The Great Gazoo asks a fantastic question:
How is what you do as a computational geometer different than what a topologist does? Or is it different?
I tend to think of myself a computational topologist these days. Computational topology sits in the intersection of—or in the wilderness between—several different fields, including classical (usually combinatorial, geometric, and/or algebraic, usually low-dimensional) topology; geometric group theory; graph theory; dynamical systems; algorithms and computational complexity theory; optimization; graphics, visualization, and geometric modeling; robotics; and (of course) discrete and computational geometry.
As I hope the abundance of links in the previous paragraph suggests, computational topology is a big and thriving field with a long history. In my opinion, the field certainly dates back at least to Dehn's 1911 algorithm for testing whether a cycle on a surface is contractible, or perhaps even to Dehn and Heegaard's 1907 algorithmic proof of the surface classification theorem. (It's rather remarkable that Dehn and Heegaard's definition of "surface"—a space constructed by gluing triangles together along common edges—so closely matches current practice in geometric modeling!) It's true that Dehn and Heegaard didn't analyze their algorithms, but the algorithmic character of these early results is inarguable. (Analysis of topological algorithms dates back at least to 1961, when Haken proved that knot triviality can be decided in at most quadruply-exponential time.)
My tiny little corner of this field is closest to computational geometry. The types of questions I work on are influenced by my amateur interest in doing "real" computational things with surfaces, like cut them into simpler pieces, paint them with bricks, or morph one into the other, and topology is a natural tool for those sorts of questions. But I'm not as interested in expanding the frontiers of topology itself, nor am I really qualified to do so. I wouldn't dare call myself a topologist without putting the word "computational" in front. Sure, I can spell "cohomology", and on a good day I can even use it correctly, but I don't really know what it means. On the other hand, I imagine most card-carrying topologists who do computational stuff would be hesitant to call themselves algorithmists, or algorithmeticians, or algorists, or dancers-with-former-vice-presidents, or whatever the word is, without using the qualifier "topological". We might be looking at the same problems, drawing the same graphs on the same donuts, and even ultimately using the same mathematics, but our interests and expertise and intuition are probably different.