Question
I live in Michigan.  I am interested in killing the grass and weeds that have emerged through a layer of landscaped stone surrounding my pool.  I prefer not to use toxic chemicals

Answer
Undiluted vingegar has been known to be a reasonably effective organic way. It is difficult on tough weeds, and will need to be repeated quite often.

You can also consider using the product called "Round-Up" (by Ortho). It is a herbicide which is taken in through the leaves of the plant which then shut down the plant's ability to photosynthesize, e.g. convert sun energy into plant energy. In a few days the plant dies (due to lack of food). The product is safe once the plant leaves have dried (a few hours). It does not reside in the plants or soil afterwards like some other products. It may need to be reapplied with regular intervals. Some other products which promisses 6,9, or 12 month coverage are based on toxic chemicals which reside in the soil. Not so with Round-Up although proper precautions should be used when applying any kind of herbicide. This includes only spraying on a calm mild day (between 40F and 75F). Use only enough to wet the leaves and wear protective clothing (layers). Avoid inhaling the spray.

Good luck.