Pseudo diagrams of knots, links and spatial graphs

Abstract

A pseudo diagram of a spatial graph is a spatial graph projection on the $2$-sphere with over/under information at some of the double points. We introduce the trivializing (resp. knotting) number of a spatial graph projection by using its pseudo diagrams as the minimum number of the crossings whose over/under information lead the triviality (resp. nontriviality) of the spatial graph. We determine the set of non-negative integers which can be realized by the trivializing (resp. knotting) numbers of knot and link projections, and characterize the projections which have a specific value of the trivializing (resp. knotting) number.