Virtual, Welded, and Ribbon Links in Arbitrary Dimensions

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.