Before revision:  We use the s-subdivision, path-addition, and node-to-circle expansion operations to show that there are fan-crossing free graphs that are not fan-crossing, k-planar, and k-gap-planar for k >= 1, respectively. A path-addition adds a long path between any two vertices to a graph . An s-subdivision replaces an edge by a path of length s, and a node-to-circle expansion substitutes a vertex by a 3-regular circle, so that each vertex of the circle inherits an edge incident to the original vertex.
After revision:  We use the s-subdivision, path-addition, and node-to-circle expansion and a node-to-circle expansion substitutes a vertex by a 3-regular circle, so that each vertex of the circle inherits an edge incident to the original vertex.