Fibered knots and virtual knots (Q2872204)

A knot \(J\) in \(S^3\) is fibered if the knot complement \(\overline{S^3-V(J)}\) fibers locally trivially over \(S^1\), where \(V(J)\) denotes a regular neighborhood of \(J\). Let \(J\) be a fibered knot in \(S^3\) with a given fibration \(p : \overline{S^3-V(J)} \to S^1\) and \(\Sigma=p^{-1}(z_0)\) for some \(z_0 \in S^1\). Let \(K\) be a knot in \(\overline{S^3-V(J)}\) such that \(\text{lk}(J,K)=0\). There is an orientation preserving homeomorphism from the infinite cyclic cover \(M_J\) of the complement of \(J\) to \(\Sigma\times (0,1)\). Let \(\pi_J : M_J \to \overline{S^3-V(J)}\) be the covering map. Let \(K'\) be a knot in \(M_J\) such that \(\pi_J(K')=K\). In this case, the lift \(K'\) of \(K\) can be considered as a knot in \(\Sigma\times [0,1]\) via the inclusion map \(\Sigma\times(0,1) \to \Sigma\times [0,1]\). Let \(\widehat{K}\) denote the virtual knot representing the stability class of \(K'\) in \(\Sigma\times [0,1]\).NEWLINENEWLINEIn the paper under review, the authors call a virtual knot \(\widehat{K}\) obtained in this way a \textit{virtual cover} of \(K\) relative to the triple \((J, p, \Sigma)\). To describe virtual covers, the authors introduce the notion of a \textit{special Seifert form} of the link \(J \cup K\), which consists of a Seifert surface \(\Sigma\) of \(J\) and finitely many \(3\)-balls in \(S^3\) satisfying certain conditions with \(K\). When knots are given in a special Seifert form, virtual covers are essentially unique up to equivalence of virtual knots. It is proved that every virtual knot is a virtual cover of some knot and that if \(\widehat{K}_1\) and \(\widehat{K}_2\) are virtual covers of two knots \(K_1\) and \(K_2\), respectively, relative to \((J,p,\Sigma)\) and if there is a smooth ambient isotopy taking \(K_1\) to \(K_2\) with fixed \(V(J)\), then \(\widehat{K}_1\) and \(\widehat{K}_2\) are equivalent as virtual knots.NEWLINENEWLINEAlso, the authors investigate what can be said about a classical knot from its virtual covers. For example, they prove that there is a pair of figure eight knots \(K_1\) and \(K_2\) in \(S^3\) and a trefoil knot \(T\) in \(\overline{S^3-(V(K_1)\cup V(K_2))}\) such that there is no ambient isotopy taking \(K_1\) to \(K_2\) fixing \(T\), and also prove a similar result for a knot \(K\) and its inverse \(K^{-1}\).NEWLINENEWLINE These results are established by applying parity arguments to virtual covers and are used to present some applications to classical knots. Accordingly, virtual covers provide a way to study classical knots with the methods of virtual knot theory. This is distinct from the usual way in which virtual knots are studied with the methods of classical knot theory. Typically, classical knots are considered as a subset of the set of virtual knots. The alternative approach advocated in the present paper allows us to exploit both the non-trivial ambient topology and the intrinsic combinatorial properties of virtual knots.