  Edge effects play an important role for many properties of graphene. While
most works have focused on the effects from isolated free edges, we present a
novel knotting phenomenon induced by the interactions between a pair of free
edges in graphene, and investigate its effect on the buckling of monolayer
graphene. Upon compression, the buckling of graphene starts gradually in the
form of two buckling waves from the warped edges. The collision of these two
buckling waves results in the creation of a knot structure in graphene. The
knot structure enables the buckled graphene to exhibit two unique post-buckling
characteristics. First, it induces a five-fold increase in graphene's
mechanical stiffness during the buckling process. Second, the knotted structure
enables graphene to exhibit a mechanically stable post-buckling regime over a
large (3%) compressive strain regime, which is significantly larger than the
critical buckling strain of about 0.5%. The combination of these two effects
enables graphene to exhibit an unexpected post-buckling stability that has
previously not been reported. We predict that numerical simulations or
experiments should observe two distinct stress strain relations for the
buckling of identical graphene samples, due to the characteristic randomness in
the formation process of the knot structure.
