Topologically adaptable snakes, or simply T-snakes, are a standard tool for automatically identifying multiple segments in an image. This work introduces a novel approach for controlling the topology of a T-snake. It focuses on the loops formed by the so-called projected curve which is obtained at every stage of the snake evolution. The idea is to make that curve the image of a piecewise linear mapping of an adequate class. Then, with the help of an additional structure-the Loop-Tree-it is possible to decide in O(1) time whether the region enclosed by each loop has already been explored by the snake. This makes it possible to construct an enhanced algorithm for evolving T-snakes whose performance is assessed by means of statistics and examples.