K11n15

K11n15 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["K11n15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−4t2 + 8t−9 + 8t−1−4t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6 + 2z4 + z2 + 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]=

{ 35, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q8 + 2q7−3q6 + 5q5−6q4 + 6q3−5q2 + 4q−2 + q−1

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−4−2z4a−2 + 5z4a−4−z4a−6−6z2a−2 + 9z2a−4−3z2a−6 + z2−4a−2 + 5a−4−2a−6 + 2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z9a−3 + z9a−5 + z8a−2 + 3z8a−4 + 2z8a−6−5z7a−3−3z7a−5 + 2z7a−7−5z6a−2−14z6a−4−7z6a−6 + 2z6a−8 + 2z5a−1 + 12z5a−3 + 4z5a−5−5z5a−7 + z5a−9 + 14z4a−2 + 29z4a−4 + 10z4a−6−6z4a−8 + z4−4z3a−1−8z3a−3 + z3a−5 + 2z3a−7−3z3a−9−15z2a−2−21z2a−4−6z2a−6 + 3z2a−8−3z2 + za−1 + za−3−za−5 + za−9 + 4a−2 + 5a−4 + 2a−6 + 2

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["K11n15"];

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]=

{ t3−4t2 + 8t−9 + 8t−1−4t−2 + t−3, −q8 + 2q7−3q6 + 5q5−6q4 + 6q3−5q2 + 4q−2 + q−1 }

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]=

{8_16, 10_156, K11n56, K11n58,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{9_12,}