In a typical (at least from what I have seen) geometric topology course, when the classification of closed & compact surfaces is introduced, what has been done first is to find equivalent surfaces by cutting and gluing operations, as it can be seen in [2].
Later in the course, as I have introduced, to classify surfaces, one uses tools like euler characteristic, homology and homotopy groups, and orient ability. However, while we were using only cutting and gluing operations, I have noticed that with only 3 simple rules, one can achieve the same result as with the cutting and gluing, only in this case we only manipulate the names of the edges given to them. I have explained the whole methodology in this 2 page long paper.
Although I did not put the method into a solid framework, this method helped me a lot while taking the course: whenever it is asked to find the type of a given surface, I was able to give the correct answer with a high confidence, since with this method, it is much more clear what I was doing compared to draw each cut and glue one by one.Plus, it is a much quicker method.
[2] Kinsey, Topology of Surfaces, p85, Lemma 4.15
Question:
First of all, is this an original work ? or I have re-discovered something, or part of something, that has already been established ?
Secondly, if we were to put this method into a formal ground (assuming there is no problem with the validity of the method), how could we do that ? What do we exactly need ?
(Optional)Thirdly, do you think this method is useful in any sense ?
Just a side note:
This was the first article (-ish) paper that I have ever tried to write, so I'm more than happy to see any kind of feedback about the paper, or its methodology.