The A2 Invariant

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KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

We compute the $A2$ (or quantum $sl(3)$) invariant using the normalization and formulas of [Khovanov], which in itself follows [Kuperberg]:

(For In[1] see Setup)

In[1]:= ?A2Invariant A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q.

10_22

10_35

As an example, let us check that the knots 10_22 and 10_35 have the same Jones polynomial but different $A2$ invariants:

In[2]:=	Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]
Out[2]=	True

In[3]:=	A2Invariant[Knot[10, 22]][q]
Out[3]=	-12 -8 -6 -4 2 4 6 8 10 12 14 -1 + q + q + q - q + -- - q - 2 q + q - q + q + q + 2 q 18 q

In[4]:=	A2Invariant[Knot[10, 35]][q]
Out[4]=	-14 -12 -10 -8 2 2 2 6 8 10 14 16 q + q - q + q - -- + -- + q - q + q - 2 q + q - q + 4 2 q q 18 20 q + q

The $A2$ invariant attains 2163 values on the 2226 knots and links known to KnotTheory:

In[5]:=

all = Join[AllKnots[], AllLinks[]];

In[6]:=	Length /@ {Union[A2Invariant[#][q]& /@ all], all}
Out[6]=	{2163, 2226}

[Khovanov] ^  M. Khovanov, $sl(3)$ link homology I, arXiv:math.QA/0304375.

[Kuperberg] ^  G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996) 109-151, arXiv:q-alg/9712003.