An invariant derived from the Alexander polynomial for handlebody-knots
Given a handlebody-knot \(H\subset S^3\) of genus \(g\), the (\(g\)-th) Alexander polynomial \(\triangle_H(t_1,\dots,t_g)\) is the greatest common divisor of all maximal minors of a presentation matrix of the \(\mathbb{Z}[t_1^{\pm 1},\dots,t_g^{\pm 1}]\)-module \(H_1(\tilde{E},p^{-1}(b))\), where \(p:\tilde{E}\rightarrow \overline{S^3\setminus H}\) is the universal abelian cover, and \(b\in \overline{S^3\setminus H}\) is a base point, and \(\{t_1,\dots,t_g\}\) is a basis of \(H_1(\overline{S^3\setminus H})\). The fact that the Alexander polynomial of a handlebody-knot is only well-defined up to basis change makes effective use of the invariant harder when \(g>1\). The paper under review circumvents the obstacle by deriving a vertex-weighted-graph-valued handlebdoy-knot invariant \(G_H\) from the Alexander polynomial \(\triangle_H(t_1,\dots,t_g)\). Suppose \[ \triangle_H(t_1,\dots,t_g)=\sum_{i=1}^m c_i t_1^{x^i_{1}}t_2^{x^i_{2}} \cdots t_g^{x^i_{g}}, \] and let \(\mathbf{P}\) be the set consisting of vectors \(\mathbf{p}(i):=(x^i_{1},\dots,x^i_{g}), i=1,\dots,m\), in \(\mathbb{R}^{g}\). Then the vertex-weighted graph \(G_H\) is given by assigning to each \(c_i\) a black vertex with a label \(c_i\), and to each \((g+1)\)-tuple \(\{\mathbf{p}(i_0),\dots,\mathbf{p}(i_g)\}\) a white vertex if the convex hull spanned by \(\{\mathbf{p}(i_0),\dots,\mathbf{p}(i_g)\}\) contains no vectors in \(\mathbf{P}\setminus \{\mathbf{p}(i_0),\dots,\mathbf{p}(i_g)\}\). The white vertex is labeled with the volume of the \(n\)-parallelotope determined by \(\{\mathbf{p}(i_0),\dots,\mathbf{p}(i_g)\}\), and is connected to black vertices corresponding to \(c_{i_0},\dots, c_{i_g}\) by edges. The first application of the vertex-weighted-graph-valued invariant is a sufficient condition for a handlebody-knot to be irreducible, while the second addresses the question of when a link is not a constituent link of a \(g\)-leafed rose spine of a given handlebody-knot. The author also shows that the invariant \(G_H\) is invariant under \(N\)-crossing change, an \(N\)-crossing change being a crossing change of a spine \(S\) of \(H\) involving only arcs that disconnect \(S\).
scientific article; zbMATH DE number 7224931
scientific article