An intersection homology invariant for knots in a rational homology 3- sphere
type: Item
instance of: scholarly article
title: An intersection homology invariant for knots in a rational homology 3- sphere
zbMATH Open document ID: 0822.57008
DOI: 10.1016/0040-9383(94)90039-6
published in: Topology
publication date: 1994-04-17T00:00:00Z
review text: To a homologically trivial knot \(K\) in a rational homology 3-sphere the authors associate a family \(\{\lambda_{n, d}, p_{n,d} (t)\}_{(n, d) \in \mathbb{N}^* \times \mathbb{N}}\) of computable homological invariants. They generalize the Casson invariant of knots. Some explicit calculations are presented. The case where \(K\) is a fibered knot is treated in detail. Invariants \(\lambda_{(n,d)}\) can be computed using the intersection Lefschetz number of the monodromy action of the moduli space of semistable holomorphic bundles of rank \(n\) and degree \(d\) and fixed determinant over a compact Riemann surface. Reasons to appeal to intersection homology are: (1) for some cases (\(n\) and \(d\) not relatively prime) this moduli space is typically singular and stratified spaces (in the sense of Goresky-MacPherson) appear instead of manifolds and (2) these invariants \(\lambda_{(n,d)}\) can be computed using intersection numbers. For the fibered case, the perversity used is the middle perversity; in the general case the situation is more complicated and stratum dependent perversities appear (instead of filtration dependent perversities).
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 55N33
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 527442
zbMATH Keywords: homologically trivial knot
zbMATH Keywords: rational homology 3-sphere
zbMATH Keywords: Casson invariant
zbMATH Keywords: fibered knot
zbMATH Keywords: intersection Lefschetz number
zbMATH Keywords: moduli space of semistable holomorphic bundles
zbMATH Keywords: stratum dependent perversities
author: Charles Frohman
author: Andrew Nicas
reviewed by: Martintxo Saralegi-Aranguren
MaRDI profile type: Publication
full work available at URL: 0040-9383(94)90039-6
OpenAlex ID: W2020898489
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description: scientific article; zbMATH DE number 527442