环面上的结节状、假结节状、辫状和假辫状
Knotoids, pseudo knotoids, Braidoids and pseudo braidoids on the Torus
摘要：
本文研究了环面T上的拟节面和拟辫面理论以及拟节面和拟辫面理论。特别是，我们在$S^2$中引入了{ \it混合knotoids}的概念，它推广了$S^3$中混合链接的概念，并且我们提出了混合knotoids的合痕定理。然后，我们推广的考夫曼括号多项式，$<;>$，混合knotoid，我们提出了一个状态和公式$<;>$。我们还引入了{ \it混合伪拟结点群}的概念，即两个分量上的多拟结点群具有一些缺失的交叉信息。更准确地说，我们提出了一个合痕定理的混合伪knotoid和我们扩大了考夫曼括号多项式的伪混合knotoid。最后，我们介绍了{ \it混合辫状体}和{ \it混合伪辫状体}理论，分别作为混合结状体和混合伪结状体的对应理论。随着使用的$L$ -移动，我们也介绍了这里的混合braidoid等价，我们制定并证明了类似的亚历山大和马尔可夫定理的混合knotoid。我们还制定和证明模拟混合伪knotoids的亚历山大定理。
英文摘要：
In this paper we study the theory of knotoids and braidoids and the theory of pseudo knotoids and pseudo braidoids on the torus T. In particular, we introduce the notion of {\it mixed knotoids} in $S^2$, that generalize the notion of mixed links in $S^3$, and we present an isotopy theorem for mixed knotoids. We then generalize the Kauffman bracket polynomial, $<;>$, for mixed knotoids and we present a state sum formula for $<;>$. We also introduce the notion of {\it mixed pseudo knotoids}, that is, multi-knotoids on two components with some missing crossing information. More precisely, we present an isotopy theorem for mixed pseudo knotoids and we extend the Kauffman bracket polynomial for pseudo mixed knotoids. Finally, we introduce the theories of {\it mixed braidoids} and {\it mixed pseudo braidoids} as counterpart theories of mixed knotoids and mixed pseudo knotoids respectively. With the use of the $L$-moves, that we also introduce here for mixed braidoid equivalence, we formulate and prove the analogue of the Alexander and the Markov theorems for mixed knotoids. We also formulate and prove the analogue of the Alexander theorem for mixed pseudo knotoids.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
