How to label knots into permutation?

Question

I'm reading a book about knot theory and I have an question about how to label a knot. The image above is captured in the book I read.For each knots are trefoil knots presented by group S3 and S4.But I do not know how the knot can be shown as S4,with 4 cross is labelled(But a trefoil knot have 3 crosses only)

Answer 1

The arcs are what is being labeled. In the left-hand figure, $(2\ 3)$ should be the label for the middle "S" arc.

(For context: the labels are giving a homomorphism $\pi_1(S^3-K)\to S_n$ with respect to the Wirtinger presentation. This means the labels of the arcs surrounding a crossing must satisfy the corresponding relation. For instance, the top crossing in the left figure satisfies $(1\ 2)(1\ 3)=(1\ 3)(2\ 3)$.)

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I somehow found the book the figure is from (Knot Theory by Charles Livingston). It appears the figure has been corrected in the edition I am looking at.

The book does not appear to have an explanation for how they found the labelings. A cursory look through the literature suggests that finding maps $\pi_1(S^3-K)\to S_n$ is somewhat difficult in general.

The trefoil knot has a fundamental group isomorphic to $\langle a,b|a^2=b^3\rangle$. There is a straightforward homomorphism to $S_3$ by sending $a\mapsto (1\ 2)$ and $b\mapsto (1\ 2\ 3)$. I haven't checked, but it is possible that something like this gives the left diagram's labelings. (I do not understand what you mean about the labeling coming from the fact that the diagram has three crossings.) Another way to get this coloring is using the isomorphism between $S^3$ and the dihedral group, then using a Dehn $3$-coloring to color the trefoil with transpositions.

A map to $S_4$ can be obtained using, say, $a\mapsto (1\ 2)$ and $b\mapsto (2\ 3\ 4)$. I do not know how they found the given labeling in particular, though it seems important to a theorem later on that the elements are all from the same conjugacy class.