What topic can be studied in knot theory?

Question

I'm a high school student from China. And I'm going to start a project about knot theory with my friend, but haven't had a topic yet. Could you guys give me some advice. Thanks a lot!

(Sorry, I'm not good at writing in English)

Answer 1

Good for you starting a project like this so early! I am not sure what your background in knot theory is like, but some interesting topics to read about include:

• Reidemeister moves: something I think would be enlightening is to show that any two knot diagrams belonging to the same knot (up to planar isotropy) can be turned into one another via a sequence of Reidemeister moves.
• Seifert surfaces: Learning about Seifert surfaces allows you to explore the intricate link between knots and manifolds. This may be a lot for a high school student, but if you are motivated it can be very rewarding.
• The Alexander polynomial (plus others): The Alexander polynomial gives an algebraic aspect to knot theory. This will have you learn about many concepts and algebraic invariants that could make for a useful project.

There are many other things that could be of interest, but hopefully these things help to give you a starting point! Have fun!