2013/07/01 by Micah W. Chrisman, Chrisman, Micah W., Vassily O. Manturov +1 · 1 citation
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.1307.0538
Improved exposition; added references
arxiv created 2013/08/09 · arxiv updated 2013/08/14
We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let K be a knot and J a knot in the complement of K with lk(J,K)=0. Suppose there is covering space πJ: Σ× (0,1) → 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 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.