Quandle cocycle invariants for spatial graphs and knotted handlebodies
The main result of this paper is the construction of a quandle cocycle invariant for spatial graphs and handlebody links. To define these invariants, the authors introduce flowed spatial graphs. An \(A\)-flow on a spatial graph \(L\) is an assignment of an element of an abelian group \(A\) to each orientation of each edge of \(L\) such that: (1) for each edge, the elements of \(A\) assigned to the two orientations are inverses of each other; (2) at each vertex, the total weight of the incoming edges equals the total weight of the outgoing edges. By considering quandle colourings for flowed spatial graphs, the authors construct quandle cocycle invariants for flowed spatial graphs. An invariant of spatial graphs is then obtained as its set of values over all \(A\)-flows of the spatial graph. As this invariant does not detect the contraction move on spatial graphs, it follows that it is also an invariant of handlebody links. Note that the invariant constructed in this paper generalizes a weight sum invariant from [\textit{A. Ishii}, Algebr. Geom. Topol. 8, No. 3, 1403--1418 (2008; Zbl 1151.57007)].
scientific article; zbMATH DE number 6023028
scientific article