Fibred knots and virtual knots

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let K be a knot and J be a knot in the complement of K with lk(J, K) = 0. Suppose there is covering space $\piJ : \Sigma \times (0, 1) \to \overline{S3 \backslash V(J)}$, where V(J) is a regular neighborhood of J satisfying V(J) ∩ im(K) = ∅ and Σ is a connected compact orientable 2-manifold. Let K′ be a knot in Σ × (0, 1) such that πJ(K′) = K. Then K′ stabilizes to a virtual knot $\hat{K}$, called a virtual cover of K relative to J. We investigate what can be said about a classical knot from its virtual covers in the case that J is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established. © 2013 World Scientific Publishing Company.