17, 2005, 14:44:34)...
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 1 + Alternating - Alternating + Alternating - Alternating + -12 -11 -10 -9 Alternating - Alternating + Alternating - Alternating + -8 -7 -6 -5 Alternating - Alternating + Alternating - Alternating + -4 -3 -2 1 Alternating - Alternating + Alternating - ----------- - Alternating 2 3 4 Alternating + Alternating - Alternating + Alternating - 5 6 7 8 Alternating + Alternating - Alternating + Alternating - 9 10 11 12 Alternating + Alternating - Alternating + Alternating - 13 14 15 16 Alternating + Alternating - Alternating + Alternating
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 2 35 3 39 4 39 q + q + Alternating q + Alternating q + Alternating q + 5 43 6 43 7 47 Alternating q + Alternating q + Alternating q + 8 47 9 51 10 51 Alternating q + Alternating q + Alternating q + 11 55 12 55 13 59 Alternating q + Alternating q + Alternating q + 14 59 15 63 16 63 Alternating q + Alternating q + Alternating q + 17 67 18 67 19 71 Alternating q + Alternating q + Alternating q + 20 71 21 75 22 75 Alternating q + Alternating q + Alternating q + 23 79 24 79 25 83 Alternating q + Alternating q + Alternating q + 26 83 27 87 28 87 Alternating q + Alternating q + Alternating q + 29 91 30 91 31 95 Alternating q + Alternating q + Alternating q + 32 95 33 99 Alternating q + Alternating q