Linking numbers in non-orientable 3-manifolds

TL;DR

This paper develops a method to define homotopy invariants of links in non-orientable 3-manifolds without relying on the manifold's orientation, expanding the scope of linking number concepts.

Contribution

It introduces a new approach to construct linking invariants in non-orientable 3-manifolds, bypassing the need for orientation.

Findings

Successfully defines homotopy invariants in non-orientable manifolds

Extends classical linking number concepts to non-orientable cases

Provides a framework for further topological studies in non-orientable 3-manifolds

Abstract

The construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.