T(21,2)

T(21,2) 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(21,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

z20 + 19z18 + 153z16 + 680z14 + 1820z12 + 3003z10 + 3003z8 + 1716z6 + 495z4 + 55z2 + 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]=

{ 21, 20 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q31 + q30−q29 + q28−q27 + q26−q25 + q24−q23 + q22−q21 + q20−q19 + q18−q17 + q16−q15 + q14−q13 + q12 + q10

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z20a−20 + 20z18a−20−z18a−22 + 171z16a−20−18z16a−22 + 816z14a−20−136z14a−22 + 2380z12a−20−560z12a−22 + 4368z10a−20−1365z10a−22 + 5005z8a−20−2002z8a−22 + 3432z6a−20−1716z6a−22 + 1287z4a−20−792z4a−22 + 220z2a−20−165z2a−22 + 11a−20−10a−22

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z20a−20 + z20a−22 + z19a−21 + z19a−23−20z18a−20−19z18a−22 + z18a−24−18z17a−21−17z17a−23 + z17a−25 + 171z16a−20 + 154z16a−22−16z16a−24 + z16a−26 + 136z15a−21 + 120z15a−23−15z15a−25 + z15a−27−816z14a−20−696z14a−22 + 105z14a−24−14z14a−26 + z14a−28−560z13a−21−455z13a−23 + 91z13a−25−13z13a−27 + z13a−29 + 2380z12a−20 + 1925z12a−22−364z12a−24 + 78z12a−26−12z12a−28 + z12a−30 + 1365z11a−21 + 1001z11a−23−286z11a−25 + 66z11a−27−11z11a−29 + z11a−31−4368z10a−20−3367z10a−22 + 715z10a−24−220z10a−26 + 55z10a−28−10z10a−30 + z10a−32−2002z9a−21−1287z9a−23 + 495z9a−25−165z9a−27 + 45z9a−29−9z9a−31 + z9a−33 + 5005z8a−20 + 3718z8a−22−792z8a−24 + 330z8a−26−120z8a−28 + 36z8a−30−8z8a−32 + z8a−34 + 1716z7a−21 + 924z7a−23−462z7a−25 + 210z7a−27−84z7a−29 + 28z7a−31−7z7a−33 + z7a−35−3432z6a−20−2508z6a−22 + 462z6a−24−252z6a−26 + 126z6a−28−56z6a−30 + 21z6a−32−6z6a−34 + z6a−36−792z5a−21−330z5a−23 + 210z5a−25−126z5a−27 + 70z5a−29−35z5a−31 + 15z5a−33−5z5a−35 + z5a−37 + 1287z4a−20 + 957z4a−22−120z4a−24 + 84z4a−26−56z4a−28 + 35z4a−30−20z4a−32 + 10z4a−34−4z4a−36 + z4a−38 + 165z3a−21 + 45z3a−23−36z3a−25 + 28z3a−27−21z3a−29 + 15z3a−31−10z3a−33 + 6z3a−35−3z3a−37 + z3a−39−220z2a−20−175z2a−22 + 9z2a−24−8z2a−26 + 7z2a−28−6z2a−30 + 5z2a−32−4z2a−34 + 3z2a−36−2z2a−38 + z2a−40−10za−21−za−23 + za−25−za−27 + za−29−za−31 + za−33−za−35 + za−37−za−39 + za−41 + 11a−20 + 10a−22

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(21,2)"];

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]=

{ 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, −q31 + q30−q29 + q28−q27 + q26−q25 + q24−q23 + q22−q21 + q20−q19 + q18−q17 + q16−q15 + q14−q13 + q12 + q10 }

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]=

{}