Quantum knot invariants

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...

Quantum knot invariants are calculated using Scott's QuantumGroups` Mathematica package. There is a subversion repository, and hopefully soon a documented release.

The quantum knots invariants in the Knot Atlas are normalised so the invariant of the unknot is the quantum dimension of the chosen representation.

Here we demonstrate the calculation of some quantum knot invariants. (For In[1] see Setup)

In[1]:= ?QuantumKnotInvariant QuantumKnotInvariant[Γ, V][K][q] calculates the quantum knot invariant of the knot K in the representation V of the quantum group Γ. This relies on the QuantumGroups` package, and you should look there for details of how Γ and V may be specified. Examples: QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1, 0}]][Knot[5, 2]][q] QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1, 0}]⊕Irrep[Subscript[G,2]][{0, 1}]][Knot[5, 2]][q]

In[2]:= QuantumKnotInvariant::about Quantum knot invariants are calculated using the mathematica package QuantumGroups`, written by Scott Morrison 2003-2006.

The Jones polynomial is a quantum knot invariant -- it corresponds to the 2-dimensional representation of the quantum group $SU(2)$, of type $A_{1}$. It's in a slightly different normalisation, however.

In[3]:=	{Jones[Knot[6,1]][q], QuantumKnotInvariant[Subscript[A,1], Irrep[Subscript[A,1]][{1}]][Knot[6,1]][q]}
Out[3]=	-4 -3 -2 2 2 -5 1 3 9 {2 + q - q + q - - - q + q , q + - - q + q } q q

In[4]:=	Simplify[(q+q^(-1))Jones[Knot[6,1]][q^(-2)] - QuantumKnotInvariant[Subscript[A,1], Irrep[Subscript[A,1]][{1}]][Knot[6,1]][q]]
Out[4]=	0

The QuantumGroups` package is capable of calculating quantum knot invariants in arbitrary representations of any quantum group. Quantum groups are specified by their Dynkin diagram. (In practice, you'll find that memory and CPU time are quite limiting.) Here are some examples.

In[5]:=	QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1,1}]][Knot[7,3]][q]
Out[5]=	-76 2 2 2 2 2 6 3 2 4 -52 q + --- - --- + --- - --- + --- - --- + --- - --- + --- - q + 72 70 68 66 64 62 60 58 56 q q q q q q q q q 4 8 4 11 4 8 6 4 2 5 2 --- - --- + --- - --- + --- - --- + --- - --- + --- + --- - --- + 50 48 46 44 42 40 38 36 34 28 26 q q q q q q q q q q q 8 6 2 2 -12 --- + --- - --- + --- + q 24 20 18 16 q q q q

In[6]:=	QuantumKnotInvariant[Subscript[A,3], Irrep[Subscript[A,3]][{0,1,0}]][Knot[8,2]][q]
Out[6]=	4 6 8 10 12 18 20 22 24 1 + 2 q + 2 q + 2 q + 3 q + 2 q - 2 q - 3 q - q - q - 26 36 38 42 44 46 48 50 52 54 56 2 q + q + 2 q + q + q - q - q + q - q - q + q

In[7]:=	QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1,0}]][Knot[5,2]][q]
Out[7]=	10 14 20 24 34 40 44 46 48 50 54 q + q + q + 2 q + q + q + q + 2 q - q + 2 q + q + 56 60 64 66 68 72 74 76 78 82 84 q + q + q - q - q - q - q - q - q - q - q - 88 90 92 94 96 100 q + q - q - q + q + q

In[8]:=	QuantumKnotInvariant[Subscript[D,4], Irrep[Subscript[D,4]][{0,1,0,0}]][Knot[5,1]][q]
Out[8]=	30 32 34 36 38 40 42 44 q + q + 4 q + 6 q + 10 q + 14 q + 19 q + 19 q + 46 48 50 52 56 58 60 22 q + 19 q + 14 q + 7 q - 11 q - 17 q - 24 q - 62 64 66 68 70 72 74 76 27 q - 27 q - 24 q - 18 q - 12 q - 4 q + 3 q + 9 q + 78 80 82 84 86 88 90 92 11 q + 14 q + 11 q + 9 q + 6 q + 2 q - q - q - 94 96 98 100 120 3 q - 3 q - q - q + q

In[9]:=	qDimension[Subscript[B,2]][Irrep[Subscript[B,2]][{0,1}]] /. q->1
Out[9]=	4

In[10]:=	BR[Knot[7,4]]
Out[10]=	KnotTheory`BR[4, {1, 1, 2, -1, 2, 2, 3, -2, 3}]

In[11]:=	QuantumKnotInvariant[Subscript[B,2], Irrep[Subscript[B,2]][{0,1}]][Knot[7,4]][q]
Out[11]=	-54 2 -48 -46 -44 -42 -40 -36 2 2 -q - --- + q - q + q - q + q - q + --- - --- + 50 34 32 q q q 2 2 3 2 2 -20 -18 -16 2 2 -8 --- - --- + --- - --- + --- + q + q + q + --- - --- + q - 30 28 26 24 22 12 10 q q q q q q q -6 -4 q + q

The representations in the examples above are all irreducibles, specified by their highest weight. (The QuantumGroups` package represents weights by the coordinates in the fundamental basis. Thus $Irrep[A_{3}][{0,1,0}]=\wedge ^{2}(C^{4})$, for example. We can also specify other representations, as direct sums and tensor products.