Kirby Elements and Quantum Invariants
نویسنده
چکیده
We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize a set of Kirby elements (in terms of the structural morphisms of a Hopf algebra in C) which is sufficiently large to recover the quantum invariants of 3-manifolds of Reshetikhin-Turaev, of Hennings-Kauffman-Radford, and of Lyubashenko. The cases of a semisimple ribbon category and of a category of representations are explored in details.
منابع مشابه
Quantum Invariants at the Sixth Root of Unity
A general topological formula is given for the 517(2) quantum invariant of a 3-manifold M at the sixth root of unity. It is expressed in terms of the homology, Witt invariants and signature defects of the various 2-fold covers of M, and thus ties in with basic 4-dimensional invariants. A discussion of the range of values of these quantum invariants is included, and explicit evaluations are made...
متن کاملLocal surgery formulas for quantum invariants and the Arf invariant
A formula for the Arf invariant of a link is given in terms of the singularities of an immersed surface bounded by the link. This is applied to study the computational complexity of quantum invariants of 3–manifolds. AMS Classification 57M27; 68Q15
متن کاملInvariants of Spin Three-manifolds from Chern-simons Theory and Finite-dimensional Hopf Algebras
A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon ∗categories which possess odd degenerate objects. This case includes the quantum groups situations corresponding to the half-integer level Chern-Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten...
متن کاملIntegrality and Symmetry of Quantum Link Invariants
0. Introduction. Quantum invariants of framed links whose components are colored by modules of a simple Lie algebra g are Laurent polynomials in v1/D (with integer coefficients), where v is the quantum parameter and D an integer depending on g. We show that quantum invariants, with a suitable normalization, are Laurent polynomials in v2. We also establish two symmetry properties of quantum link...
متن کاملQuantum Supergroups And Topological Invariants of Three - Manifolds
The Reshetikhin Turaeve approach to topological invariants of three manifolds is generalized to quantum supergroups. A general method for constructing three manifold invariants is developed, which requires only the study of the eigenvalues of certain central elements of the quantum supergroup in irreducible representations. To illustrate how the method works, Uq(gl(2|1)) at odd roots of unity i...
متن کامل