Do you have a tool which is sold here in Germany by the name of Dremel - looks like it is available worldwide: Dremel?

And if you have it, do you have a small "diamond cutting wheel" (or whatever the correct translation for Diamant-Trennscheibe is ;-))

If so, empty the bottle.

Get yourself protective glasses to prevent tiny pieces of glass from jumping into your eyes!

Now, the complicated part:
get yourself (with Dremel and bottle) to a wash basin.
Get the water dripping fast or a very thin continuous stream.
Firmly hold the bottle in such a way that the water is dripping/running on the position where you are going to cut.

First do a cut all around the bottle, but only a fraction of the glass' thickness. Then work in deeper and deeper.

Use high speed, almost no pressure!
And always keep the water dripping on the cut!