Knotology is a new type of weaving invented by Heinz Strobl that combines tabby weave and a corrugated form of kagome. The weaving elements are rectangular strips of standard dimensions, creased in a repeating pattern of right triangles. The surfaces of knotology baskets are composed entirely of squares that are diagonally creased or folded. From such simple means great variety is achieved. This paper examines the scope of this new type of weaving. I derive a correspondence between knotology baskets and topological maps, establishing the scope of knotology weaving at a topological level, but leaving unresolved the crucial issue of geometric realizability, i.e., whether such a basket has at least one conformation in three-dimensional space without forbidden folds. Experiments with paper models corresponding to some of the smallest topological maps are reported.