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We consider diagrams of links in $S^2$ obtained by projection from $S^3$ withthe Hopf map and the minimal crossing number for such diagrams. Knots admittingdiagrams with at most one crossing are classified. Some properties of theseknots are exhibited. In particular, we establish which of these knots arealgebraic and, for such knots, give an answer to a problem posed by Fiedler.

Knots with Hopf crossing number at most one