Quandle cocycle invariants for spatial graphs and knotted handlebodies
type: Item
instance of: scholarly article
zbMATH Open document ID: 1245.57014
DOI: 10.4153/CJM-2011-035-0
author: Atsushi Ishii
author: Masahide Iwakiri
published in: Canadian Journal of Mathematics
publication date: 2012-04-12T00:00:00Z
full work available at URL: CJM-2011-035-0
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6023028
zbMATH Keywords: spatial graph
zbMATH Keywords: knotted handlebody
zbMATH Keywords: flow
zbMATH Keywords: quandle cocycle invariant
MaRDI profile type: Publication
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title: Quandle cocycle invariants for spatial graphs and knotted handlebodies
review text: The main result of this paper is the construction of a quandle cocycle invariant for spatial graphs and handlebody links. To define these invariants, the authors introduce flowed spatial graphs. An \(A\)-flow on a spatial graph \(L\) is an assignment of an element of an abelian group \(A\) to each orientation of each edge of \(L\) such that: (1) for each edge, the elements of \(A\) assigned to the two orientations are inverses of each other; (2) at each vertex, the total weight of the incoming edges equals the total weight of the outgoing edges. By considering quandle colourings for flowed spatial graphs, the authors construct quandle cocycle invariants for flowed spatial graphs. An invariant of spatial graphs is then obtained as its set of values over all \(A\)-flows of the spatial graph. As this invariant does not detect the contraction move on spatial graphs, it follows that it is also an invariant of handlebody links. Note that the invariant constructed in this paper generalizes a weight sum invariant from [\textit{A. Ishii}, Algebr. Geom. Topol. 8, No. 3, 1403--1418 (2008; Zbl 1151.57007)].
reviewed by: Iain Moffatt
description: scientific article; zbMATH DE number 6023028