Extensions of Augmented Racks and Surface Ribbon Cocycle Invariants

TL;DR

This paper explores the structure of augmented racks, their extensions, and introduces new surface ribbon cocycle invariants for knots and surfaces, advancing the algebraic tools for knot theory.

Contribution

It characterizes extensions of augmented racks using fibrant and additive cohomology and constructs new invariants for surfaces in 3-space.

Findings

Characterization of augmented rack extensions via fibrant and additive cohomology

Development of coloring and cocycle invariants for surfaces with boundary

Relations between rack and group 2-cocycles in simultaneous extensions

Abstract

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 {\it 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 {\it 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.