A construction of multiple group racks

A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e. oriented compact surfaces with boundaries embedded in the [Formula: see text]-sphere [Formula: see text]. A G-family of racks is a set with a family of binary operations indexed by the elements of a group G. There are two known methods for constructing multiple group racks. One is via a G-family of racks. The resulting multiple group rack is called the associated multiple group rack of the G-family of racks. The other is by taking an abelian extension of a multiple group rack. In this paper, we introduce a new method for constructing multiple group racks by using a G-family of racks and a normal subgroup N of G. We show that this construction yields multiple group racks that are neither the associated multiple group racks of any G-family of racks nor their abelian extensions when the right conjugation action of G on N is nontrivial. As an application, we present a pair of spatial surfaces that cannot be distinguished by invariants derived from the associated multiple group racks of any G-family of racks, yet can be distinguished using invariants obtained from a multiple group rack introduced in this paper.