标题： 手柄链和空间图的不变量 显示英文标题

标题： Invariants of Handlebody-Links and Spatial Graphs

摘要： 一个$G-$族的偶尔代数是一种代数结构，由 A. Ishii、M. Iwakiri、Y. Jang 和 K. Oshiro 在 2013 年提出。 这些代数系统的公理受到手柄结理论的启发。 在本工作中，我们研究了可能推广$G-$偶尔代数族以及其他类似构造（例如，$Q-$和$(G,*,f)-$偶尔代数族）的方法。 我们给出了必要条件，使得所得对象（称为$(X,G,{*_g},f,\otimes,\oplus)-$系统）成为扭结手柄体的颜色不变量。 我们还讨论了所提出的构造的几种其他修改，提供具有任意（有限）顶点度数集合的空间图的不变量。 此外，我们考虑了几个例子，特别是展示了空间三值图和手柄体链理论之间的差异。 显示英文摘要

摘要： 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 $G-$family of quandles and other similar constructions (for example, $Q-$ and $(G,*,f)-$families of quandles). We provide the necessary conditions under which the resulting object (called an $(X,G,{*_g},f,\otimes,\oplus)-$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 in particular showcase the differences between spatial trivalent graph and handlebody-link theories.