Numbers 21:13

From the Emorite border. The boundary, the edge of their border. Similarly, “the border of Moav”, meaning the edge, the end.

Opposite Arnon. They circled the entire southern and eastern territory of Moav, until they reached the other side, at Arnon, inside Emorite territory, north of Moav.

Extending from the Emorite border. A strip extends from the Emorite border, and it belongs to the Emorites, but penetrates the boundary of Moav until Arnon, the Moavite border. The Israelites camped there without entering the border of Moav, Arnon being the Moav border, and they did not grant them permission to pass through their land. Although Moshe did not state this clearly, Yiftach did, as was said by Yiftach, “he (Moshe) also sent [a request] to the king of Moav, but he was unwilling.” Moshe alluded to this: “Just as they did to me, the children of Eisav, who dwell in Seir, and the Moavites, who dwell in Ar,” Just as these [children of Eisav] did not permit them to pass through their land, but they circled around them, so it was with Moav.

And they journeyed from thence, and encamped beyond the Arnon, in a passage of the desert that stretcheth from the coast of the Amoraah; for Arnon is the border of Moab, situate between Moab and the Amoraah; and therein dwelt a priesthood of the worshippers of idols.

FOR ARNON IS THE BORDER OF MOAB, BETWEEN MOAB AND THE AMORITES. Arnon was a city which belonged to Moab, [located] near the end of the Amorite border and the beginning of the border of Moab, and it was situated on brooks that flowed through it..

מעבר ארנון...היוצא, in that region where the river Arnon borders the region where the desert starts.

כי ארנון גבול מואב, the border between the Emorite and Moav is a joint border only at that point where the river separates the two nations. The area where the Israelites had entered the territory of the Emorite was not anywhere near its boundary with Moav.

ויחנו מעבר ארנון, “they encamped across the river Arnon. They encamped on the bank that used to belong to King Sichon after having made a detour around the land of Moav. They had not been allowed by G–d to cross the territory belonging to Moav at that time. They were only allowed to straddle its border, without crossing it. (Compare Judges, 11,17) where Yiftach recounts the history of the people’s wanderings. They had approached that spot coming from the east, leaving the land of Moav on their west, bordered by the territory that belonged to King Sichon.

כי ארנון גבול מואב בין מואב ובין האמורי, “for the river Arnon was the border between Moav and the Emorite.” In practice this means that the river separated the borders from one another. Originally, both banks of the river were part of the land of Moav, but Sichon had conquered all the land on the north bank of that river. The Israelites took possession of all the formerly Moabite lands that Sichon had conquered from them. That land was no longer considered out of bounds to them when they came out of Egypt. G–d rearranges nations’ boundaries from time to time, as He sees fit. The respective rulers during such warfare who appear as winners or losers are pawns in His hands.