T(35,2)

T(35,2) 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(35,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t17−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 + t−17

In[5]:=

Conway[K][z]

Out[5]=

z34 + 33z32 + 496z30 + 4495z28 + 27405z26 + 118755z24 + 376740z22 + 888030z20 + 1562275z18 + 2042975z16 + 1961256z14 + 1352078z12 + 646646z10 + 203490z8 + 38760z6 + 3876z4 + 153z2 + 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]=

{ 35, 34 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q52 + q51−q50 + 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 + q17

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z34a−34−34z32a−34−z32a−36 + 528z30a−34 + 32z30a−36−4960z28a−34−465z28a−36 + 31465z26a−34 + 4060z26a−36−142506z24a−34−23751z24a−36 + 475020z22a−34 + 98280z22a−36−1184040z20a−34−296010z20a−36 + 2220075z18a−34 + 657800z18a−36−3124550z16a−34−1081575z16a−36 + 3268760z14a−34 + 1307504z14a−36−2496144z12a−34−1144066z12a−36 + 1352078z10a−34 + 705432z10a−36−497420z8a−34−293930z8a−36 + 116280z6a−34 + 77520z6a−36−15504z4a−34−11628z4a−36 + 969z2a−34 + 816z2a−36−18a−34−17a−36

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z34a−34 + z34a−36 + z33a−35 + z33a−37−34z32a−34−33z32a−36 + z32a−38−32z31a−35−31z31a−37 + z31a−39 + 528z30a−34 + 497z30a−36−30z30a−38 + z30a−40 + 465z29a−35 + 435z29a−37−29z29a−39 + z29a−41−4960z28a−34−4525z28a−36 + 406z28a−38−28z28a−40 + z28a−42−4060z27a−35−3654z27a−37 + 378z27a−39−27z27a−41 + z27a−43 + 31465z26a−34 + 27811z26a−36−3276z26a−38 + 351z26a−40−26z26a−42 + z26a−44 + 23751z25a−35 + 20475z25a−37−2925z25a−39 + 325z25a−41−25z25a−43 + z25a−45−142506z24a−34−122031z24a−36 + 17550z24a−38−2600z24a−40 + 300z24a−42−24z24a−44 + z24a−46−98280z23a−35−80730z23a−37 + 14950z23a−39−2300z23a−41 + 276z23a−43−23z23a−45 + z23a−47 + 475020z22a−34 + 394290z22a−36−65780z22a−38 + 12650z22a−40−2024z22a−42 + 253z22a−44−22z22a−46 + z22a−48 + 296010z21a−35 + 230230z21a−37−53130z21a−39 + 10626z21a−41−1771z21a−43 + 231z21a−45−21z21a−47 + z21a−49−1184040z20a−34−953810z20a−36 + 177100z20a−38−42504z20a−40 + 8855z20a−42−1540z20a−44 + 210z20a−46−20z20a−48 + z20a−50−657800z19a−35−480700z19a−37 + 134596z19a−39−33649z19a−41 + 7315z19a−43−1330z19a−45 + 190z19a−47−19z19a−49 + z19a−51 + 2220075z18a−34 + 1739375z18a−36−346104z18a−38 + 100947z18a−40−26334z18a−42 + 5985z18a−44−1140z18a−46 + 171z18a−48−18z18a−50 + z18a−52 + 1081575z17a−35 + 735471z17a−37−245157z17a−39 + 74613z17a−41−20349z17a−43 + 4845z17a−45−969z17a−47 + 153z17a−49−17z17a−51 + z17a−53−3124550z16a−34−2389079z16a−36 + 490314z16a−38−170544z16a−40 + 54264z16a−42−15504z16a−44 + 3876z16a−46−816z16a−48 + 136z16a−50−16z16a−52 + z16a−54−1307504z15a−35−817190z15a−37 + 319770z15a−39−116280z15a−41 + 