Which object allows a falconer to hold the bird on the fist?

The object that allows a falconer to hold the bird on the fist is the Alymeri jesses. These are straps that are attached to the bird's legs and typically secured around the falconer's glove or fist. The design of the jesses provides a safe and secure way to manage the bird while also allowing for some mobility. When falconers have birds on their fists, the jesses enable them to control the bird's movements and prevent it from flying away unexpectedly.

While the other options mentioned have their own specific uses in falconry, they do not serve the primary purpose of facilitating a falconer's grip on the bird. For instance, the block perch is a stationary object used for resting the bird when it's not being flown, while the bow perch is designed for holding birds in transit. Brails are used to restrict the bird's flight during training but do not aid in physically holding the bird on the falconer's fist. Thus, the Alymeri jesses are the correct answer for providing that essential connection between the falconer and the bird during the handling process.