The column group and its link invariants (Q2881364)

A \textit{birack} is a pair, denoted by \((X, B)\), of a set \(X\) and a map \(B : X\times X\to X\times X\) satisfying certain axioms which correspond to the Reidemeister moves. Special cases of biracks include quandles, racks, and so on. The birack counting invariants can be defined for a link in a similar way as in the case of quandles. The main results of the paper are to define the column group \(\mathrm{CG}(X)\) of a finite birack \((X, B)\), and the \textit{column group enhanced birack multiset invariant} \(\phi_{(X, B)}^{\mathrm{CG}, M}(L)\) and the column group enhanced birack polynomial invariant \(\phi_{(X, B)}^{\mathrm{CG}}(L)\) for an oriented link \(L\), which are enhancements of the birack counting invariants. In Example 4.5 and 4.6, the authors show effectiveness of their invariants.NEWLINENEWLINEThe map \(B=(B_1(x, y), B_2(x, y))\) is strongly invertible (cf.\ Definition 2.1) if it is (i) invertible, (ii) sideways invertible, and (iii) diagonally invertible. The condition (i) implies \(B\) is bijective, and it corresponds to the type II move. The conditions (ii), (iii) and the kink map in Definition 2.2 correspond to the type I move. The set-theoretic Yang-Baxter equation in Definition 2.3 corresponds to the type III move. Since the kink map \(\pi : X\to X\) of \((X, B)\) is not always the identity, a birack coloring for a link diagram is an invariant of a blackboard-framed link which is a pair of a \(c\)-component link \(L\) and a writhe vector \(\mathbf{w}\in \mathbb{Z}^c\). If there exists an integer \(N\geq 1\) such that \(\pi^N\) is the identity, then a birack coloring is stable for a writhe vector modulo \(N\). If \(N=1\), then a birack is a strong biquandle.NEWLINENEWLINEFor a finite birack \((X, B)\) with \(X=\{ x_1, \ldots, x_n\}\), the birack matrix of \((X, B)\) is defined as an \(n\times (2n)\) matrix \(M=[M_{B_1}|\;M_{B_2}]\) consisting of two \(n\times n\) blocks \(M_{B_1}\) and \(M_{B_2}\) where the \((i, j)\) entries of \(M_{B_1}\) and \(M_{B_2}\) are \(B_1(x_j, x_i)\) and \(B_2(x_i, x_j)\) respectively. Then every column of \(M\) is a permutation of \(X\). The column group \(\mathrm{CG}(X)\) of \((X, B)\) is a subgroup of the \(n\)-th symmetric group \(S_n\) generated by permutations of the columns of \(M\). For a subbirack \(S\) of \(X\), the column subgroup \(\mathrm{CG}(S\subset X)\) is a subgroup \(\mathrm{CG}(X)\) generated by permutations of the columns corresponding to the elements in \(S\).NEWLINENEWLINESuppose that \((X, B)\) is a finite birack and there exists an integer \(N\geq 1\) such that \(\pi^N\) is the identity. The birack counting invariants of an oriented link \(L\) are weighted sums of the numbers of birack colorings such as the integral birack counting invariant \(\Phi_{(X, B)}^{\mathbb{Z}}(L)\), the image enhanced birack counting invariant \(\Phi_{(X, B)}^{\mathrm{Im}}(L)\), the writhe enhanced blackboard birack counting invariant \(\Phi_{(X, B)}^W(L)\), the column group enhanced birack multiset invariant \(\phi_{(X, B)}^{\mathrm{CG}, M}(L)\), and the column group enhanced birack polynomial invariant \(\phi_{(X, B)}^{\mathrm{CG}}(L)\). The authors remark that \(\Phi_{(X, B)}^{\mathrm{Im}}(L)\), \(\Phi_{(X, B)}^W(L)\), \(\phi_{(X, B)}^{\mathrm{CG}, M}(L)\) and \(\phi_{(X, B)}^{\mathrm{CG}}(L)\) dominate \(\Phi_{(X, B)}^{\mathbb{Z}}(L)\), and that their invariants can be defined for virtual links, surface knots and tangles in the same way. In the final section, they collect questions for future research.