What do you mean by "abstracts"?
Hive and Tak are are high on that list, and absolutely match what I mean..
Tak is fucking amazing for what it is, and I love that (unlike go) it seems to have this property of unpredictableness (despite the perfect information aspect), which means that even though I've played maybe a hundred games, a newish player can beat me quite often, just because I missed one possible path. I also love the story behind it.
I'd be suuuper keen on something like that that is good for 3-6 players.