SoI take it that roundup is not an option.....
This is a family forum, Seed, watch your language.

I use Glyphosate in certain places, such as the cracks in driveways, fence rows, or other specific areas to control certain things. This weed, nutsedge, is all over those areas so it gets hit, turns yellow, and dries up. I don't know if it actually kills the nut or not. You'd think it should but it doesn't take long for anther nutsedge to show up. That could be from seeds or other nuts in the ground. Those can last for years. To try to actually control it chemically would require a lot of spraying, I'm not saturating my soil like that.. I try to control it with mowing and weed-eating as much as I can but it still reproduces. Seeding will take a little longer but the nutlets and rhizomes start developing about when it has six leaves. It doesn't take long for it to develop six leaves. Doesn't take it much longer to go to seed.
I used to rant and rave about Bermuda grass. You probably haven't noticed that lately, I consider this stuff worse.. Besides since I moved I don't have Bermuda grass. I left that in Arkansas.
I've looked at herbicides that are generally pointed toward nutsedge, mainly comparing withdrawal periods and cautions about certain vegetables. Glyphosate is one of the milder ones. But as a general rule I try to keep herbicides out of my vegetable garden. I don't have the patience some people do that use a small paint brush and paint Glyphosate on certain plants.
When I built my raised beds I removed the top 5" to 6" of dirt to remove the vast majority of nutsedge seeds and nuts. It worked, before I did that there was a mat of nutsedge growing where some of them are. Obviously I did not get them all, you can see how deep I would have had to go, but I got them down to a manageable number. With my other plants in there I usually can't dig like this to get them out so I try too pull the growing part off before they have a chance to reproduce. This one just happened to be in an area where I could dig and I was gentle enough to not break off that stem growing from it. That stem can be delicate and if it breaks you are not going to find that nut.