Automorphisms, cohomology and extensions of symmetric quandles

TL;DR

This paper explores the cohomology of symmetric quandles, establishing a four-term exact sequence linking automorphisms, cocycles, and extensions, and highlighting the role of second cohomology in automorphism lifting.

Contribution

It introduces a new exact sequence connecting 1-cocycles, second cohomology, and automorphisms of symmetric quandle extensions, advancing the understanding of their algebraic structure.

Findings

Derived a four-term exact sequence relating cohomology and automorphisms.

Identified second cohomology as the obstruction to automorphism lifting.

Discussed general aspects of dynamical cocycles and extensions.

Abstract

It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles, we derive a four-term exact sequence that relates 1-cocycles, second cohomology, and a specific group of automorphisms associated with the extensions of symmetric quandles. This exact sequence shows that the obstruction to lifting and extending automorphisms is found in the second symmetric quandle cohomology. Additionally, some general aspects of dynamical cocycles and extensions are discussed.