Somewhat related to this.
I also understand this might be wishful thinking but I'm looking for a textbook/notes/video giving a simple/minimal framework of the topology of surfaces for nonspecialists more specifically graph theory/discrete optimization. Videos would be especially nice. Something like here are all surfaces up to... and a list of theorems you will likely use a kind of topological toolbox for non-topologists so to speak. Proofs are not really important here.
e.g. making sense of and proving Euler's formula for a torus or higher genus surface. Classifying the types of cycles/curves of graphs embedded on a torus or higher genus. Given 2 cycles of the same "type" on a torus that intersect at two points and letting P1,P2 and Q1,Q2 be the two paths between the two points on each cycle, either P1,Q1 divide the torus into two regions or P1,Q2 do, further if P1,Q1 do then so do P2,Q2 and generalizations of this to higher genus.
Another particular thing I would like to prove is for a circle D on a torus and 5 loops (closed curves) that intersect D at 5 distinct points non of which divide the torus into two regions some 2 of the 5 loops intersect in at least 2 points.