I already identified the letters of the alphabet up to homeomorphism and the useful characteristic was cut-points and their preservation under homeomorphism. As a visual representation you can imagine trying to bend the letters in $R^2$ until they form the new letter you're trying to take in their equivalence class.
Is there a similar invariant I can use for homotopic equivalence? What about another visual way to see the letter's equivalence?
Thanks.
Very roughly speaking, two objects are homotopy equivalent if they both result from "squishing" some larger object. So if you are a visual learner, grab some clay and squish it into the letters!
BTW, according to me, you should get the following sets of homotopy equivalent letters: $$\{A,R,D,O,P,Q\},\{B\},\{C,E,F,G,H,I,J,K,L,M,N,S,T,U,V,W,X,Y,Z\}$$