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.
After revision:  We use the subdivision, 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.