First of all, I want to say that I've made this hanging kitchen towel thingy many times and have never given it a picot edging. I've never started and ended the edging with a slip stitch before either. But, there's always a first time for everything.
-- I have always made it with an edging, though.
When adding an edging in knitting, one has to be careful about row and stitch gauge. A knit stitch is about 1 1/2 times wider than it is tall (at least for a gauge of 4 sts and 6 rows per inch). So, the number of stitches to be picked up along a vertical edge is different than the number of rows along that edge.
On the other hand, a single crochet stitch is pretty much as tall as it is wide. When picking up stitches along a vertical edge, one would pick up 1 st off of a row of sc's and 2 sts off of a row of dc's and 3 sts off of a row of tr's, etc. (A dc is about twice as tall as a sc, and a tr is about 3 times as tall.) The picture shows a picked up edge (plus some picots).
After finishing off the buttonhole and weaving in the ends, I started with another color of thread. As usual, for starting any crochet project, I made a slip stitch knot around the hook. Then since I wanted an almost invisible start to the edging, I did a slip stitch stitch into the bottom of the first sc in the project (as in the picture at the right). -- To get the yo through the starting loop more easily, I held one edge of the loop (the part of the loop with the loose end) with my left hand. That completed the slip stitch. Then, I did a sc into the edge of the next row, a sc row.
Then comes the first picot.
I used the first method of making picots (as described in the American Thread Company brochure excerpt). I chained 3 and then did a slip stitch into the top of the last sc. NexStitch has a nice video of the stitch.
As you can see, it is a bit difficult to do. That's probably why there is an alternative way to do the picot (as mentioned by both the booklet and NexStitch) -- namely, instead of doing a slip stitch, make another sc (or whatever stitch the picot is on top of) "in the same space" or, in other words, as if one were doing an increase.
Just doing a sc without either the slip stitch or the "increase" would make the picot too open.
One isn't limited to doing just 3 or 4 chains for a picot. It just depends on how large you want it to be. However, with more chain stitches, the inside of the loop of chain stitches becomes more visible, and it's naturally called a chain loop. One is also not limited to doing picots on top of sc's. It can be done on top of most any stitch. Picots also don't have to be restricted to edgings.
To finish this edging, repeat *sc 4, picot* around. For the buttonhole, pick up as many stitches as were skipped at the bottom of the buttonhole. End by making a slip stitch.