T(33,2)

T(33,2) Quantum Invariants.

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]=

t16−t15 + t14−t13 + t12−t11 + t10−t9 + t8−t7 + t6−t5 + t4−t3 + t2−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]=

z32 + 31z30 + 435z28 + 3654z26 + 20475z24 + 80730z22 + 230230z20 + 480700z18 + 735471z16 + 817190z14 + 646646z12 + 352716z10 + 125970z8 + 27132z6 + 3060z4 + 136z2 + 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]=

−q49 + q48−q47 + q46−q45 + q44−q43 + q42−q41 + q40−q39 + q38−q37 + q36−q35 + q34−q33 + q32−q31 + q30−q29 + q28−q27 + q26−q25 + q24−q23 + q22−q21 + q20−q19 + q18 + q16

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z32a−32−32z30a−32−z30a−34 + 465z28a−32 + 30z28a−34−4060z26a−32−406z26a−34 + 23751z24a−32 + 3276z24a−34−98280z22a−32−17550z22a−34 + 296010z20a−32 + 65780z20a−34−657800z18a−32−177100z18a−34 + 1081575z16a−32 + 346104z16a−34−1307504z14a−32−490314z14a−34 + 1144066z12a−32 + 497420z12a−34−705432z10a−32−352716z10a−34 + 293930z8a−32 + 167960z8a−34−77520z6a−32−50388z6a−34 + 11628z4a−32 + 8568z4a−34−816z2a−32−680z2a−34 + 17a−32 + 16a−34

