  It is known that every three dimensional Gorenstein toric singularity has a
crepant resolution. Although it is not unique, all crepant resolutions are
connected by repeating the operation "flop". On the other hand, this
singularity also has a non-commutative crepant resolution (= NCCR) which is
constructed from a consistent dimer model. Such an NCCR is given as the
endomorphism ring of a certain module which we call splitting maximal modifying
module. In this paper, we show that all splitting maximal modifying modules are
connected by repeating the operation "mutation" of splitting maximal modifying
modules for the case of toric singularities associated with reflexive polygons.

Representation Theory;Commutative Algebra;Algebraic Geometry;Combinatorics