I have a three strand upholstery rope trim for a chair that I need to attach to the other end of same rope. What is the best way to do this without fraying it?

Depending on whether the join is going to be visible or hidden, there are two options. For both options, the two ends of the rope need to be treated with glue. Sobo glue does not yellow and dries clear and is waterproof when dry. E6000 glue will not soak into the fiber on its own. By soaking the rope with a little acetone, the glue will thin out and be absorbed. If the join will be hidden, the ends can be overlapped and wrapped with another thread. If the join is visible, overlap (side by side) the glued ends and, using a razor blade, cut both cords diagonally. Glue the diagonal cut edges together. The join can also be wrapped with invisible thread or thin fishing line for strength.