Novikov homologies in knot theory
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 \(H_ 1(\widetilde{M})\).
scientific article; zbMATH DE number 484275