WASHINGTON — Most nations, including many close allies of the United States, require up to a week’s notice before American warplanes are allowed to cross their territory. Not Egypt, which offers near-automatic approval for military overflights, to resupply the war effort in Afghanistan or to carry out counterterrorism operations in the Middle East, Southwest Asia or the Horn of Africa.

Losing that route could significantly increase flight times to the region.

American warships are also allowed to cut to the front of the line through the Suez Canal in times of crisis, even when oil tankers are stacked up like cars on an interstate highway at rush hour. Without Egypt’s cooperation, military missions could take days longer.

Those are some of the largely invisible ways the Egyptian military has assisted the United States as it pursues its national security interests across the region — and why the generals now in charge in Cairo are not without their own leverage in dealing with Washington in the aftermath of President Obama’s condemnation Thursday of the military’s bloody crackdown on supporters of the former president, Mohamed Morsi.

In his first overtly punitive step, Mr. Obama canceled the Bright Star military exercise, the largest and most visible sign of cooperation between the armed forces of the two nations. But given the growing violence in Egypt, it might have been impossible to guarantee the safety of the thousands of American troops scheduled to deploy for the war game, and the decision to call it off might have been the wise move regardless of the politics.