T(23,2)

T(23,2) 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(23,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

z22 + 21z20 + 190z18 + 969z16 + 3060z14 + 6188z12 + 8008z10 + 6435z8 + 3003z6 + 715z4 + 66z2 + 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]=

{ 23, 22 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q34 + q33−q32 + q31−q30 + q29−q28 + q27−q26 + q25−q24 + q23−q22 + q21−q20 + q19−q18 + q17−q16 + q15−q14 + q13 + q11

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z22a−22 + 22z20a−22−z20a−24 + 210z18a−22−20z18a−24 + 1140z16a−22−171z16a−24 + 3876z14a−22−816z14a−24 + 8568z12a−22−2380z12a−24 + 12376z10a−22−4368z10a−24 + 11440z8a−22−5005z8a−24 + 6435z6a−22−3432z6a−24 + 2002z4a−22−1287z4a−24 + 286z2a−22−220z2a−24 + 12a−22−11a−24

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z22a−22 + z22a−24 + z21a−23 + z21a−25−22z20a−22−21z20a−24 + z20a−26−20z19a−23−19z19a−25 + z19a−27 + 210z18a−22 + 191z18a−24−18z18a−26 + z18a−28 + 171z17a−23 + 153z17a−25−17z17a−27 + z17a−29−1140z16a−22−987z16a−24 + 136z16a−26−16z16a−28 + z16a−30−816z15a−23−680z15a−25 + 120z15a−27−15z15a−29 + z15a−31 + 3876z14a−22 + 3196z14a−24−560z14a−26 + 105z14a−28−14z14a−30 + z14a−32 + 2380z13a−23 + 1820z13a−25−455z13a−27 + 91z13a−29−13z13a−31 + z13a−33−8568z12a−22−6748z12a−24 + 1365z12a−26−364z12a−28 + 78z12a−30−12z12a−32 + z12a−34−4368z11a−23−3003z11a−25 + 1001z11a−27−286z11a−29 + 66z11a−31−11z11a−33 + z11a−35 + 12376z10a−22 + 9373z10a−24−2002z10a−26 + 715z10a−28−220z10a−30 + 55z10a−32−10z10a−34 + z10a−36 + 5005z9a−23 + 3003z9a−25−1287z9a−27 + 495z9a−29−165z9a−31 + 45z9a−33−9z9a−35 + z9a−37−11440z8a−22−8437z8a−24 + 1716z8a−26−792z8a−28 + 330z8a−30−120z8a−32 + 36z8a−34−8z8a−36 + z8a−38−3432z7a−23−1716z7a−25 + 924z7a−27−462z7a−29 + 210z7a−31−84z7a−33 + 28z7a−35−7z7a−37 + z7a−39 + 6435z6a−22 + 4719z6a−24−792z6a−26 + 462z6a−28−252z6a−30 + 126z6a−32−56z6a−34 + 21z6a−36−6z6a−38 + z6a−40 + 1287z5a−23 + 495z5a−25−330z5a−27 + 210z5a−29−126z5a−31 + 70z5a−33−35z5a−35 + 15z5a−37−5z5a−39 + z5a−41−2002z4a−22−1507z4a−24 + 165z4a−26−120z4a−28 + 84z4a−30−56z4a−32 + 35z4a−34−20z4a−36 + 10z4a−38−4z4a−40 + z4a−42−220z3a−23−55z3a−25 + 45z3a−27−36z3a−29 + 28z3a−31−21z3a−33 + 15z3a−35−10z3a−37 + 6z3a−39−3z3a−41 + z3a−43 + 286z2a−22 + 231z2a−24−10z2a−26 + 9z2a−28−8z2a−30 + 7z2a−32−6z2a−34 + 5z2a−36−4z2a−38 + 3z2a−40−2z2a−42 + z2a−44 + 11za−23 + za−25−za−27 + za−29−za−31 + za−33−za−35 + za−37−za−39 + za−41−za−43 + za−45−12a−22−11a−24

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(23,2)"];

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]=

{ 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, −q34 + q33−q32 + q31−q30 + q29−q28 + q27−q26 + q25−q24 + q23−q22 + q21−q20 + q19−q18 + q17−q16 + q15−q14 + q13 + q11 }

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]=

{}