# Why is there no contradiction by construction of alternating knots? [duplicate]

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??

-

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

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

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.