There are no surviving Middle Eastern/Arabic sword traditions that I am aware of. Ditto for the Turks (Saljuq, Ottoman, etc).
The Turks primarily made use of sabers of various types. The original curved saber used by various steppe peoples was known in Europe by the 8th century AD/CE, and was known there as the gladius Hunniscus. The classic Persian saber was the shamshir (which became corrupted as simitara in Italian, and then "scimitar" in English). In use by the Ottomans was the shorter, heavier kilij. The yataghan--one of the most beautiful short swords ever made--featured a forward-swept blade which was still functional for thrusting (a really practical design). Finally, there was the mec, a specialized, narrow-bladed thrusting sword of European inspiration.
IIRC, the Mamluks practiced test-cutting by slicing into mounds of wet clay.
The Turkish "manly arts" included horsemanship, archery, swordsmanship, & wrestling. Their wrestling has at least survived, and the Turks are still strong in international competition.