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	$t^{16}-t^{15}+t^{14}-t^{13}+t^{12}-t^{11}+t^{10}-t^{9}+t^{8}-t^{7}+t^{6}-t^{5}+t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}-t^{-5}+t^{-6}-t^{-7}+t^{-8}-t^{-9}+t^{-10}-t^{-11}+t^{-12}-t^{-13}+t^{-14}-t^{-15}+t^{-16}$
Conway polynomial	$z^{32}+31z^{30}+435z^{28}+3654z^{26}+20475z^{24}+80730z^{22}+230230z^{20}+480700z^{18}+735471z^{16}+817190z^{14}+646646z^{12}+352716z^{10}+125970z^{8}+27132z^{6}+3060z^{4}+136z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 32 }
Jones polynomial	$-q^{49}+q^{48}-q^{47}+q^{46}-q^{45}+q^{44}-q^{43}+q^{42}-q^{41}+q^{40}-q^{39}+q^{38}-q^{37}+q^{36}-q^{35}+q^{34}-q^{33}+q^{32}-q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}+q^{16}$
HOMFLY-PT polynomial (db, data sources)	$z^{32}a^{-32}-32z^{30}a^{-32}-z^{30}a^{-34}+465z^{28}a^{-32}+30z^{28}a^{-34}-4060z^{26}a^{-32}-406z^{26}a^{-34}+23751z^{24}a^{-32}+3276z^{24}a^{-34}-98280z^{22}a^{-32}-17550z^{22}a^{-34}+296010z^{20}a^{-32}+65780z^{20}a^{-34}-657800z^{18}a^{-32}-177100z^{18}a^{-34}+1081575z^{16}a^{-32}+346104z^{16}a^{-34}-1307504z^{14}a^{-32}-490314z^{14}a^{-34}+1144066z^{12}a^{-32}+497420z^{12}a^{-34}-705432z^{10}a^{-32}-352716z^{10}a^{-34}+293930z^{8}a^{-32}+167960z^{8}a^{-34}-77520z^{6}a^{-32}-50388z^{6}a^{-34}+11628z^{4}a^{-32}+8568z^{4}a^{-34}-816z^{2}a^{-32}-680z^{2}a^{-34}+17a^{-32}+16a^{-34}$
Kauffman polynomial (db, data sources)	$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^{9}a^{-33}-75582z^{9}a^{-35}+43758z^{9}a^{-37}-24310z^{9}a^{-39}+12870z^{9}a^{-41}-6435z^{9}a^{-43}+3003z^{9}a^{-45}-1287z^{9}a^{-47}+495z^{9}a^{-49}-165z^{9}a^{-51}+45z^{9}a^{-53}-9z^{9}a^{-55}+z^{9}a^{-57}+293930z^{8}a^{-32}+218348z^{8}a^{-34}-31824z^{8}a^{-36}+19448z^{8}a^{-38}-11440z^{8}a^{-40}+6435z^{8}a^{-42}-3432z^{8}a^{-44}+1716z^{8}a^{-46}-792z^{8}a^{-48}+330z^{8}a^{-50}-120z^{8}a^{-52}+36z^{8}a^{-54}-8z^{8}a^{-56}+z^{8}a^{-58}+50388z^{7}a^{-33}+18564z^{7}a^{-35}-12376z^{7}a^{-37}+8008z^{7}a^{-39}-5005z^{7}a^{-41}+3003z^{7}a^{-43}-1716z^{7}a^{-45}+924z^{7}a^{-47}-462z^{7}a^{-49}+210z^{7}a^{-51}-84z^{7}a^{-53}+28z^{7}a^{-55}-7z^{7}a^{-57}+z^{7}a^{-59}-77520z^{6}a^{-32}-58956z^{6}a^{-34}+6188z^{6}a^{-36}-4368z^{6}a^{-38}+3003z^{6}a^{-40}-2002z^{6}a^{-42}+1287z^{6}a^{-44}-792z^{6}a^{-46}+462z^{6}a^{-48}-252z^{6}a^{-50}+126z^{6}a^{-52}-56z^{6}a^{-54}+21z^{6}a^{-56}-6z^{6}a^{-58}+z^