Virtual, Welded, and Ribbon Links in Arbitrary Dimensions

TL;DR

This paper extends the concept of virtual links to higher dimensions, introduces combinatorial invariants, and explores their properties and classifications, including welded and ribbon links in arbitrary dimensions.

Contribution

It generalizes virtual link theory to arbitrary dimensions, develops new invariants like the biquandle, and analyzes classification results for ribbon knots in higher dimensions.

Findings

Many classical invariants extend to virtual links in higher dimensions

A combinatorial biquandle invariant for virtual 2-links is constructed

Ribbon knots in dimension 4 or greater are classified by their knot quandle

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. For ribbon knots in dimension $4$ or greater, we show that the knot quandle is a complete classifying invariant up to taking connected sums with a trivially knotted $S^1\times S^{n-1}$, and that all the isotopies involved may be taken to be generated by stable equivalences of ribbon knots.