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I am trying to understand the operation "gluing straight segments" (or "gluing marks") in order to construct a surface (English is not my mother tongue so it makes it a little bit more difficult for me) .

To be more specific I am reading these papers based on the infinite loch ness monster:

https://arxiv.org/pdf/1701.07151.pdf (pg. 4)

https://arxiv.org/pdf/1603.00503.pdf (pg. 10)

I am a little bit confused about "cutting along" and "gluing segments back". Any help for understanding the topic or material for reference would be really useful for me.

I am sorry if my question seems obvious but I find differential geometry a bit difficult.

Thank you in advance.

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It's loose language in topology that you see for example in the classification of compact topological surfaces (usually text have a lot of pictures). Cutting along a curve means removing the curve from your surface (and possibly considering only one of the connected components), while gluing or pasting is a synonymous of identifying, so that gluing two surfaces along a curve (one curve on each surface) means considering the disjoint union and identify the two curves.

For example you can cut off one disk from a ball and one disk from a torus, then if you glue them together you end up with a torus!

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    http://people.maths.ox.ac.uk/hitchin/hitchinnotes/Geometry_of_surfaces/Chapter_1_Topology.pdf at the end of p. 13 there is an example where they do the 'connected sum' of two tori, they cut off a disk from each of them and glue together; http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Teo.pdf here at end of p. 10 you have another example. – 2017-02-19
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    I really thank you for your answer. I appreciate it! So, if I understood right, the logic in the papers I mentioned is : "Cutting along" the straight segments (which means removing the segments from the surface) and then create a half-torus by gluing them back in order to get the desired new loch ness surface we want.. – 2017-02-19
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    Yes and no, essentially you take off two _open_ disks, so that the boundary is still in your space (this is important); then you have two holes in E^2, and identify the boundaries (two closed curves, each of them isomorphic to $S^1$), so you get that torus pictured by one point (I'm referring to the first paper). – 2017-02-19
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    You make this in order to create the torus pictured by one point . But if you want to create the picture with the handle (the picture at the left of the torus) you have to follow the operation with the straight segments.. Right ? – 2017-02-19
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    The plane with one handle and the pictured torus are the same space, just as the whole plane and an open disk are(and a sphere minus one point); the loch ness monster is a plane with infinite handles. If you look at picture 8, when he says to cut along a segment he really means to take off the interior of the two circles, you need the boundary if you want to glue something there :) – 2017-02-19
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    Ι think I got the feeling, I have to familiarise though. I really thank you for your time and patience! You helped me a lot ! – 2017-02-19