Quasi-triviality of quandles for link-homotopy

TL;DR

This paper introduces quasi-trivial quandles and their homology, showing that quandle cocycle invariants derived from them are preserved under link-homotopy, leading to new numerical invariants for links.

Contribution

It defines quasi-trivial quandles and their homology, establishing invariance of certain cocycle invariants under link-homotopy, a novel approach in knot theory.

Findings

Quandle cocycle invariants are invariant under link-homotopy.

Introduction of homology for quasi-trivial quandles.

Generation of numerous numerical link-homotopy invariants.

Abstract

We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.