Template:Large Knot Page

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Polynomial invariants

Alexander polynomial	Data:Large Knot Page/Alexander Polynomial
Conway polynomial	Data:Large Knot Page/Conway Polynomial
2nd Alexander ideal (db, data sources)	Data:Large Knot Page/2nd AlexanderIdeal
Determinant and Signature	{ Data:Large Knot Page/Determinant, Data:Large Knot Page/Signature }
Jones polynomial	Data:Large Knot Page/Jones Polynomial
HOMFLY-PT polynomial (db, data sources)	Data:Large Knot Page/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:Large Knot Page/Kauffman Polynomial
The A2 invariant	Data:Large Knot Page/QuantumInvariant/A2/1,0
The G2 invariant	Data:Large Knot Page/QuantumInvariant/G2/1,0

Further Quantum Invariants

Large Knot Page Quantum Invariants.

Computer Talk

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["Large Knot Page"];

In[4]:=

Alexander[K][t]

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

Out[4]=

Data:Large Knot Page/Alexander Polynomial

In[5]:=

Conway[K][z]

Out[5]=

Data:Large Knot Page/Conway Polynomial

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]=

Data:Large Knot Page/2nd AlexanderIdeal

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ Data:Large Knot Page/Determinant, Data:Large Knot Page/Signature }

In[8]:=

Jones[K][q]

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

Out[8]=

Data:Large Knot Page/Jones Polynomial

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Data:Large Knot Page/HOMFLYPT Polynomial

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Data:Large Knot Page/Kauffman Polynomial

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$Data:Large Knot Page/Signature is the signature of Large Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:Large Knot Page/KhovanovTable

Integral Khovanov Homology (db, data source)

Data:Large Knot Page/Integral Khovanov Homology