Battle of Manzikert

As far as I understand, Manzikert was not a very total defeat - the majority of the Byzantine army emerged intact, and we have records of units involved at Manzikert fighting in other actions through Anatolia and beyond.

The problems which led to the loss of Anatolia for the Romans were twofold: the migrations of Turks, and the growing weakness of the Byzantine military establishment in the face of this. Manzikert was a symptom, not the underlying problem. While obviously a Byzantine victory at Manzikert would be beneficial, it would not be a panacea. In fact, it might well prevent any administrative reformation and lead to problems of a greater magnitude down the line. However, at least in the short term the Byzantines not having a crisis of leadership could only be a boon.
 
The specific First Crusade we know is almost certainly butterflied. Various alternatives for the armed strength of Europe seem rather obvious - either alternate armed pilgrimages to Jerusalem and other holy sites, eastward expansion, or a more foreign reconquesta. Likely a combination of all three.

Byzantium might ask the West for help if it suffered similar losses as OTL, but even in OTL that sort of plea fell on deaf ears more often than not. The Popes in Rome often had more immediate concerns with their own backyard. Turkish nomads carving out petty states and possibly a single "Rum" polity is fairly plausible in any case. Alexios himself might be able to exploit a power vacuum. It all depends on how you want the victory to go. With the limited information the OP has given almost anything is possible. All I think can be said is that the broader trends would point to continued Turkish migration and Byzantine weakness. If there are too many weak Emperors in a row, the Byzantine state will be in really rough shape, but even still its improbable to me that the Turkish invaders will cross the straits.
 
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