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z32a−32 + z32a−34 + z31a−33 + z31a−35−32z30a−32−31z30a−34 + z30a−36−30z29a−33−29z29a−35 + z29a−37 + 465z28a−32 + 436z28a−34−28z28a−36 + z28a−38 + 406z27a−33 + 378z27a−35−27z27a−37 + z27a−39−4060z26a−32−3682z26a−34 + 351z26a−36−26z26a−38 + z26a−40−3276z25a−33−2925z25a−35 + 325z25a−37−25z25a−39 + z25a−41 + 23751z24a−32 + 20826z24a−34−2600z24a−36 + 300z24a−38−24z24a−40 + z24a−42 + 17550z23a−33 + 14950z23a−35−2300z23a−37 + 276z23a−39−23z23a−41 + z23a−43−98280z22a−32−83330z22a−34 + 12650z22a−36−2024z22a−38 + 253z22a−40−22z22a−42 + z22a−44−65780z21a−33−53130z21a−35 + 10626z21a−37−1771z21a−39 + 231z21a−41−21z21a−43 + z21a−45 + 296010z20a−32 + 242880z20a−34−42504z20a−36 + 8855z20a−38−1540z20a−40 + 210z20a−42−20z20a−44 + z20a−46 + 177100z19a−33 + 134596z19a−35−33649z19a−37 + 7315z19a−39−1330z19a−41 + 190z19a−43−19z19a−45 + z19a−47−657800z18a−32−523204z18a−34 + 100947z18a−36−26334z18a−38 + 5985z18a−40−1140z18a−42 + 171z18a−44−18z18a−46 + z18a−48−346104z17a−33−245157z17a−35 + 74613z17a−37−20349z17a−39 + 4845z17a−41−969z17a−43 + 153z17a−45−17z17a−47 + z17a−49 + 1081575z16a−32 + 836418z16a−34−170544z16a−36 + 54264z16a−38−15504z16a−40 + 3876z16a−42−816z16a−44 + 136z16a−46−16z16a−48 + z16a−50 + 490314z15a−33 + 319770z15a−35−116280z15a−37 + 38760z15a−39−11628z15a−41 + 3060z15a−43−680z15a−45 + 120z15a−47−15z15a−49 + z15a−51−1307504z14a−32−987734z14a−34 + 203490z14a−36−77520z14a−38 + 27132z14a−40−8568z14a−42 + 2380z14a−44−560z14a−46 + 105z14a−48−14z14a−50 + z14a−52−497420z13a−33−293930z13a−35 + 125970z13a−37−50388z13a−39 + 18564z13a−41−6188z13a−43 + 1820z13a−45−455z13a−47 + 91z13a−49−13z13a−51 + z13a−53 + 1144066z12a−32 + 850136z12a−34−167960z12a−36 + 75582z12a−38−31824z12a−40 + 12376z12a−42−4368z12a−44 + 1365z12a−46−364z12a−48 + 78z12a−50−12z12a−52 + z12a−54 + 352716z11a−33 + 184756z11a−35−92378z11a−37 + 43758z11a−39−19448z11a−41 + 8008z11a−43−3003z11a−45 + 1001z11a−47−286z11a−49 + 66z11a−51−11z11a−53 + z11a−55−705432z10a−32−520676z10a−34 + 92378z10a−36−48620z10a−38 + 24310z10a−40−11440z10a−42 + 5005z10a−44−2002z10a−46 + 715z10a−48−220z10a−50 + 55z10a−52−10z10a−54 + z10a−56−167960z9a−33−75582z9a−35 + 43758z9a−37−24310z9a−39 + 12870z9a−41−6435z9a−43 + 3003z9a−45−1287z9a−47 + 495z9a−49−165z9a−51 + 45z9a−53−9z9a−55 + z9a−57 + 293930z8a−32 + 218348z8a−34−31824z8a−36 + 19448z8a−38−11440z8a−40 + 6435z8a−42−3432z8a−44 + 1716z8a−46−792z8a−48 + 330z8a−50−120z8a−52 + 36z8a−54−8z8a−56 + z8a−58 + 50388z7a−33 + 18564z7a−35−12376z7a−37 + 8008z7a−39−5005z7a−41 + 3003z7a−43−1716z7a−45 + 924z7a−47−462z7a−49 + 210z7a−51−84z7a−53 + 28z7a−55−7z7a−57 + z7a−59−77520z6a−32−58956z6a−34 + 6188z6a−36−4368z6a−38 + 3003z6a−40−2002z6a−42 + 1287z6a−44−792z6a−46 + 462z6a−48−252z6a−50 + 126z6a−52−56z6a−54 + 21z6a−56−6z6a−58 + z6a−60−8568z5a−33−2380z5a−35 + 1820z5a−37−1365z5a−39 + 1001z5a−41−715z5a−43 + 495z5a−45−330z5a−47 + 210z5a−49−126z5a−51 + 70z5a−53−35z5a−55 + 15z5a−57−5z5a−59 + z5a−61 + 11628z4a−32 + 9248z4a−34−560z4a−36 + 455z4a−38−364z4a−40 + 286z4a−42−220z4a−44 + 165z4a−46−120z4a−48 + 84z4a−50−56z4a−52 + 35z4a−54−20z4a−56 + 10z4a−58−4z4a−60 + z4a−62 + 680z3a−33 + 120z3a−35−105z3a−37 + 91z3a−39−78z3a−41 + 66z3a−43−55z3a−45 + 45z3a−47−36z3a−49 + 28z3a−51−21z3a−53 + 15z3a−55−10z3a−57 + 6z3a−59−3z3a−61 + z3a−63−816z2a−32−696z2a−34 + 15z2a−36−14z2a−38 + 13z2a−40−12z2a−42 + 11z2a−44−10z2a−46 + 9z2a−48−8z2a−50 + 7z2a−52−6z2a−54 + 5z2a−56−4z2a−58 + 3z2a−60−2z2a−62 + z2a−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 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]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(33,2)"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

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

Out[4]=

{ t16−t15 + t14−t13 + t12−t11 + t10−t9 + t8−t7 + t6−t5 + t4−t3 + t2−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, −q49 + q48−q47 + q46−q45 + q44−q43 + q42−q41 + q40−q39 + q38−q37 + q36−q35 + q34−q33 + q32−q31 + q30−q29 + q28−q27 + q26−q25 + q24−q23 + q22−q21 + q20−q19 + q18 + q16 }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}