The PSU(3) invariant of the Poincaré homology sphere
type: Item
instance of: scholarly article
title: The PSU(3) invariant of the Poincaré homology sphere
zbMATH Open document ID: 1022.57005
DOI: 10.1016/S0166-8641(02)00057-3
published in: Topology and its Applications
publication date: 2003-04-28T00:00:00Z
review text: In the paper under discussion the author uses the \(R\)-matrix presentation of link and 3-manifold invariants to compute the \(PSU(3)\) pertubative invariant for the Poincaré homology sphere. The computation is based on a certain description of the \(\mathfrak{sl}_{3}\) colored Jones polynomial of a link and of the \(PSU(3)\) invariant of 3-manifolds (see also [\textit{Reshetikhin} and \textit{Turaev}, Commun. Math. Phys. 127, 1-26 (1990; Zbl 0768.57003)] and [\textit{T. Le}, Topology 39, 813-849 (2000; Zbl 0955.57010)]). From this description the author obtains a certain form for the colored Jones polynomial of the trefoil knot. This is applied to give an algorithm to compute the coefficients in the \(PSU(3)\) pertubative invariant for the Poincaré homology sphere.
reviewed by: Michael Heusener
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 1901932
zbMATH Keywords: Ohtsuki series
zbMATH Keywords: pertubative invariant
zbMATH Keywords: quantum topology
zbMATH Keywords: quantum groups
zbMATH Keywords: Poincaré homology sphere
author: Ruth Lawrence
describes a project that uses: Mathematica
MaRDI profile type: Publication
cites work: Hecke algebras, modular categories and 3-manifolds quantum invariants
cites work: q-Weyl group and a multiplicative formula for universal R-matrices
cites work: Symmetry of Witten’s 3-manifold invariants for sl(n, C)
cites work: Q4892004
cites work: On perturbative \(PSU(n)\) invariants of rational homology 3-spheres
cites work: Some applications of the quantum Weyl groups
cites work: A simple proof of integrality of quantum invariants at prime roots of unity
cites work: Integral modular categories and integrality of quantum invariants at roots of unity of prime order
cites work: Quantum SU(3) Invariant of 3-Manifolds via Linear Skein Theory
cites work: Ribbon graphs and their invariants derived from quantum groups
cites work: An analogue of P.B.W. theorem and the universal R-matrix for \(U_ h\mathfrak{sl}(N+1)\)
cites work: ON QUANTUM PSU(n)-INVARIANTS FOR LENS SPACES
cites work: On Quantum PSU(n) Invariants for Seifert Manifolds
cites work: THE PSU(N) INVARIANTS OF 3-MANIFOLDS ARE ALGEBRAIC INTEGERS
cites work: QUANTUM INVARIANTS OF 3-MANIFOLDS ASSOCIATED WITH CLASSICAL SIMPLE LIE ALGEBRAS
cites work: Quantum field theory and the Jones polynomial
description: scientific article; zbMATH DE number 1901932