Sweet potatoes will happily let you trellis them, but you will need to wind the runners through the trellis you use, they will not climb by themselves. Will help you with the gophers to some extent by keeping the ground around where you plant them clear, cannot help re baskets as no gophers down under. Would imagine it depends a lot on varieties planted, are they "nest" formers or do tubers form on runners and how big do you expect tubers to get.
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