Invariants of Handlebody-links and Spatial Graphs

A G-family of quandles is an algebraic construction which was proposed by A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro in 2013. The axioms of these algebraic systems were motivated by handlebody-knot theory. In the present work we investigate possible constructions which generalise [Formula: see text]-family of quandles and other similar constructions (for example, [Formula: see text]- and [Formula: see text]-families of quandles). We provide the necessary conditions under which the resulting object (called an [Formula: see text]-system) gives a colouring invariant of knotted handlebodies. We also discuss several other modifications of the proposed construction, providing invariants of spatial graphs with an arbitrary (finite) set of values of vertex valency. Besides, we consider several examples which showcase the differences between spatial trivalent graph and handlebody-link theories.