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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!

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    $\begingroup$ The Seifert circles might be nested, leading to nested disks, so to speak. The genus of the resulting surface is an upper bound for the genus of a knot, and there are some knots where the minimal genus Seifert surface does not arise from Seifert's algorithm for any diagram! $\endgroup$ – Kyle Miller Aug 1 '19 at 18:52
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    $\begingroup$ Some references are Rolfsen's Knots and Links and Lickorish's Introduction to Knot Theory (theorem 2.2). I was also made aware of Prasolov's Intuitive Topology recently, which has some nice pictures. $\endgroup$ – Kyle Miller Aug 1 '19 at 19:34
  • $\begingroup$ @KyleMiller for your last part in your first comment are you speaking about calculating the lower bound of the genus through half the span of the Alexander polynomail? $\endgroup$ – Secretly Aug 4 '19 at 19:47
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    $\begingroup$ No, I'm not speaking about algebraic invariants. The "canonical genus" is the minimal genus of all Seifert surfaces that come from Seifert's algorithm, and this might be bigger than the actual minimal genus of a Seifert surface --- a Seifert surface is any embedded compact oriented surface whose boundary is the knot, not just those that come from Seifert's algorithm. I brought it up to point out that Seifert's algorithm does not compute genus, which is what you said you were trying to do (though it does provide upper bounds). $\endgroup$ – Kyle Miller Aug 4 '19 at 20:42
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    $\begingroup$ Yes, an orientable surface, and by "does not compute genus" I meant in particular that the surface you get might not have minimal genus. Smooth 4-genus is the minimal genus of a compact oriented surface that is allowed to be smoothly embedded in $B^4$ rather than $S^3=\partial B^4$. This is also known as smooth slice genus. A (smoothly) slice knot is a knot whose smooth 4-genus is $0$. $\endgroup$ – Kyle Miller Aug 4 '19 at 21:23
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Here's how you can get a Seifert surface using Seifert's algorithm for $6_1$'s usual diagram:

Getting a Seifert surface for 6_1

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:

The Seifert surface in another form

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

Constructing a Seifert surface for 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.

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  • $\begingroup$ Does the orientation of the half twisted strips matters? Because the above mentioned algorithm did not mention anything about that. $\endgroup$ – Emptymind Aug 5 '19 at 9:06
  • $\begingroup$ @Intuition Yes, the orientations matter (if you did the wrong orientation, you would get a Seifert surface of a crossing-changed version of the knot). $\endgroup$ – Kyle Miller Aug 5 '19 at 16:39

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