T(15,2)

T(15,2) 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(15,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t7−t6 + t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5−t−6 + t−7

In[5]:=

Conway[K][z]

Out[5]=

z14 + 13z12 + 66z10 + 165z8 + 210z6 + 126z4 + 28z2 + 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]=

{ 15, 14 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q22 + q21−q20 + q19−q18 + q17−q16 + q15−q14 + q13−q12 + q11−q10 + q9 + q7

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z14a−14 + 14z12a−14−z12a−16 + 78z10a−14−12z10a−16 + 220z8a−14−55z8a−16 + 330z6a−14−120z6a−16 + 252z4a−14−126z4a−16 + 84z2a−14−56z2a−16 + 8a−14−7a−16

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z14a−14 + z14a−16 + z13a−15 + z13a−17−14z12a−14−13z12a−16 + z12a−18−12z11a−15−11z11a−17 + z11a−19 + 78z10a−14 + 67z10a−16−10z10a−18 + z10a−20 + 55z9a−15 + 45z9a−17−9z9a−19 + z9a−21−220z8a−14−175z8a−16 + 36z8a−18−8z8a−20 + z8a−22−120z7a−15−84z7a−17 + 28z7a−19−7z7a−21 + z7a−23 + 330z6a−14 + 246z6a−16−56z6a−18 + 21z6a−20−6z6a−22 + z6a−24 + 126z5a−15 + 70z5a−17−35z5a−19 + 15z5a−21−5z5a−23 + z5a−25−252z4a−14−182z4a−16 + 35z4a−18−20z4a−20 + 10z4a−22−4z4a−24 + z4a−26−56z3a−15−21z3a−17 + 15z3a−19−10z3a−21 + 6z3a−23−3z3a−25 + z3a−27 + 84z2a−14 + 63z2a−16−6z2a−18 + 5z2a−20−4z2a−22 + 3z2a−24−2z2a−26 + z2a−28 + 7za−15 + za−17−za−19 + za−21−za−23 + za−25−za−27 + za−29−8a−14−7a−16

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(15,2)"];

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]=

{ t7−t6 + t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5−t−6 + t−7, −q22 + q21−q20 + q19−q18 + q17−q16 + q15−q14 + q13−q12 + q11−q10 + q9 + q7 }

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]=

{}