T(17,2)

T(17,2) 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(17,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

z16 + 15z14 + 91z12 + 286z10 + 495z8 + 462z6 + 210z4 + 36z2 + 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]=

{ 17, 16 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q25 + q24−q23 + q22−q21 + q20−q19 + q18−q17 + q16−q15 + q14−q13 + q12−q11 + q10 + q8

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z16a−16 + 16z14a−16−z14a−18 + 105z12a−16−14z12a−18 + 364z10a−16−78z10a−18 + 715z8a−16−220z8a−18 + 792z6a−16−330z6a−18 + 462z4a−16−252z4a−18 + 120z2a−16−84z2a−18 + 9a−16−8a−18

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z16a−16 + z16a−18 + z15a−17 + z15a−19−16z14a−16−15z14a−18 + z14a−20−14z13a−17−13z13a−19 + z13a−21 + 105z12a−16 + 92z12a−18−12z12a−20 + z12a−22 + 78z11a−17 + 66z11a−19−11z11a−21 + z11a−23−364z10a−16−298z10a−18 + 55z10a−20−10z10a−22 + z10a−24−220z9a−17−165z9a−19 + 45z9a−21−9z9a−23 + z9a−25 + 715z8a−16 + 550z8a−18−120z8a−20 + 36z8a−22−8z8a−24 + z8a−26 + 330z7a−17 + 210z7a−19−84z7a−21 + 28z7a−23−7z7a−25 + z7a−27−792z6a−16−582z6a−18 + 126z6a−20−56z6a−22 + 21z6a−24−6z6a−26 + z6a−28−252z5a−17−126z5a−19 + 70z5a−21−35z5a−23 + 15z5a−25−5z5a−27 + z5a−29 + 462z4a−16 + 336z4a−18−56z4a−20 + 35z4a−22−20z4a−24 + 10z4a−26−4z4a−28 + z4a−30 + 84z3a−17 + 28z3a−19−21z3a−21 + 15z3a−23−10z3a−25 + 6z3a−27−3z3a−29 + z3a−31−120z2a−16−92z2a−18 + 7z2a−20−6z2a−22 + 5z2a−24−4z2a−26 + 3z2a−28−2z2a−30 + z2a−32−8za−17−za−19 + za−21−za−23 + za−25−za−27 + za−29−za−31 + za−33 + 9a−16 + 8a−18

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(17,2)"];

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]=

{ 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, −q25 + q24−q23 + q22−q21 + q20−q19 + q18−q17 + q16−q15 + q14−q13 + q12−q11 + q10 + q8 }

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]=

{}