You need to seek care from a reproductive surgeon or reproductive endocrinologist to determine the best option for you. Your dr. Will need to know the specific technique used to cut your fallopian tubes (ideally they would like to review the operation report), your general health and some reproductive fertility
Sterilization reversal is one of the few reproductive surgeries that offers a high success rate. The surgery requires an operative microscope in order to remove the scar tissue around the ends of the tube that was cut and to sew them together. Not every surgeon is trained to perform microsurgery therefore you should seek out a surgeon with experience.
The success rate after tubal reversal surgery is determined by your age (always important for fertility), the length of tube after reversal (less than 4 cm is very poor), the type of sterilization (cautery is worse than loop or clip, for example), how many children
do you want (if you want more than 1 then surgery may be better than ivf) and the amount of time you can invest to achieve pregnancy (it can take 12-24 months to get the cumulative pregnancy rates quoted). You also need to have a normal male partner and good egg quality.
Ivf does not require that your fallopian tubes be open. Ivf is a good option for situations in which the sterilization surgery removed a large amount of fallopian tube, the woman's age is older than 35-38 years or or there is also a male factor involved such that the sperm
has a low potential for fertilizing the egg.
In order to choose between these 2 treatment options, you need to review the issues i mentioned with your fertility specialist. We often cannot reliably determine whether surgery or ivf would be the best option. You will get differing opions on the best way to proceed.
We also need to look at the costs of both surgery and ivf with respect to the absolute costs and what insurance
will pay for. Sterilization reversal surgery can cost almost as much as an ivf cycle because the surgery requires anywhere from 2-4 hours to perform in the operating room is very expensive part of the hospital. Ivf can be expensive too but many ivf centers offer refund guarantees, so that if you do not get a baby, then you will get most of your money back (if you do not have insurance for the ivf).
In conclusion, you probably have 2 good options for achieving another child. I predict that when you meet with your physician, at the answer to some of these questions will become more obvious. Good luck.