Fetching details from the dictionary...

figure; design
twill weave; pattern of diagonal stripes
style (of writing); figure (of speech)
design; plot; plan
minor market fluctuation; technical correction
cat's cradle
lease rod (in a loom)
あやのある生地は光の角度で表情が変わる。
Figured fabric changes appearance depending on the light angle.
この布にはきれいなあやが入っている。
This cloth has a beautiful woven pattern.
あやの入った絹が美しい。
The silk with the pattern is beautiful.