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Mathematical Notes
Article . 1992 . Peer-reviewed
License: Springer Nature TDM
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 1992
Data sources: zbMATH Open
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Novikov homologies in Knot theory

Novikov homologies in knot theory
Authors: Lazarev, A. Yu.;

Novikov homologies in Knot theory

Abstract

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 \(H_ 1(\widetilde{M})\).

Related Organizations
Keywords

Critical points and critical submanifolds in differential topology, knot in \(S^ 3\), Alexander module, Knots and links in the \(3\)-sphere, Novikov complex, knot space, Novikov homologies

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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