T(31,2)

T(31,2) 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(31,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

z30 + 29z28 + 378z26 + 2925z24 + 14950z22 + 53130z20 + 134596z18 + 245157z16 + 319770z14 + 293930z12 + 184756z10 + 75582z8 + 18564z6 + 2380z4 + 120z2 + 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]=

{ 31, 30 }

In[8]:=

Jones[K][q]

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

Out[8]=

−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 + q17 + q15

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z30a−30−30z28a−30−z28a−32 + 406z26a−30 + 28z26a−32−3276z24a−30−351z24a−32 + 17550z22a−30 + 2600z22a−32−65780z20a−30−12650z20a−32 + 177100z18a−30 + 42504z18a−32−346104z16a−30−100947z16a−32 + 490314z14a−30 + 170544z14a−32−497420z12a−30−203490z12a−32 + 352716z10a−30 + 167960z10a−32−167960z8a−30−92378z8a−32 + 50388z6a−30 + 31824z6a−32−8568z4a−30−6188z4a−32 + 680z2a−30 + 560z2a−32−16a−30−15a−32

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z30a−30 + z30a−32 + z29a−31 + z29a−33−30z28a−30−29z28a−32 + z28a−34−28z27a−31−27z27a−33 + z27a−35 + 406z26a−30 + 379z26a−32−26z26a−34 + z26a−36 + 351z25a−31 + 325z25a−33−25z25a−35 + z25a−37−3276z24a−30−2951z24a−32 + 300z24a−34−24z24a−36 + z24a−38−2600z23a−31−2300z23a−33 + 276z23a−35−23z23a−37 + z23a−39 + 17550z22a−30 + 15250z22a−32−2024z22a−34 + 253z22a−36−22z22a−38 + z22a−40 + 12650z21a−31 + 10626z21a−33−1771z21a−35 + 231z21a−37−21z21a−39 + z21a−41−65780z20a−30−55154z20a−32 + 8855z20a−34−1540z20a−36 + 210z20a−38−20z20a−40 + z20a−42−42504z19a−31−33649z19a−33 + 7315z19a−35−1330z19a−37 + 190z19a−39−19z19a−41 + z19a−43 + 177100z18a−30 + 143451z18a−32−26334z18a−34 + 5985z18a−36−1140z18a−38 + 171z18a−40−18z18a−42 + z18a−44 + 100947z17a−31 + 74613z17a−33−20349z17a−35 + 4845z17a−37−969z17a−39 + 153z17a−41−17z17a−43 + z17a−45−346104z16a−30−271491z16a−32 + 54264z16a−34−15504z16a−36 + 3876z16a−38−816z16a−40 + 136z16a−42−16z16a−44 + z16a−46−170544z15a−31−116280z15a−33 + 38760z15a−35−11628z15a−37 + 3060z15a−39−680z15a−41 + 120z15a−43−15z15a−45 + z15a−47 + 490314z14a−30 + 374034z14a−32−77520z14a−34 + 27132z14a−36−8568z14a−38 + 2380z14a−40−560z14a−42 + 105z14a−44−14z14a−46 + z14a−48 + 203490z13a−31 + 125970z13a−33−50388z13a−35 + 18564z13a−37−6188z13a−39 + 1820z13a−41−455z13a−43 + 91z13a−45−13z13a−47 + z13a−49−497420z12a−30−371450z12a−32 + 75582z12a−34−31824z12a−36 + 12376z12a−38−4368z12a−40 + 1365z12a−42−364z12a−44 + 78z12a−46−12z12a−48 + z12a−50−167960z11a−31−92378z11a−33 + 43758z11a−35−19448z11a−37 + 8008z11a−39−3003z11a−41 + 1001z11a−43−286z11a−45 + 66z11a−47−11z11a−49 + z11a−51 + 352716z10a−30 + 260338z10a−32−48620z10a−34 + 24310z10a−36−11440z10a−38 + 5005z10a−40−2002z10a−42 + 715z10a−44−220z10a−46 + 55z10a−48−10z10a−50 + z10a−52 + 92378z9a−31 + 43758z9a−33−24310z9a−35 + 12870z9a−37−6435z9a−39 + 3003z9a−41−1287z9a−43 + 495z9a−45−165z9a−47 + 45z9a−49−9z9a−51 + z9a−53−167960z8a−30−124202z8a−32 + 19448z8a−34−11440z8a−36 + 6435z8a−38−3432z8a−40 + 1716z8a−42−792z8a−44 + 330z8a−46−120z8a−48 + 36z8a−50−8z8a−52 + z8a−54−31824z7a−31−12376z7a−33 + 8008z7a−35−5005z7a−37 + 3003z7a−39−1716z7a−41 + 924z7a−43−462z7a−45 + 210z7a−47−84z7a−49 + 28z7a−51−7z7a−53 + z7a−55 + 50388z6a−30 + 38012z6a−32−4368z6a−34 + 3003z6a−36−2002z6a−38 + 1287z6a−40−792z6a−42 + 462z6a−44−252z6a−46 + 126z6a−48−56z6a−50 + 21z6a−52−6z6a−54 + z6a−56 + 6188z5a−31 + 1820z5a−33−1365z5a−35 + 1001z5a−37−715z5a−39 + 495z5a−41−330z5a−43 + 210z5a−45−126z5a−47 + 70z5a−49−35z5a−51 + 15z5a−53−5z5a−55 + z5a−57−8568z4a−30−6748z4a−32 + 455z4a−34−364z4a−36 + 286z4a−38−220z4a−40 + 165z4a−42−120z4a−44 + 84z4a−46−56z4a−48 + 35z4a−50−20z4a−52 + 10z4a−54−4z4a−56 + z4a−58−560z3a−31−105z3a−33 + 91z3a−35−78z3a−37 + 66z3a−39−55z3a−41 + 45z3a−43−36z3a−45 + 28z3a−47−21z3a−49 + 15z3a−51−10z3a−53 + 6z3a−55−3z3a−57 + z3a−59 + 680z2a−30 + 575z2a−32−14z2a−34 + 13z2a−36−12z2a−38 + 11z2a−40−10z2a−42 + 9z2a−44−8z2a−46 + 7z2a−48−6z2a−50 + 5z2a−52−4z2a−54 + 3z2a−56−2z2a−58 + z2a−60 + 15za−31 + za−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−61−16a−30−15a−32

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(31,2)"];

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]=

{ 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, −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 + q17 + q15 }

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]=

{}