Novikov homologies in knot theory
type: Item
instance of: scholarly article
title: Novikov homologies in knot theory
zbMATH Open document ID: 0796.57003
DOI: 10.1007/BF01206388
author: A. Yu. Lazarev
published in: Mathematical Notes
publication date: 1994-02-08T00:00:00Z
review text: Let \(K\) be a knot in \(S^ 3\). To define Novikov homologies of the knot space \(M = S^ 3 - K\), one should take a closed 1-form \(\omega\) on \(M\) such that \(\omega\) can be lifted to the differential of a Morse function \(f:\widetilde M \to R\) where \(\widetilde{M}\) is an infinite cyclic covering of \(M\). As in usual Morse theory the incidence coefficients between critical points of neighboring indices may be determined and the Novikov complex \(C_ 0 \leftarrow C_ 1 \leftarrow C_ 2 \leftarrow C_ 3\) may be constructed. The homologies of this complex do not depend on the form \(\omega\). It follows from the definition that Novikov homologies of \(M\) should be closely related to the Alexander module \(H_ 1(\widetilde{M})\). The author makes this statement precise by showing that Novikov homologies of \(M\) can be explicitly expressed via polynomial invariants of the module \(H_ 1(\widetilde{M})\).
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57R70
zbMATH DE Number: 484275
zbMATH Keywords: knot in \(S^ 3\)
zbMATH Keywords: Novikov homologies
zbMATH Keywords: knot space
zbMATH Keywords: Novikov complex
zbMATH Keywords: Alexander module
reviewed by: Sergei Vladimirovich Matveev
MaRDI profile type: Publication
cites work: Q3791912
cites work: Topological properties of extremal boundaries of convex compact sets in \(\ell ^ 2\)
cites work: Q5528469
full work available at URL: https://doi.org/10.1007/bf01206388
OpenAlex ID: W1986053970
description: scientific article; zbMATH DE number 484275