If you say something archly, you say it with a small, knowing amusement — as if you see the joke, or the real meaning, a little sooner than other people. An arch look is not a big laugh. It is a raised-eyebrow smile: playful, a bit pleased with itself, sometimes teasing, as if you understand something better than the person you are talking to. Though actually, this has nothing to do with the stone arch over a door. This arch is the old “chief / sly” arch, the one in arch-enemy, which slid into meaning roguish, too pleased with how clever we are. Archly is that adjective plus -ly. It is a bookish, slightly old-fashioned adverb. You meet it in novels (“she looked at him archly”) more than in student chat. Young people might just say she gave a knowing look, or she was being teasing. Use it when you want that particular mix of amusement and “I already see it.”

促狹地、得意地（帶著「我比較懂」的笑）