Scrub potatoes well and add to a large pot, covering with water by at least 2 inches. Add a pinch of salt to the water and bring to a boil over high heat.
Mix until you have a sauce (add more oil or lemon as needed). Set aside. Once potatoes are finished, drain and let cool for a few minutes before handling.
When you can handle the potatoes, start cutting into cubes. Toss them into your serving bowl. Mix potatoes with the pesto and add sliced radish and arugula.