Quantum knot invariants: Difference between revisions
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<!--QuantumKnotInvariant[Subscript[D,4], Irrep[Subscript[D,4]][{0,1,0,0}]][Knot[5,1]][q] |
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<!--QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q] |
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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 <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products. |
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 <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products. |
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<!--QuantumKnotInvariant[Subscript[A,1], TensorProduct[DirectSum[Irrep[Subscript[A,1]][{1}], Irrep[Subscript[A,1]][{1}]],Irrep[Subscript[A,1]][{2}]]][Knot[5,1]][q] |
<!--QuantumKnotInvariant[Subscript[A,1], TensorProduct[DirectSum[Irrep[Subscript[A,1]][{1}], Irrep[Subscript[A,1]][{1}]],Irrep[Subscript[A,1]][{2}]]][Knot[5,1]][q]--> |
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(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.) |
(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.) |
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Revision as of 13:37, 24 June 2006
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)
The Jones polynomial is a quantum knot invariant -- it corresponds to the 2-dimensional representation of the quantum group , of type . It's in a slightly different normalisation, however.
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.
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 , for example. We can also specify other representations, as direct sums and tensor products.
![{\displaystyle Irrep[A_{3}][{0,1,0}]=\wedge ^{2}(C^{4})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/931bbf0e542aa16358310eddca541c2ac2f19d35)
(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <esc>c*<esc> and <esc>c+<esc>.)
