Must graphs in a mathematical PhD thesis be drawn nicely? I am writing my PhD thesis in mathematics and there are several graphs to be included (on knots, braids, and other related objects). I know that there are several software available for drawing graphs like this, but I am not experienced with any one of those. Also I do not have much time to learn a new software to draw graphs. I have asked a related question before but am frustrated by the time that is needed in learning the drawing software. Am also considering using hand-drawn graphs.

My question: It is generally a requirement for the graphs to be drawn very nicely in a mathematical PhD thesis? (I know that different universities have different rules and here I am asking about the general principle.) Is it generally considered acceptable if I draw some graphs by hand, scan them and include them in the thesis?
Even if it's not a formal requirement, consider how important first impressions are. People will judge a paper with sloppy diagrams, and assume that the results and writing will also be sloppy. This is especially true for a PhD thesis. Since you've had 4-5 years to work on it, people will assume that a thesis is well-written and every aspect carefully planned. That includes the figures. A thesis with hand-drawn figures will not be seen as professional.

I personally recommend Inkscape for drawing things. I think it's easier than Illustrator. I believe for my undergraduate thesis I picked up most of the knowledge i needed in 2 hours of watching Youtube tutorials and trying them out. Fortunately, math diagrams usually only require knowledge of the most basic tools in a program.

If you are truly desperate, I believe it is usually acceptable to pay someone to take your pencil drawings and turn them into professional ones, as long as you give proper credit (check your university guidelines though). There is probably an undergrad at your university with experience in Inkscape or Illustrator who wants to make some extra money.