Miscellanea

Q: What is your favorate knot?
- A: Right-handed trefoil. Because this is the only non-trivial knot I can draw with confidence on the blackboard.
- p.s. Not even figure-eight: Once I was teaching students to compute the fundamental group of knots via knot diagrams. After I present the method of Wirtinger presentations, I presented two examples. The first example was trefoil and everything went through smoothly. The second example was supposed to be the figure-eight, and I draw a knot which I thought to be figure-eight. Then we perform the method of Wirtinger presentation but the result turned out to be the same as the previous example of trefoil. I was confused at the moment, because knot complements are Haken, and hence by Waldhausen's theorem plus Gordon-Luecke's theorem, knots are determined by their fundamental groups. Then after a careful check, the knot diagram I drew for figure-eight was actually a trefoil.
- p.s. Even for the trefoil, it seems I cannot achieve the 100% successful rate. For example, in the first version of this paper, the trefoil I draw in Figure 2 is actually an unknot.

Eureka!