Extended quandle spaces and shadow homotopy invariants of classical links

TL;DR

This paper introduces extended quandle spaces and shadow homotopy invariants for classical links, generalizing previous invariants and establishing their relationship with quandle order.

Contribution

It defines Cayley-type graphs and extended quandle spaces, and constructs shadow homotopy invariants, linking them to existing quandle homotopy invariants.

Findings

Shadow homotopy invariant equals quandle homotopy invariant times quandle order.

Extended quandle spaces generalize previous quandle and rack spaces.

The paper establishes a new connection between invariants and quandle structure.

Abstract

In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle.