Characterizations of knot groups and knot symmetric quandles of surface-links

TL;DR

This paper extends the characterization of knot groups and symmetric quandles from classical links to surface-links, including non-orientable cases, using plat presentations and demonstrating realizability of dihedral quandles.

Contribution

It generalizes existing characterizations to all surface-links and introduces a method to realize dihedral quandles as knot symmetric quandles.

Findings

Characterization of knot groups of surface-links including non-orientable cases.

Use of plat presentations to prove the characterization.

Realization of dihedral quandles as knot symmetric quandles.

Abstract

The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method. Gonz\'alez-Acu\~na and Kamada independently extended this characterization to the knot groups of orientable surface-links. Kamada applied the closed 2-dimensional braid method to show this result. In this paper, we generalize these results to characterize the knot groups of surface-links, including non-orientable ones. We use a plat presentation for surface-links to prove it. Furthermore, we show a similar characterization for the knot symmetric quandles of surface-links. As an application, we show that every dihedral quandle with an arbitrarily good involution can be realized as the knot symmetric quandle of a surface-link.