This experimental genus 6 surface adds a handle to Alan Schoen’s I6 surface.
It keeps the figure 8 pattern of squares but replaces the Callahan-Hoffman-Meeks pattern with a higher genus variant. The other limit of this 1-parameter family become singly periodic Scherk surfaces with different frequencies.
The divisor of the square Gauss map on the quotient torus by the order two rotation about a vertical axis uses a completely unexplored pattern with point on the imaginary axis. There are likely many other surfaces like this.