Knot Groups and Symbolic Dynamical Systems

研究人员开发了一种新的方法，将符号动力系统的概念应用于纽结和链群的研究。定向纽结群的换位子群或定向链环群的扩张子群的有限群的表示集具有有限移位的结构，这是一种特殊类型的动力系统，可以用有限图完全描述。有向链群的换位子群的有限阿贝尔群的表示集是有限类型的高维移位。这种位移的动力学不变量，如拓扑熵、方向熵和Zeta函数，产生了新的可计算的纽结不变量，并给出了关于分支循环覆盖空间的有用信息。自由群自同构的基本和有效的共轭障碍是一个副产品。该项目将扩展研究人员以前的工作，使用这些新技术来解决纽结理论中的几个未决问题。DNA、太阳等离子体细丝和花园软管都有一个共同的特征：每一个都表现出打结和连接。结和环的数学理论始于19世纪中期，今天它的重要性在物理、化学、生物学和工程学的各个领域都得到了承认。直到最近，这个领域所采用的方法都来自于拓扑学和代数。两位主要研究人员发现了符号动力学领域的新工具，这些工具对于研究结和链接是新颖和有效的。研究数据阵列的符号动力学是信息和通信理论的核心，在混沌系统分析和材料科学中有重要的应用。该项目将扩展研究人员的技术，解决纽结理论中尚未回答的问题，并在学科之间建立新的桥梁。

The investigators have developed a new method for applying concepts of symbolic dynamical systems to the study of knot and link groups. The set of representations into a finite group of the commutator subgroup of an oriented knot group or the augementation subgroup of an oriented link group has the structure of a shift of finite type, a special type of dynamical system that can be completely described by a finite graph. The set of representations into a finite abelian group of the commutator subgroup of an oriented link group is a higher dimensional shift of finite type. Dynamical invariants of the shift such as topological entropy, directional entropy and the zeta function produce new, computable knot invariants, and give useful information about branched cyclic covering spaces. Elementary and effective obstructions to conjugacy for free group automorphisms result as a byproduct. The project will expand the investigators' previous work, using these new techniques to address several open questions in knot theory. DNA, solar plasma filaments and garden hoses have a common feature: each exhibits knotting and linking. The mathematical theory of knots and links began in the mid-nineteenth century, and today its importance is recognized in diverse areas of physics, chemistry, biology and engineering. Until recently, the methods employed in this field have come from topology and algebra. The two principal investigators have discovered tools from the field of symbolic dynamics that are novel and effective for studying knots and links. Symbolic dynamics, which studies arrays of data, is central to information and communication theory and has important applications in the analysis of chaotic systems and the science of materials. This project will extend the investigators' techniques, address unanswered questions in knot theory and establish new bridges between disciplines.