论文标题
同源性辣椒类型的结节不变
Homological Casson type invariant of knotoids
论文作者
论文摘要
我们考虑了众所周知的卡森结的类似物。我们从经典结构的直接类似物开始，该结构提供了两个不同的整数值的结节不变性，然后专注于其同源性扩展。扩展的值是第一个同源组$ h_1（σ）$的子组的正式总和，其中$σ$是一个定向的表面，具有（也许是）非空的边界。为了使球形结的扩展信息提供信息，足以通过删除其端点周围的小磁盘来将$ S^2 $的初始缝合图转换为环形图中的一个结图。作为不变性的应用，我们证明了两个定理：一个结节的交叉数的急剧下限（估计与M.Polyak和O.Viro在2001年证明的经典结的原型不同），并且有足够的条件，可以使$ S^2 $的结节型（或与pure natogoid of Terragoid of Terraigotic of Terraigation of Turaa exev）在$ s^2 $中的吻合。最后，我们给出了一个包含用于所有球形素数适当的打结的不变式值的表，最多为$ 5 $交叉。
We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. Value of the extension is a formal sum of subgroups of the first homology group $H_1(Σ)$ where $Σ$ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in $S^2$ into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M.Polyak and O.Viro in 2001) and a sufficient condition for a knotoid in $S^2$ to be a proper knotoid (or pure knotoid with respect to Turaev's terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most $5$ crossings.
