T(27,2)

T(27,2) 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(27,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

z26 + 25z24 + 276z22 + 1771z20 + 7315z18 + 20349z16 + 38760z14 + 50388z12 + 43758z10 + 24310z8 + 8008z6 + 1365z4 + 91z2 + 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]=

{ 27, 26 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q40 + q39−q38 + q37−q36 + q35−q34 + q33−q32 + q31−q30 + q29−q28 + q27−q26 + q25−q24 + q23−q22 + q21−q20 + q19−q18 + q17−q16 + q15 + q13

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z26a−26−26z24a−26−z24a−28 + 300z22a−26 + 24z22a−28−2024z20a−26−253z20a−28 + 8855z18a−26 + 1540z18a−28−26334z16a−26−5985z16a−28 + 54264z14a−26 + 15504z14a−28−77520z12a−26−27132z12a−28 + 75582z10a−26 + 31824z10a−28−48620z8a−26−24310z8a−28 + 19448z6a−26 + 11440z6a−28−4368z4a−26−3003z4a−28 + 455z2a−26 + 364z2a−28−14a−26−13a−28

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z26a−26 + z26a−28 + z25a−27 + z25a−29−26z24a−26−25z24a−28 + z24a−30−24z23a−27−23z23a−29 + z23a−31 + 300z22a−26 + 277z22a−28−22z22a−30 + z22a−32 + 253z21a−27 + 231z21a−29−21z21a−31 + z21a−33−2024z20a−26−1793z20a−28 + 210z20a−30−20z20a−32 + z20a−34−1540z19a−27−1330z19a−29 + 190z19a−31−19z19a−33 + z19a−35 + 8855z18a−26 + 7525z18a−28−1140z18a−30 + 171z18a−32−18z18a−34 + z18a−36 + 5985z17a−27 + 4845z17a−29−969z17a−31 + 153z17a−33−17z17a−35 + z17a−37−26334z16a−26−21489z16a−28 + 3876z16a−30−816z16a−32 + 136z16a−34−16z16a−36 + z16a−38−15504z15a−27−11628z15a−29 + 3060z15a−31−680z15a−33 + 120z15a−35−15z15a−37 + z15a−39 + 54264z14a−26 + 42636z14a−28−8568z14a−30 + 2380z14a−32−560z14a−34 + 105z14a−36−14z14a−38 + z14a−40 + 27132z13a−27 + 18564z13a−29−6188z13a−31 + 1820z13a−33−455z13a−35 + 91z13a−37−13z13a−39 + z13a−41−77520z12a−26−58956z12a−28 + 12376z12a−30−4368z12a−32 + 1365z12a−34−364z12a−36 + 78z12a−38−12z12a−40 + z12a−42−31824z11a−27−19448z11a−29 + 8008z11a−31−3003z11a−33 + 1001z11a−35−286z11a−37 + 66z11a−39−11z11a−41 + z11a−43 + 75582z10a−26 + 56134z10a−28−11440z10a−30 + 5005z10a−32−2002z10a−34 + 715z10a−36−220z10a−38 + 55z10a−40−10z10a−42 + z10a−44 + 24310z9a−27 + 12870z9a−29−6435z9a−31 + 3003z9a−33−1287z9a−35 + 495z9a−37−165z9a−39 + 45z9a−41−9z9a−43 + z9a−45−48620z8a−26−35750z8a−28 + 6435z8a−30−3432z8a−32 + 1716z8a−34−792z8a−36 + 330z8a−38−120z8a−40 + 36z8a−42−8z8a−44 + z8a−46−11440z7a−27−5005z7a−29 + 3003z7a−31−1716z7a−33 + 924z7a−35−462z7a−37 + 210z7a−39−84z7a−41 + 28z7a−43−7z7a−45 + z7a−47 + 19448z6a−26 + 14443z6a−28−2002z6a−30 + 1287z6a−32−792z6a−34 + 462z6a−36−252z6a−38 + 126z6a−40−56z6a−42 + 21z6a−44−6z6a−46 + z6a−48 + 3003z5a−27 + 1001z5a−29−715z5a−31 + 495z5a−33−330z5a−35 + 210z5a−37−126z5a−39 + 70z5a−41−35z5a−43 + 15z5a−45−5z5a−47 + z5a−49−4368z4a−26−3367z4a−28 + 286z4a−30−220z4a−32 + 165z4a−34−120z4a−36 + 84z4a−38−56z4a−40 + 35z4a−42−20z4a−44 + 10z4a−46−4z4a−48 + z4a−50−364z3a−27−78z3a−29 + 66z3a−31−55z3a−33 + 45z3a−35−36z3a−37 + 28z3a−39−21z3a−41 + 15z3a−43−10z3a−45 + 6z3a−47−3z3a−49 + z3a−51 + 455z2a−26 + 377z2a−28−12z2a−30 + 11z2a−32−10z2a−34 + 9z2a−36−8z2a−38 + 7z2a−40−6z2a−42 + 5z2a−44−4z2a−46 + 3z2a−48−2z2a−50 + z2a−52 + 13za−27 + za−29−za−31 + za−33−za−35 + za−37−za−39 + za−41−za−43 + za−45−za−47 + za−49−za−51 + za−53−14a−26−13a−28

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(27,2)"];

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]=

{ 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, −q40 + q39−q38 + q37−q36 + q35−q34 + q33−q32 + q31−q30 + q29−q28 + q27−q26 + q25−q24 + q23−q22 + q21−q20 + q19−q18 + q17−q16 + q15 + q13 }

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]=

{}