I don't know how to
cut bottles into something that looks like that, but you could always try
casting the rings from a wax model.
Make whatever shape you want from wax, cover it in a mould mix (except for a hole with a funnel for pouring the glass in), wait for the mould to dry and burn the wax out. Then crush the glass bottles (wear a mask) and pour the crushed glass into the mould. Put the whole thing into the kiln long enough to melt the glass to the shape (depending on which glass you use and which mould mix the temp could differ). You may need to wait for it to cool and do it again if your glass fragments were big-ish.
They sell special tools for crushing glass, but a thick bag and a hammer works ok. Again, wear a mask, glass dust is bad for the lungs. As for the mould mix, google it. There are ready made moulds availiable that don't need the wax step at all, but if you want to design your own with the wax method I described you'll have to look a bit harder for a mould mix that'll work for you. It's worth it though, you can make all sorts of artsy things beyong rings with it depending on the size of your kiln.
There's a wikipedia article
here. It's the process called kiln casting.
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