Is there a way to create a weave pattern from semitransparent lines?

Eli,

Maybe too silly and simple, but fully reversible and editable; you can:

1) Select the bottommost set of lines and Group them all (Group the Groups if applicanle) if  not already;

2) Repeat 1) for the original topmost set of lines;

3) Copy the bottom line Group to the very top (Ctrl/Cmd+C+F+X+F or move it to the top in the epanded Layer in the Layers panel);

3) Create a square with a black fill and no stroke (with the Rectangle Tool) that encloses each of the squares where you want the lines from 1) on top and Group all these:

Now you have four Groups (line, line, line, square from bottom up);

4) Copy the square Group in front of itself (Ctrl/Cmd+C+F);

5) In the expanded Layer in the Layers panel, move one of the square Groups down to be between the line Groups from 1) and 2);

Now you have four Groups (line, square, line, square from bottom up);

Now you have five Groups (line, square, line, line, square from bottom up);

6) Select the bottom square Group and the bottom line Group and in the Transparency panel click Make Opacity Mask with both Clip and Invert Mask unticked;

7) Select the top square Group and the top line Group (the copy of the bottom line Group) and in the Transparency panel click Make Opacity Mask with both Clip and Invert Mask ticked.

6) will make those parts of the original bottom line Group invisible, and 7) will make the opposite parts of the top copy of the original bottom line Group invisible, so everything only appears once.

Without the semitransparency, 4) - 6) would be unneeded.