Step 1: Locate the loops that are dropped but still intact. Locate the broken ends that caused the hole.
Step 2: Slip a toothpick, sewing needle, or knitting needle through the loops to prevent them from dropping more. You will need to do this on both sides of the hole.
You will also need to pull out a couple more stitches where the broken ends are, and catch the newly freed loops. This will allow more to work with when securing the broken ends when you're finished.
Step 3: Using a third needle/toothpick/etc, pull the strand without a loop through the first loop. For knitters: you are just knitting it back together.
Step 4: Do the same thing for the next loop on the same needle. Keep the first loop (from step 3) on the third needle. Use the same loose strand for the second loop that you did for the first.
Repeat these steps for the dropped loops at the other side of the hole.
For the next step, try to find a yarn in your stash that is a close match to the color and weight of the sweater yarn you're mending. Otherwise try to find thread that is the same color.
I mended this at work, with limited supplies, so my thread was lighter than the sweater and shows a little in the end.
Step 5: I strongly recommend using the Kitchener stitch to join the two halves of the hole together. If using yarn to mend, this will create a near-seamless look. If using thread, well, the mend will be subtle.
The Kitchener stitch video and photo tutorial from KnitPicks.
Finishing: Turn the sweater over to the wrong side/inside.
Find the broken or frayed ends that caused the hole in the first place.
Weave these ends in. Weave in the ends of the thread or yarn you used to mend the hole.
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