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Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links

机译：代数缠结和链节的补集中的代数缠结和闭合不可压缩曲面的有理结构

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Let F be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle (B, T). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B, T) and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead to an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link K. In particular we show that if K is a knot, then the complement of if does not contain such a surface.
机译：令F为不可压缩，子午不可压缩且边界不平行于代数缠结（B，T）的边界的平行曲面。然后F将B中的T字符串分开，并且F的边界斜率由（B，T）唯一确定，因此我们可以定义代数缠结的斜率。除了康威的缠结和，我们还定义了两个缠结的自然乘积。代数缠结上的斜率和二元运算导致代数结构与有理数同构。我们引入了一个新的结和链接类，即代数交替的结和链接，粗略地说，这些结和链接是由交替的结和链接构成的，方法是用代数缠结代替一些交叉。我们给出了一个闭合面在代数交替结或链节K的补集中不可压缩和子午不可压缩的充要条件。特别是，我们表明，如果K是一个结，则if的补不包含这样的表面。

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