Homological characterization of the unknot (Q1861476)

A \textsl{quandle}, as introduced by \textit{D. Joyce} [J. Pure Appl. Algebra 23, 37-65 (1982; Zbl 0474.57003)] is a set \(Q\) with two binary operations whose axioms model conjugation in a group, or equivalently, the Reidemeister moves of knot diagrams. Given a knot \(K\) in the \(3\)--sphere, the Wirtinger presentation of the knot group \(\pi_K:=\pi_1(\mathbb S^3 \setminus K)\) involves only conjugation and thus may be re-interpreted as defining a quandle. The quandle \(Q_K\) so presented is called the fundamental quandle of the knot \(K\). Besides closed knots \(k:\mathbb S^1 \to \mathbb S^3\) the author considers long knots \(\ell:\mathbb R\to \mathbb R^3\), i.e. smooth embeddings such that \(\ell (t)=(t,0,0)\) for all parameters \(t\) outside some compact interval, and considers the image \(L=\ell(\mathbb R)\) in \(\mathbb R^3\) up to isotopy with compact support. The fundamental quandle \(Q_L\) is defined as \(Q_K\) with the exception that the first and the last arc of \(L\) are not identified. The main result of the paper is that if \(K\) is not the trivial knot, then \(H_2(Q_K)\cong \mathbb Z\) and the orientation of \(K\) defines a canonical generator \([K]\in H_2(Q_K)\). Dually, \(H^2(Q_K)\cong \mathbb Z\) and the central extension \(Q_L \rightarrow Q_K\) defines a canonical generator \([L]\in H^2(Q_K)\) whose evaluation yields \(\langle [L][K]\rangle=1\). This result answers a question raised in \textit{J. S. Carter, S. Kamada} and \textit{M. Saito} [J. Knot Theory Ramifications 10, 345-386 (2001; Zbl 1002.57019)]: the orientation class \([K]\) vanishes if and only if \(K\) is trivial. An analogous result holds for links, thus characterizing trivial components.