Knot Groups and Symbolic Dynamical Systems

结群和符号动力系统

• 批准号: 9704399
• 负责人: Daniel Silver
• 金额: $ 9.02万
• 依托单位: University of South Alabama
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704399&HistoricalAwards=false
• 关键词: Knot; Groups; Symbolic; Dynamical; Systems

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.

研究人员已经开发出一种新的方法，将符号动力系统的概念应用于结群和链群的研究。定向纽结群的换位子群或定向链环群的增广子群在有限群中的表示集具有有限型移位的结构，这是一种特殊类型的动力系统，可以完全用有限图来描述。定向环群的换位子群到有限交换群的表示集是有限型的高维移位。动态不变量的移位，如拓扑熵，方向熵和zeta函数产生新的，可计算的结不变量，并提供有用的信息，分支循环覆盖空间。自由群自同构共轭的基本和有效的障碍是副产品。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结理论中的几个开放问题。DNA、太阳能等离子体细丝和花园软管有一个共同的功能：每一个都表现出打结和连接。 结和链接的数学理论始于世纪中期，今天它的重要性在物理学、化学、生物学和工程学的各个领域都得到了认可。直到最近，在这一领域采用的方法来自拓扑学和代数。 两位主要研究者发现了符号动力学领域的工具，这些工具对于研究节点和链接是新颖而有效的。 符号动力学研究数据阵列，是信息和通信理论的核心，在混沌系统和材料科学的分析中有重要的应用。 该项目将扩展研究人员的技术，解决纽结理论中未回答的问题，并在学科之间建立新的桥梁。