Quantum knot invariants

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.