I have read all the replies, and agree with most of the info.
Your exposure from a few gaskets' cutting is minimal, negligible. The main problem with asbestos material is its "friability" which means its capacity to become airborne fibers that you can breathe in, entrench in your lungs, and, maybe eventually develop into mesothelioma. Or not.
Small chance. Some asbestos material has a higher probability of becoming friable than others. Gasket material is pretty stable by comparison to insulation wrap, etc. Personally, I would not lose any sleep over it. If you've ANY concerns, invest in a respirator with appropriate filters and work carefully. But...really, the worst case of lung disease linked to asbestos I ever read of was the wife of a fellow who worked with it for DECADES...and his wife did his laundry for the same period, SHAKING IT ALL OUT AS BEST SHE COULD BEFORE LAUNDERING.
He was okay. She wasn't. Suggests other variables too, no?
In all honesty, I have cut/replaced/tooled, etc. asbestos roof and/or siding tiles for years prior to knowing the "issues." As have lots of others in the 50-60 year old range (or more), without ramifications. Not to say that in the last 10-25 years of life they may not manifest.
But then, I wasn't just cutting gaskets.