Layers of knot region colorings and higher differentials (Q2187152)

The main results are as follow. For a knot (or knotted surface) diagram, a knot-theoretic ternary quasigroup (KTQ) and a KTQ-coloring, the author gives new KTQ-colorings, which form a coloring system called a layered coloring. Using a layered coloring, the author constructs a homology that is a generalization of the standard KTQ homology, whose cycles give a new invariant for knots or knotted surfaces. Examples of computation are given. The author gives a layered coloring as a system of region colorings by using colorings called right coloring and left coloring. We review the main terms. A ternary quasigroup is a set \(X\) with a ternary operation \([ \,]: X^3 \to X\) such that for \(x_1, x_2, x_3, x_4 \in X\) satisfying \([x_1x_2x_3]=x_4\), the specification of any three elements determines the remaining one element uniquely. A knot-theoretic ternary quasigroup (KTQ) is a ternary quasigroup satisfying certain conditions derived from the Reidemeister move III. A KTQ-coloring is a map which assigns an element of a KTQ to each region of a knot (or knotted surface) diagram, where a region is a connected component of the complement of the diagram. And the standard KTQ homology theory has been established in the author's earlier work [``Homology of ternary algebras yielding invariants of knots and knotted surfaces'', Preprint, \url{arXiv:1706.04307}, to appear in Algebr. Geom. Topol.].