  Cavity optomechanics, a promising platform to investigate macroscopic quantum
effects, has been widely used to study nonreciprocal entanglement with Sagnec
effect. Here we propose an alternative way to realize nonreciprocal
entanglemment among magnons, photons, and phonons in a hybrid cavity-magnon
optomechanics, where magnon Kerr effect is used. We show that the Kerr effect
gives rise to a magnon frequency shift and an additional two-magnon effect.
Both of them can be tuned from positive to negative via tuning the magectic
field direction, leading to nonreciprocity. By tuning system parameters such as
magnon frequency detuning or the coefficient of the two-magnon effect,
bipartite and tripartite entanglements can be nonreciprocally enhanced. By
further studying the defined bidirectional contrast ratio, we find that
nonreciprocity in our system can be switch on and off, and can be engineered by
the bath temperature. Our proposal not only provides a potential path to
demonstrate nonreciprocal entanglement with the magnon Kerr effect, but also
opens a direction to engineer and design diverse nonreciprocal devices in
hybrid cavity-magnon optomechanics with nonlinear effects.
