Outside of New Hampshire, it is called a gap or pass….
but once you cross our state line, it is called a notch.
The “Chronicles of the White Mountains,” written in 1916 by Frederick Wilkinson states: “The pass now called Crawford Notch was known to the native American tribes of the area, but may not have been frequently used by them, however some say war parties of Canadian Indians used this passage in making raids on the New England coast, and in carrying prisoners up to Canada. 1771 is the year generally accepted when the “Notch” became known to the colonists. Reportedly a hunter, Timothy Nash, came upon it and reported it to Governor Wentworth, who offered him a grant of land, if he could bring a horse through it.With the help of Benjamin Sawyer he did so, and in returned they received “Nash and Sawyer’s Location.” It was not until 1803 when a turnpike was constructed through the Notch at the expense of $40,000.”
P.S. Admittedly a few other locations outside of New Hampshire have “notches” but I’d guess that a New Englander probably had something to do with their names. Examples are Vermont Notches: Smuggler’s, Hazen’s, and Nebraska; Wilmington New York’s Wilmington Notch; Amherst Massachusett’s Notch Traverse; Maine’s Grafton Notch, and Utah’s Blue Notch.