38760z15a−43−11628z15a−45 + 3060z15a−47−680z15a−49 + 120z15a−51−15z15a−53 + z15a−55 + 3268760z14a−34 + 2451570z14a−36−497420z14a−38 + 203490z14a−40−77520z14a−42 + 27132z14a−44−8568z14a−46 + 2380z14a−48−560z14a−50 + 105z14a−52−14z14a−54 + z14a−56 + 1144066z13a−35 + 646646z13a−37−293930z13a−39 + 125970z13a−41−50388z13a−43 + 18564z13a−45−6188z13a−47 + 1820z13a−49−455z13a−51 + 91z13a−53−13z13a−55 + z13a−57−2496144z12a−34−1849498z12a−36 + 352716z12a−38−167960z12a−40 + 75582z12a−42−31824z12a−44 + 12376z12a−46−4368z12a−48 + 1365z12a−50−364z12a−52 + 78z12a−54−12z12a−56 + z12a−58−705432z11a−35−352716z11a−37 + 184756z11a−39−92378z11a−41 + 43758z11a−43−19448z11a−45 + 8008z11a−47−3003z11a−49 + 1001z11a−51−286z11a−53 + 66z11a−55−11z11a−57 + z11a−59 + 1352078z10a−34 + 999362z10a−36−167960z10a−38 + 92378z10a−40−48620z10a−42 + 24310z10a−44−11440z10a−46 + 5005z10a−48−2002z10a−50 + 715z10a−52−220z10a−54 + 55z10a−56−10z10a−58 + z10a−60 + 293930z9a−35 + 125970z9a−37−75582z9a−39 + 43758z9a−41−24310z9a−43 + 12870z9a−45−6435z9a−47 + 3003z9a−49−1287z9a−51 + 495z9a−53−165z9a−55 + 45z9a−57−9z9a−59 + z9a−61−497420z8a−34−371450z8a−36 + 50388z8a−38−31824z8a−40 + 19448z8a−42−11440z8a−44 + 6435z8a−46−3432z8a−48 + 1716z8a−50−792z8a−52 + 330z8a−54−120z8a−56 + 36z8a−58−8z8a−60 + z8a−62−77520z7a−35−27132z7a−37 + 18564z7a−39−12376z7a−41 + 8008z7a−43−5005z7a−45 + 3003z7a−47−1716z7a−49 + 924z7a−51−462z7a−53 + 210z7a−55−84z7a−57 + 28z7a−59−7z7a−61 + z7a−63 + 116280z6a−34 + 89148z6a−36−8568z6a−38 + 6188z6a−40−4368z6a−42 + 3003z6a−44−2002z6a−46 + 1287z6a−48−792z6a−50 + 462z6a−52−252z6a−54 + 126z6a−56−56z6a−58 + 21z6a−60−6z6a−62 + z6a−64 + 11628z5a−35 + 3060z5a−37−2380z5a−39 + 1820z5a−41−1365z5a−43 + 1001z5a−45−715z5a−47 + 495z5a−49−330z5a−51 + 210z5a−53−126z5a−55 + 70z5a−57−35z5a−59 + 15z5a−61−5z5a−63 + z5a−65−15504z4a−34−12444z4a−36 + 680z4a−38−560z4a−40 + 455z4a−42−364z4a−44 + 286z4a−46−220z4a−48 + 165z4a−50−120z4a−52 + 84z4a−54−56z4a−56 + 35z4a−58−20z4a−60 + 10z4a−62−4z4a−64 + z4a−66−816z3a−35−136z3a−37 + 120z3a−39−105z3a−41 + 91z3a−43−78z3a−45 + 66z3a−47−55z3a−49 + 45z3a−51−36z3a−53 + 28z3a−55−21z3a−57 + 15z3a−59−10z3a−61 + 6z3a−63−3z3a−65 + z3a−67 + 969z2a−34 + 833z2a−36−16z2a−38 + 15z2a−40−14z2a−42 + 13z2a−44−12z2a−46 + 11z2a−48−10z2a−50 + 9z2a−52−8z2a−54 + 7z2a−56−6z2a−58 + 5z2a−60−4z2a−62 + 3z2a−64−2z2a−66 + z2a−68 + 17za−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−za−67 + za−69−18a−34−17a−36

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(35,2)"];

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]=

{ t17−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 + t−17, −q52 + q51−q50 + 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 + q17 }

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]=

{}