Alternating Link crossings. Consider a link consisting of $2n$ components such that each components passes over or under each other component exactly $2$ times. Prove that you can swap any of the crossings any number of times so that the link becomes alternating.  
Can someone help? I'm not sure where to start.
I think the way to solve this is to prove such an alternating link exists that satisfies the conditions and from there show a way to make that alternating link just by swapping crossings. I'm not sure how to continue from here unfortunately. 
I'm mainly looking for an algorithm that would transform the link into an alternating one. 
 A: To show alternating projections exist for any link, first take a planar doodle where all link crossings are transverse double points. Step 1 is to show that such a doodle can be $2$-colored. [To see this, define a coloring function by drawing a line from a region out to infinity that intersects the doodle in finitely many double points. The parity of the number of intersections gives you a $2$-coloring. The fact that any two lines give the same parity is a consequence of the Jordan separation theorem.]
 Now fix a $2$-coloring of the doodle. So all crossings have an alternating white and shaded pattern as you move around them. Now, as you approach each crossing insist that the strand coming in from your right is an over-crossing.  Verify that this is independent of which of the two shaded regions you approach from. Then also verify that this gives an alternating diagram. 
In the picture below, I illustrate how to turn an intersection into a crossing using the coloring data. The second picture illustrates how two adjacent crossings going around a shaded region are forced to alternate.

I first learned this argument in Louis Kauffman's book "On Knots."
