A hook knife is a small, recessed blade mounted on a rig or jumpsuit. The blade sits inside a hooked slot so it can cut a line drawn into it without presenting an exposed cutting edge.
What it is for
Entanglements — with lines, with a cutaway canopy, or with another jumper's equipment — are situations where being able to cut something is the difference between a problem and a serious one.
The recessed design exists so a jumper can use it under load without cutting the wrong thing, or themselves.
Carrying one is not the same as knowing when to use it
When and whether to use a hook knife is a judgement covered in training and in currency briefings. Like everything else here, that is a conversation with instructors rather than something to work out from a definition.
These pages explain what things are so the language makes sense. They are not instruction and cannot substitute for it — skydiving is taught in person by rated instructors at a drop zone, and anything involving equipment handling or emergency procedures belongs there rather than on a web page.