Surface Distance on Knots

Let K be the set of all oriented knots up to isotopy. There are several distance functions on K, such as the Gordian distance and the ♯-Gordian distance [9]. In general, given an ‘unknotting operation’, that is, a method to untie every knot, such as the ∆-unknotting operation introduced by S. Matveev [8] (see also [10]), one can define the corresponding distance function as the minimal number of the ‘unknotting operations’ needed to deform one knot to the other. The corresponding ‘unknotting number’ is the distance to the unknot. For example, crossing change is the most familiar ‘unknotting operation’, and it defines the Gordian distance and the (ordinary) unknotting number. In this paper we introduce yet another distance function by using surfaces bounding given two knots. We also study its relations with other distance functions.