Links with finite $n$-quandles

TL;DR

This paper proves a conjecture linking the finiteness of the $n$-quandle of a link in the 3-sphere to the finiteness of the fundamental group of its $n$-fold cyclic branched cover, establishing a deep connection between algebraic and topological properties.

Contribution

It proves Przytycki's conjecture that the $n$-quandle's finiteness is equivalent to the fundamental group's finiteness of the branched cover.

Findings

Established the equivalence between $n$-quandle finiteness and branched cover fundamental group finiteness.

Confirmed the conjecture for all links in the 3-sphere.

Bridged algebraic invariants with topological properties of links.

Abstract

We prove a conjecture of Przytycki which asserts that the $n$-quandle of a link $L$ in the 3-sphere is finite if and only if the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere, branched over $L$, is finite.