Virtual Links in Arbitrary Dimensions

TL;DR

This paper generalizes virtual links to arbitrary dimensions, extending classical invariants and diagrams, and introduces combinatorial and algebraic tools for their study, including biquandle invariants and higher-dimensional welded links.

Contribution

It provides a comprehensive framework for virtual links in any dimension, extending classical invariants, diagrams, and introducing new algebraic invariants like biquandles.

Findings

Many classical homotopy invariants extend to virtual links.

Defined virtual link diagrams and Gauss codes for higher dimensions.

Constructed a biquandle invariant for virtual 2-links.

Abstract

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtual links, and use such diagrams to construct a combinatorial biquandle invariant for virtual $2$-links. In the case of $2$-links, we also explore generalizations of Fox-Milnor movies to the virtual case. In addition, we discuss definitions extending the notion of welded links to higher dimensions.