Extensions of augmented racks and surface ribbon cocycle invariants

• Published in: Topology and its applications 2023-08, Vol.335, p.108555, Article 108555
• Main Authors: Saito, Masahico, Zappala, Emanuele
• Format: Article
• Language: English
• Subjects: Augmented rack cohomology; Compact orientable surface with boundary
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2023.108555

A rack is a set with a binary operation that is right-invertible and self-distributive, properties diagrammatically corresponding to Reidemeister moves II and III, respectively. A rack is said to be an augmented rack if the operation is written by a group action. Racks and their cohomology theories have been extensively used for knot and knotted surface invariants. Similarly to group cohomology, rack 2-cocycles relate to extensions, and a natural question that arises is to characterize the extensions of augmented racks that are themselves augmented racks. In this paper, we characterize such extensions in terms of what we call fibrant and additive cohomology of racks. Simultaneous extensions of racks and groups are considered, where the respective 2-cocycles are related through a certain formula. Furthermore, we construct coloring and cocycle invariants for compact orientable surfaces with boundary in ribbon forms embedded in 3-space.