Seifert surfaces for knots $6_1, 6_2, 6_3$.

Question

I have been trying to calculate the genera of these knots, but the first step in doing so is to convert them into orientable knots by constructing Seifert surfaces for those knots. I started to do this using the algorithm here https://users.math.yale.edu/~ml859/seifert_talk.pdf but the algorithm is not clear in the case of those 3 knots.

I got stuck (I will upload the pictures I got as long as I know how to do this as they are on my phone right know).

My Questions:

1. Can anyone mention a more rigorous reference for Constructing Seifert surfaces, knowing that the talk mentioned before refer to Massey "Algebraic topology: an introduction" but I did not find a rigorous algorithm in it.

2. Can anyone show me the shape of this Seifert surfaces?

Thanks!

Answer 1

Here's how you can get a Seifert surface using Seifert's algorithm for $6_1$'s usual diagram:

First you choose an orientation of the knot -- either gives the same result. Then you smooth the crossings in the way that preserves orientations, giving a collection of Seifert circles/circuits (drawn in black). Thinking of the diagram as sitting in a plane, start with an innermost Seifert circle and attach a disk to it underneath the plane. Keep taking an innermost circle that has yet to have a disk attached and continue this process. Then, glue in half-twisted strips at the crossings in the way that follows the knot (drawn in blue).

For $6_1$, there is one disk nested in the other. The resulting Seifert surface has genus $1$, and since by other reasons $6_1$ is non-trivial (for example, this is a reduced alternating diagram so has the minimal crossing number), its Seifert genus must be exactly $1$.

You can isotope the surface to look like the following, which possibly might illustrate the surface a little better:

For sake of another example, here is $6_2$:

How do we calculate genus? The genus of a connected surface with exactly one boundary component satisfies the following formula, where $d$ is the number of Seifert circuit disks and $c$ is the number of crossing bands: $d-c+1=2-2g$, hence $g=(1-d+c)/2$. In this example, we get a Seifert surface with genus $(1-3+6)/2=2$. According to KnotInfo, this is the genus of $6_2$, but a priori we only know that $2$ is an upper bound for the genus.