On homotopy groups of quandle spaces and the quandle homotopy invariant of links
Let \(L\) be an oriented link in the \(3\)-sphere \(S^3\), \(Q_L\) the link quandle of \(L\) which is the conjugacy class of \(\pi_1(S^3\setminus L)\) including the meridians of \(L\), and \(X\) a quandle. The author studies the quandle homotopy invariant and the quandle cocycle invariants related with \(X\)-colorings, \(\text{Hom}(Q_L, X)\), of \(L\). Following the definition of the rack space due to [\textit{R.~Fenn}, \textit{C.~Rourke} and \textit{B.~Sanderson}, Appl. Categ. Struct. 3, No. 4, 321--356 (1995; Zbl 0853.55021)], the author defines the quandle space \(BX\) of \(X\), the invariant \({\Xi}_X(L; C)\) of an \(X\)-coloring \(C\) valued in \(\pi_2(BX)\), and the quandle homotopy invariant \({\Xi}_X(L)\) as the formal sum of \(\{{\Xi}_X(L; C)\}\) running over all \(C\). When \(X\) is a finite quandle, \(BX\) is a path connected CW complex and \({\Xi}_X(L)\) is valued in \(\mathbb{Z}[\pi_2(BX)]\) (i.e.\ \(\text{Hom}(Q_L, X)\) is a finite set). By using universality of the quandle homotopy invariant, an alternative definition of the quandle cocycle invariant due to [\textit{J.~S.~Carter}, \textit{D.~Jelsovski}, \textit{S.~Kamada}, \textit{L.~Langford} and \textit{M.~Saito}, Trans. Am. Math. Soc. 355, No.~10, 3947--3989 (2003; Zbl 1028.57003)] is given. In Section 3, the structure of \(\pi_2(BX)\) is studied, and it is applied to obtain some properties of the quandle cocycle invariants. Suppose that \(X\) is finite. Then \(\pi_2(BX)\) is finitely generated (Theorem 3.1), and its rank and the order of its torsion part are estimated (Theorem 3.6). In particular, it is finite if and only if \(X\) is connected (i.e.\ For any \(x, y\in X\), there is an element \(z\in X\) such that \(x=y\ast z\)). In Section 4, \(\pi_2(BX)\) and its generator are determined concretely for \(X\) such as the dihedral quandle of odd prime order, the tetrahedral quandle, the conjugacy quandle of a conjugacy class of the \(4\)-th symmetric group, and the knot quandle. For some cases, they are determined up to two possibilities. In Section 5, the quandle homotopy invariant and the quandle cocycle invariant of the connected sum of two knots are described under a certain generic situation (i.e., for any \(x, y\in X\), if there exists an element \(z\in X\) such that \(x=y\ast z\), then it is unique). The effect on the quandle homotopy invariant of an oriented link of taking the mirror image with opposite orientation is also mentioned. In an appendix, the author points out that the set of any \(X\)-colorings of any oriented diagrams modulo concordance relations corresponds bijectively to \(\pi_2(BX)\) via the quandle homotopy invariants, following the arguments due to Fenn, Rourke and Sanderson for racks. The integral quandle second homology group of a certain quandle is calculated. More general results are obtained in successive work by the author. One of the important aspects of the present paper is to describe quandle theory by the language of homological algebra.
scientific article; zbMATH DE number 5888202
scientific article