Minimal cobordisms between thin and thick torus knots

We determine the locally flat cobordism distance between torus knots with small and large braid index, up to high precision. Here small means 2, 3, 4, or 6. As an application, we derive a surprising fact about torus knots that are met along the way of almost minimal cobordisms between two-stranded torus knots and the trivial knot.