- journal articles metadata
European Mathematical Society Publishing House
2024-03-19 07:14:20
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=QT&vol=5&iss=2&update_since=2024-03-19
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
5
2014
2
Integrality of Kauffman brackets of trivalent graphs
Francesco
Costantino
Université Paul Sabatier, TOULOUSE, FRANCE
Spin networks, quantum invariants, Jones polynomials, trivalent graphs
We show that Kauffman brackets of colored framed graphs (also known as quantum spin networks) can be renormalized to a Laurent polynomial with integer coefficients by multiplying it by a coefficient which is a product of quantum factorials depending only on the abstract combinatorial structure of the graph. Then we compare the shadow-state sums and the state-sums based on $R$-matrices and Clebsch–Gordan symbols, reprove their equivalence and comment on the integrality of the weight of the states. We also provide short proofs of most of the standard identities satisfied by quantum $6j$-symbols of $U_q(\operatorname{sl}_2)$.
Manifolds and cell complexes
General
143
184
10.4171/QT/49
http://www.ems-ph.org/doi/10.4171/QT/49
An untwisted cube of resolutions for knot Floer homology
Ciprian
Manolescu
UCLA, LOS ANGELES, UNITED STATES
Knots, knot Floer homology, Khovanov–Rozansky homology, spectral sequences
Ozsváth and Szabó gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the $t=1$ specialization of their construction. The associated spectral sequence converges to knot Floer homology, and we conjecture that its $E_1$ page is isomorphic to the HOMFLY-PT chain complex of Khovanov and Rozansky. At the level of each $E_1$ summand, this conjecture can be stated in terms of an isomorphism between certain Tor groups. As evidence for the conjecture, we prove that such an isomorphism exists in degree zero.
Manifolds and cell complexes
General
185
223
10.4171/QT/50
http://www.ems-ph.org/doi/10.4171/QT/50
Irreducible factors of modular representations of mapping class groups arising in Integral TQFT
Patrick
Gilmer
Louisiana State University, BATON ROUGE, UNITED STATES
Gregor
Masbaum
Université Paris 7, Denis Diderot, PARIS CEDEX 05, FRANCE
Lollipop basis, Topological Quantum Field Theory, skein theory, symplectic group, Verlinde formula
We find decomposition series of length at most two for modular representations in positive characteristic of mapping class groups of surfaces induced by an integral version of the Witten–Reshetikhin–Turaev SO(3)-TQFT at the $p$-th root of unity, where $p$ is an odd prime. The dimensions of the irreducible factors are given by Verlinde-type formulas.
Manifolds and cell complexes
General
225
258
10.4171/QT/51
http://www.ems-ph.org/doi/10.4171/QT/51