标题： 有向定向Reidemeister移动的最小生成集 显示英文标题

标题： Minimal generating sets of directed oriented Reidemeister moves

摘要： 波利亚克证明了集合$\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$是有向Reidemeister移动的最小生成集。 人们可以区分向前和向后的移动，得到$32$种不同的移动，我们称之为有向有向Reidemeister移动。 本文证明了集合$8$的有向Polyak移动$\{ \Omega{1a}^\uparrow, \Omega{1a}^\downarrow, \Omega{1b}^\uparrow, \Omega{1b}^\downarrow, \Omega{2a}^\uparrow, \Omega{2a}^\downarrow, \Omega{3a}^\uparrow, \Omega{3a}^\downarrow \}$是有向有向Reidemeister移动的最小生成集。 我们还专门研究了这个问题，引入了链环$L$的$L$-生成集的概念。 被证明是任意具有至少$2$个分量的链环$L$的最小$L$-生成集。最后，我们讨论了在研究任意纽结$K$的$K$-生成集时出现的纽结图不变量，重点强调了类型$\Omega3$上升和下降移动之间的区别。 显示英文摘要

摘要： Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $8$ directed Polyak moves $\{ \Omega{1a}^\uparrow, \Omega{1a}^\downarrow, \Omega{1b}^\uparrow, \Omega{1b}^\downarrow, \Omega{2a}^\uparrow, \Omega{2a}^\downarrow, \Omega{3a}^\uparrow, \Omega{3a}^\downarrow \}$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a $L$-generating set for a link $L$. The same set is proven to be a minimal $L$-generating set for any link $L$ with at least $2$ components. Finally, we discuss knot diagram invariants arising in the study of $K$-generating sets for an arbitrary knot $K$, emphasizing the distinction between ascending and descending moves of type $\Omega3$.