On two crossing numbers of algebraic knots under Hopf fibration

Abstract

We answer a question posed by Fielder concerning two notions of crossing number for algebraic knots K under Hopf fibration. One notion, denoted h(K), is topological and the other, denoted Calg(K), comes from the realization of such knots around complex singularities. We show that Calg(K)−h(K) can be arbitrarily large. We also give an upper bound for h on some families of knots such as torus knots T(2,n), twist knots and their mirror images.