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I have got a question. From definition alternating diagram $D$ of a knot $K$ is a diagram such passes alternately over and under crossings. A knot $K$ with such a diagram $D$ is called a alternating knot.

Now we can construct such a knot $K$ on a easy way. Given a arbitrary projection in the plane (such that the projection is regular) we can start at a crossing point on the curve and run along the projection imposing alternation of crossings.

This method works, but my question is: Why does it work and never give a contradiction when one returs to a crosing for the second time?

Can you give the answere with help from graph-theory??

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marked as duplicate by Gerry Myerson, Micah, Quixotic, Did, Davide Giraudo Feb 8 '13 at 10:27

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Duplicate? Warp-like pattern in a closed curve –  Rahul Feb 8 '13 at 5:28

1 Answer 1

Here's a somewhat sketchy proof. It's topological rather than graph-theoretical.

Start with a projection $p: S^1 \rightarrow S^2$, an immersed circle in general position. $S^2 - p(S^1)$ is a collection $R$ of regions $\{R_i\}$. Color them like a checkerboard, as follows. For any $R_i$, the winding number of $p$ around $x \in R_i$ does not depend on the choice of $x$. So we have a map $w: R \rightarrow \mathbb{Z}$. Color the regions white or black depending on whether $w$ is even or odd. Now every edge of the embedded graph $p(S^1)$ has a black region on one side and a white region on the other side.

So the colors alternate around each vertex of the graph. Each edge of the graph belongs to the closure of exactly one black region. Also, each vertex has degree 4, and each region is an open disk.

Now assign a value to each end of each edge with the following rule: traveling clockwise around the inside boundary of a black region, as we travel along an edge, mark the beginning "over" and the end "under". It's easy to check that this assigns crossings consistently to the diagram: at each vertex of the graph, four ends of edges meet, with two opposite ones marked "over" and the other two marked "under". Also, it's clear that the crossings alternate.

So this is the set of crossings you got by simply alternating crossings (up to swapping over and under), and it's consistent.

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