T(13,2)

T(13,2) 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(13,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

z12 + 11z10 + 45z8 + 84z6 + 70z4 + 21z2 + 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]=

{ 13, 12 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q19 + q18−q17 + q16−q15 + q14−q13 + q12−q11 + q10−q9 + q8 + q6

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z12a−12 + 12z10a−12−z10a−14 + 55z8a−12−10z8a−14 + 120z6a−12−36z6a−14 + 126z4a−12−56z4a−14 + 56z2a−12−35z2a−14 + 7a−12−6a−14

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z12a−12 + z12a−14 + z11a−13 + z11a−15−12z10a−12−11z10a−14 + z10a−16−10z9a−13−9z9a−15 + z9a−17 + 55z8a−12 + 46z8a−14−8z8a−16 + z8a−18 + 36z7a−13 + 28z7a−15−7z7a−17 + z7a−19−120z6a−12−92z6a−14 + 21z6a−16−6z6a−18 + z6a−20−56z5a−13−35z5a−15 + 15z5a−17−5z5a−19 + z5a−21 + 126z4a−12 + 91z4a−14−20z4a−16 + 10z4a−18−4z4a−20 + z4a−22 + 35z3a−13 + 15z3a−15−10z3a−17 + 6z3a−19−3z3a−21 + z3a−23−56z2a−12−41z2a−14 + 5z2a−16−4z2a−18 + 3z2a−20−2z2a−22 + z2a−24−6za−13−za−15 + za−17−za−19 + za−21−za−23 + za−25 + 7a−12 + 6a−14

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(13,2)"];

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]=

{ t6−t5 + t4−t3 + t2−t + 1−t−1 + t−2−t−3 + t−4−t−5 + t−6, −q19 + q18−q17 + q16−q15 + q14−q13 + q12−q11 + q10−q9 + q8 + q6 }

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]=

{}