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To try to restate that. Someone owns currency A then buys B which is a cost, but it has a good exchange rate with currency C and C has a good exchange rate with A. So the B-C-A move yields a profit which would be representative as a negative weight. Is that right?

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Yes, if trading BCA results in a profit, either BC or CA has negative weight. Negative weight here just means you can sell a currency more than its cost basis, and or buy a currency for cheaper than its cost basis.

Though with arbitrage, you typically want to find a cycle like BCAB so you have more of the initial currency you started off with.

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