isotopologically

Isotopologically is an adverb that describes something possessing the same topological properties. Topology is a branch of mathematics concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, and bending, but not tearing or gluing. Two objects are considered topologically equivalent if one can be continuously transformed into the other. Therefore, if two objects are isotopological, they can be deformed into each other without altering their fundamental topological characteristics. This means they would have the same number of holes, be connected in the same way, and exhibit similar general shapes in a topological sense, even if their geometric appearances are quite different. For example, a coffee mug and a donut are considered isotopological because they both have one hole and can be continuously deformed into one another. The concept is fundamental in fields like knot theory, where the isotopology of knots determines whether they can be deformed into one another without passing through themselves. In essence, to describe something as isotopologically equivalent is to state that their underlying topological structures are identical.