Fiedler type combinatorial formulas for generalized Fiedler type invariants of knots in \(M^2\times R^1\)
type: Item
instance of: scholarly article
title: Fiedler type combinatorial formulas for generalized Fiedler type invariants of knots in \(M^2\times R^1\)
zbMATH Open document ID: 1207.57018
author: S. A. Grishanov
author: V. A. Vassiliev
published in: Topology and its Applications
publication date: 2009-08-20T00:00:00Z
review text: By using Gauss diagrams for knots and links, \textit{M. Polyak} and \textit{O. Viro} developed combinatorial formulas for finite type invariants [Int. Math. Res. Not. 1994, No.~11, 445--453 (1994; Zbl 0851.57010)]. Fiedler introduced and studied Gauss diagram invariants of low degree for knots in a trivial surface bundle \(M^2\times\mathbb R\). These invariants are parametrized by non-trivial homology classes in \(H_1(M^2\times\mathbb R)\) [\textit{T. Fiedler}, Gauss diagram invariants for knots and links. Mathematics and its Applications (Dordrecht). 532. Dordrecht: Kluwer Academic Publishers (2002; Zbl 1009.57001)]. In the paper under discussion the authors construct Gauss diagram invariants of higher order indexed by ordered collections of non-trivial free loop homotopy classes of \(M^2\times\mathbb R\). The authors show how their invariants are related in low degree to Fiedler's invariants. They also give examples of knots in the torus bundle \(T^2\times\mathbb R\) which are not separated by any degree 1 and 2 invariants but can be separated by their new invariants.
reviewed by: Michael Heusener
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5597509
zbMATH Keywords: knot invariant
zbMATH Keywords: weight system
zbMATH Keywords: Gauss diagram
zbMATH Keywords: combinatorial formula
MaRDI profile type: Publication
full work available at URL: j.topol.2009.06.002
OpenAlex ID: W1979844248
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DOI: 10.1016/J.TOPOL.2009.06.002
description: scientific article; zbMATH DE number 5597509