{6}a^{-60}-8568z^{5}a^{-33}-2380z^{5}a^{-35}+1820z^{5}a^{-37}-1365z^{5}a^{-39}+1001z^{5}a^{-41}-715z^{5}a^{-43}+495z^{5}a^{-45}-330z^{5}a^{-47}+210z^{5}a^{-49}-126z^{5}a^{-51}+70z^{5}a^{-53}-35z^{5}a^{-55}+15z^{5}a^{-57}-5z^{5}a^{-59}+z^{5}a^{-61}+11628z^{4}a^{-32}+9248z^{4}a^{-34}-560z^{4}a^{-36}+455z^{4}a^{-38}-364z^{4}a^{-40}+286z^{4}a^{-42}-220z^{4}a^{-44}+165z^{4}a^{-46}-120z^{4}a^{-48}+84z^{4}a^{-50}-56z^{4}a^{-52}+35z^{4}a^{-54}-20z^{4}a^{-56}+10z^{4}a^{-58}-4z^{4}a^{-60}+z^{4}a^{-62}+680z^{3}a^{-33}+120z^{3}a^{-35}-105z^{3}a^{-37}+91z^{3}a^{-39}-78z^{3}a^{-41}+66z^{3}a^{-43}-55z^{3}a^{-45}+45z^{3}a^{-47}-36z^{3}a^{-49}+28z^{3}a^{-51}-21z^{3}a^{-53}+15z^{3}a^{-55}-10z^{3}a^{-57}+6z^{3}a^{-59}-3z^{3}a^{-61}+z^{3}a^{-63}-816z^{2}a^{-32}-696z^{2}a^{-34}+15z^{2}a^{-36}-14z^{2}a^{-38}+13z^{2}a^{-40}-12z^{2}a^{-42}+11z^{2}a^{-44}-10z^{2}a^{-46}+9z^{2}a^{-48}-8z^{2}a^{-50}+7z^{2}a^{-52}-6z^{2}a^{-54}+5z^{2}a^{-56}-4z^{2}a^{-58}+3z^{2}a^{-60}-2z^{2}a^{-62}+z^{2}a^{-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}$
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]=

$t^{16}-t^{15}+t^{14}-t^{13}+t^{12}-t^{11}+t^{10}-t^{9}+t^{8}-t^{7}+t^{6}-t^{5}+t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}-t^{-5}+t^{-6}-t^{-7}+t^{-8}-t^{-9}+t^{-10}-t^{-11}+t^{-12}-t^{-13}+t^{-14}-t^{-15}+t^{-16}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{32}+31z^{30}+435z^{28}+3654z^{26}+20475z^{24}+80730z^{22}+230230z^{20}+480700z^{18}+735471z^{16}+817190z^{14}+646646z^{12}+352716z^{10}+125970z^{8}+27132z^{6}+3060z^{4}+136z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, 32 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{49}+q^{48}-q^{47}+q^{46}-q^{45}+q^{44}-q^{43}+q^{42}-q^{41}+q^{40}-q^{39}+q^{38}-q^{37}+q^{36}-q^{35}+q^{34}-q^{33}+q^{32}-q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}+q^{16}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{32}a^{-32}-32z^{30}a^{-32}-z^{30}a^{-34}+465z^{28}a^{-32}+30z^{28}a^{-34}-4060z^{26}a^{-32}-406z^{26}a^{-34}+23751z^{24}a^{-32}+3276z^{24}a^{-34}-98280z^{22}a^{-32}-17550z^{22}a^{-34}+296010z^{20}a^{-32}+65780z^{20}a^{-34}-657800z^{18}a^{-32}-177100z^{18}a^{-34}+1081575z^{16}a^{-32}+346104z^{16}a^{-34}-1307504z^{14}a^{-32}-490314z^{14}a^{-34}+1144066z^{12}a^{-32}+497420z^{12}a^{-34}-705432z^{10}a^{-32}-352716z^{10}a^{-34}+293930z^{8}a^{-32}+167960z^{8}a^{-34}-77520z^{6}