在置换的背景下理解三维流形
Understanding 3-manifolds in the context of permutations
摘要：
我们演示了如何3流形，Heegaard图，和一组演示文稿可以被解释为一对有符号的排列在对称群$S_d.$我们展示了权力的排列数据编程和讨论我们已经开发的算法，以置换数据作为输入，并确定数据是否代表一个封闭的3流形。因此，我们有一个群的不变量，即给定任何群表示，我们可以确定该固定表示是否表示一个封闭的3-流形。(The所提出的技术以一对有符号置换开始，并建立有限群表示。有限群表示得到了有限类相关联的3-流形。请注意，否定的回答只意味着固定表示在这种构造下不会导致闭的3-流形，但没有说群表示的同构形式。
英文摘要：
We demonstrate how a 3-manifold, a Heegaard diagram, and a group presentation can each be interpreted as a pair of signed permutations in the symmetric group $S_d.$ We demonstrate the power of permutation data in programming and discuss an algorithm we have developed that takes the permutation data as input and determines whether the data represents a closed 3-manifold. We therefore have an invariant of groups, that is given any group presentation, we can determine if that fixed presentation presents a closed 3-manifold. (The proposed techniques begin with a pair of signed permutations and builds a finite group presentation. The finite group presentation results in a finite class of associated 3-manifolds. Notice that a negative answer only implies the fixed presentation does not result in a closed 3-manifold under this construction, but says nothing about an isomorphic form of the group presentation.)
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。
