T(19,2)

T(19,2) 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(19,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

z18 + 17z16 + 120z14 + 455z12 + 1001z10 + 1287z8 + 924z6 + 330z4 + 45z2 + 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]=

{ 19, 18 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q28 + q27−q26 + q25−q24 + q23−q22 + q21−q20 + q19−q18 + q17−q16 + q15−q14 + q13−q12 + q11 + q9

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z18a−18 + 18z16a−18−z16a−20 + 136z14a−18−16z14a−20 + 560z12a−18−105z12a−20 + 1365z10a−18−364z10a−20 + 2002z8a−18−715z8a−20 + 1716z6a−18−792z6a−20 + 792z4a−18−462z4a−20 + 165z2a−18−120z2a−20 + 10a−18−9a−20

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z18a−18 + z18a−20 + z17a−19 + z17a−21−18z16a−18−17z16a−20 + z16a−22−16z15a−19−15z15a−21 + z15a−23 + 136z14a−18 + 121z14a−20−14z14a−22 + z14a−24 + 105z13a−19 + 91z13a−21−13z13a−23 + z13a−25−560z12a−18−469z12a−20 + 78z12a−22−12z12a−24 + z12a−26−364z11a−19−286z11a−21 + 66z11a−23−11z11a−25 + z11a−27 + 1365z10a−18 + 1079z10a−20−220z10a−22 + 55z10a−24−10z10a−26 + z10a−28 + 715z9a−19 + 495z9a−21−165z9a−23 + 45z9a−25−9z9a−27 + z9a−29−2002z8a−18−1507z8a−20 + 330z8a−22−120z8a−24 + 36z8a−26−8z8a−28 + z8a−30−792z7a−19−462z7a−21 + 210z7a−23−84z7a−25 + 28z7a−27−7z7a−29 + z7a−31 + 1716z6a−18 + 1254z6a−20−252z6a−22 + 126z6a−24−56z6a−26 + 21z6a−28−6z6a−30 + z6a−32 + 462z5a−19 + 210z5a−21−126z5a−23 + 70z5a−25−35z5a−27 + 15z5a−29−5z5a−31 + z5a−33−792z4a−18−582z4a−20 + 84z4a−22−56z4a−24 + 35z4a−26−20z4a−28 + 10z4a−30−4z4a−32 + z4a−34−120z3a−19−36z3a−21 + 28z3a−23−21z3a−25 + 15z3a−27−10z3a−29 + 6z3a−31−3z3a−33 + z3a−35 + 165z2a−18 + 129z2a−20−8z2a−22 + 7z2a−24−6z2a−26 + 5z2a−28−4z2a−30 + 3z2a−32−2z2a−34 + z2a−36 + 9za−19 + za−21−za−23 + za−25−za−27 + za−29−za−31 + za−33−za−35 + za−37−10a−18−9a−20

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(19,2)"];

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]=

{ 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, −q28 + q27−q26 + q25−q24 + q23−q22 + q21−q20 + q19−q18 + q17−q16 + q15−q14 + q13−q12 + q11 + q9 }

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]=

{}