a^{-32}-50388z^{6}a^{-34}+11628z^{4}a^{-32}+8568z^{4}a^{-34}-816z^{2}a^{-32}-680z^{2}a^{-34}+17a^{-32}+16a^{-34}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$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^{9}a^{-33}-75582z^{9}a^{-35}+43758z^{9}a^{-37}-24310z^{9}a^{-39}+12870z^{9}a^{-41}-6435z^{9}a^{-43}+3003z^{9}a^{-45}-1287z^{9}a^{-47}+495z^{9}a^{-49}-165z^{9}a^{-51}+45z^{9}a^{-53}-9z^{9}a^{-55}+z^{9}a^{-57}+293930z^{8}a^{-32}+218348z^{8}a^{-34}-31824z^{8}a^{-36}+19448z^{8}a^{-38}-11440z^{8}a^{-40}+6435z^{8}a^{-42}-3432z^{8}a^{-44}+1716z^{8}a^{-46}-792z^{8}a^{-48}+330z^{8}a^{-50}-120z^{8}a^{-52}+36z^{8}a^{-54}-8z^{8}a^{-56}+z^{8}a^{-58}+50388z^{7}a^{-33}+18564z^{7}a^{-35}-12376z^{7}a^{-37}+8008z^{7}a^{-39}-5005z^{7}a^{-41}+3003z^{7}a^{-43}-1716z^{7}a^{-45}+924z^{7}a^{-47}-462z^{7}a^{-49}+210z^{7}a^{-51}-84z^{7}a^{-53}+28z^{7}a^{-55}-7z^{7}a^{-57}+z^{7}a^{-59}-77520z^{6}a^{-32}-58956z^{6}a^{-34}+6188z^{6}a^{-36}-4368z^{6}a^{-38}+3003z^{6}a^{-40}-2002z^{6}a^{-42}+1287z^{6}a^{-44}-792z^{6}a^{-46}+462z^{6}a^{-48}-252z^{6}a^{-50}+126z^{6}a^{-52}-56z^{6}a^{-54}+21z^{6}a^{-56}-6z^{6}a^{-58}+z^{6}a^{-60}-8568z^{5}a^{-33}-2380z^{5}a^{-35}+1820z^{5}a^{-37}-1365z^{5}a^{-39}+1001z^{5}a^{-41}-715z^{5}a^{-43}+495z^{5}a^{-45}-330z^{5}a^{-47}+210z^{5}a^{-49}-126z^{5}a^{-51}+70z^{5}a^{-53}-35z^{5}a^{-55}+15z^{5}a^{-57}-5z^{5}a^{-59}+z^{5}a^{-61}+11628z^{4}a^{-32}+9248z^{4}a^{-34}-560z^{4}a^{-36}+455z^{4}a^{-38}-364z^{4}a^{-40}+286z^{4}a^{-42}-220z^{4}a^{-44}+165z^{4}a^{-46}-120z^{4}a^{-48}+84z^{4}a^{-50}-56z^{4}a^{-52}+35z^{4}a^{-54}-20z^{4}a^{-56}+10z^{4}a^{-58}-4z^{4}a^{-60}+z^{4}a^{-62}+680z^{3}a^{-33}+120z^{3}a^{-35}-105z^{3}a^{-37}+91z^{3}a^{-39}-78z^{3}a^{-41}+66z^{3}a^{-43}-55z^{3}a^{-45}+45z^{3}a^{-47}-36z^{3}a^{-49}+28z^{3}a^{-51}-21z^{3}a^{-53}+15z^{3}a^{-55}-10z^{3}a^{-57}+6z^{3}a^{-59}-3z^{3}a^{-61}+z^{3}a^{-63}-816z^{2}a^{-32}-696z^{2}a^{-34}+15z^{2}a^{-36}-14z^{2}a^{-38}+13z^{2}a^{-40}-12z^{2}a^{-42}+11z^{2}a^{-44}-10z^{2}a^{-46}+9z^{2}a^{-48}-8z^{2}a^{-50}+7z^{2}a^{-52}-6z^{2}a^{-54}+5z^{2}a^{-56}-4z^{2}a^{-58}+3z^{2}a^{-60}-2z^{2}a^{-62}+z^{2}a^{-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}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$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)

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

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