I am not a mathematician. I am just trying to understand the difference between algebraic geometry and algebraic topology, categorically.
spatial objects: surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.
University of Oxford Mathematical Institute: “As its name suggests, algebraic geometry deals with curves or surfaces (or more abstract generalisations of these) which can be viewed both as geometric objects and as solutions of algebraic (specifically, polynomial) equations.” See more here.
Wolfram MathWorld: “Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.”
Topology studies the connectivity characteristics of spatial objects. The concept of “connectivity” necessarily involves graph theory. In addition to graph theory many other mathematical techniques can be used to study connectivity. Categorization of the hole structures of spatial objects is big part of this subject.
Wolfram MathWorld: “Algebraic topology is the study of intrinsic qualitative aspects of spatial objects (e.g., surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.) that remain invariant under both-directions continuous one-to-one (homeomorphic) transformations. The discipline of algebraic topology is popularly known as “rubber-sheet geometry” and can also be viewed as the study of disconnectivities. Algebraic topology has a great deal of mathematical machinery for studying different kinds of hole structures, and it gets the prefix “algebraic” since many hole structures are represented best by algebraic objects like groups and rings.”