The second quandle homology group of the knot $n$-quandle

Abstract

The knot quandle is a complete invariant for oriented classical knots in the $3$-sphere up to orientation. Eisermann computed the second quandle homology group of the knot quandle and showed that it characterizes the unknot. In this paper, we compute the second quandle homology group of the knot $n$-quandle completely, where the knot $n$-quandle is a certain quotient of the knot quandle for each integer $n$ greater than one. Although the knot $n$-quandle is weaker than the knot quandle, the second quandle homology group of the former is found to have more information than that of the latter. As one of the consequences, it follows that the second quandle homology group of the knot $3$-quandle characterizes the unknot, the trefoil and the cinquefoil.