The quandle of the trefoil knot as the Dehn quandle of the torus

Abstract

We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space $\mathbb{Z} \oplus \mathbb{Z}$. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures.