Brent S. answered 10/21/22
Doctor of Physical Therapy tutoring Anatomy, Physiology, and more
Possible diagnoses for this injury (listed from most superficial to deepest) include deep laceration to dorsal hand, full or partial thickness rupture of superficial extrinsic hand muscles (extensor digitorum is at particular risk), partial or full thickness tear/rupture of deep intrinsic hand muscles (lumbricals, dorsal interossei), and metarcarpal fracture(s) requiring further imaging to dictate type.
Considering that this injury occurred with a blade, it is likely that the soft tissue will heal by primary intention - leading to quicker healing than would be experienced if the individual was injured with a blunt object, burned, or scraped. This is due to the relatively smooth edges that have the potential to reattach and scar nicely with less of an inflammatory reaction. This will likely require stitches to approximate the edges of the wound.
With the deeper tissue, such as the rupture of the extrinsic extensors of the hand, this would require a surgical reconstruction, as when tendons are fully lacerated, they will often retract due to continued muscle tension without having an appropriate anchor point. This will be a specialized surgery done by a hand surgeon. In addition to tendon injury, it is also likely that some part of the radial nerve will be damaged, which may or may not be able to be surgery repaired.
With regards to the fracture, solutions depend on type and severity of fracture. Due to the accidental nature of the injury, it is unlikely to be a severe fracture requiring setting or surgical fixation. In this case, it is likely that the hand would be immobilized, which is also a requirement following tendon reconstruction, and bone healing would be monitored via diagnostic imaging (likely x-ray).