The quandle and group for higher-dimensional and virtual knots

TL;DR

This paper extends the relationship between quandles and groups from classical knots to higher-dimensional, virtual, and welded knots, showing how these algebraic structures can be derived from each other.

Contribution

It generalizes Joyce's results to higher dimensions and virtual knots, establishing new connections between quandles and groups in these broader contexts.

Findings

Fundamental quandle can be derived from the fundamental group in higher dimensions.

Group and peripheral structure can be recovered from the quandle for virtual knots.

Results extend classical knot theory to virtual and higher-dimensional knots.

Abstract

Joyce has shown that the fundamental quandle of a classical knot can be derived from consideration of the fundamental group and the peripheral structure of the knot, and also that the group and much of the peripheral structure can be recovered from the quandle. We generalize these results to arbitrary dimensions, and also to virtual and welded knots and arcs.