Crossing of strings

Question

There are two strings of color red and blue. They are made to cross each other odd number of times (greater than one) without any self crossing. Is it always possible that there will be pair of crossings which are adjacent on both the strings ?

Answer 1

Complete Answer updated :

This is wrong for all $n>3$. Robert gave a solution for 4, and I gave a solution for 5, there is also a solution for 6. From those solutions, you can always add 3 points to get another larger solution (and you can repeat and add another 3 points as long as you want), which lead to all possible n without adjacent crossings.

Here a seven crossings from the four crossings solution :

You can see that you can always extend the blue string (from the end that is at the third crossing from left) and add 3 more points (two points will be between the second and third crossing, and the middle point will be right after the last rightmost crossing). There are similar solutions from 5 points and 6 points.

For 5 crossings, a counter-example :

Answer 2

For an even number of crossings, try

(This is not an answer to the original question, but I don't know how to put an image in a comment)