My PhD research was about something called knot theory. The basic question in this topic can be explained, essentially, as “if you loop up a piece of elasticated string and glue the ends together, can you tell if it’s fundamentally the same or different from one that I make?”
It’s not hard to visualise, I think, but wow is it hard to actually get answers!
You need 1-variable polynomials of unoriented knots and 2-variable polynomials of oriented links. You need about 150 years of theory and new people all the time researching niche cases. You need invariants, polynomials, equations and Reidemeister moves. You could use arc invariants, stacked k-tangles and skein relations. You need to know your homology from your homotopy and you need to know what, when and why you use all these things.
Sorry. You don’t need to know all this. I did!
It was a lot to keep straight at times. And not all of it was directly relevant to my research, my thesis or my viva. As I was writing my thesis and then preparing for my viva, it helped me to untangle all of these terms and be sure of what they meant, what I needed them for and – in some cases – what I didn’t need.
You don’t need to know all of my stuff. You need to know your stuff for your viva. So take some time to think through what you need to be clear about. What do you need to check? What basics do you need to go over one more time? And what can you file away as probably important?
You need to know your stuff. So make sure you do.