Vogel's Algorithm - Why can we read braid words from nested coherent Seifert surfaces?

Question

I was reading this paper from R. Goldstein-Rose: http://math.uchicago.edu/~may/REU2017/REUPapers/GoldsteinRose.pdf

In Figure 12 it was mentioned that if a Seifert surface is coherent and nested, then we can read off braid words from it. I did an example with the left-hand trefoil knot (see below image link) which verifies it, as the crossings in the Seifert surface reads $\sigma_{1}^3$, of which the closure is equivalent to the original trefoil knot. However I am not sure why it is always true.

Constructing Seifert surface from a left-hand trefoil knot

Answer 1

Very quickly, nested coherent Seifert circles means that we are actually looking at the link as braid diagram. Braids wrap around an axis which is the center of the Seifert circles. Coherent means that the circles are oriented in the same direction, which is required for a braid closure to always "flow down." You can prove this as an exercise:

Given a diagram $D$, $D$ is a braid if and only if all the Seifert circles of $D$ are nested and coherent.

Good luck.