Mathematical Sciences: Theoretical and Computational Problems Associated with the Tabulation of Knots

数学科学：与结列表相关的理论和计算问题

• 批准号: 9401139
• 负责人: Morwen Thistlethwaite
• 金额: $ 7.5万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401139&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Theoretical; Computational; Problems

9401139 Thistlethwaite Using the vastly improved computer power now available on workstations, the investigator has recently classified knots up to 14 crossings. He will continue the compilation of knots through 15 or 16 crossings. This computational work is being done jointly with J. Hoste (Pitzer College). The investigator will endeavor to answer various theoretical problems which arise naturally from the classification process. In particular, he will seek to complete the computation of the mapping class group of an alternating link pair, and he will look into questions concerning primality of knots and questions concerning homomorphic images of knot groups or subgroups of knot groups. He will search for a second proof of the Tait Conjecture and for possible extensions of this conjecture. These extensions could be used to answer conjectures concerning the asymptotic density of alternating knots. Thistlethwaite's research treats the topology of knots and links. A knot is a simple closed curve, which one may think of as a closed loop of string, situated somehow in space; a link is a more general kind of object, consisting of one or more separate loops. The mathematical interest lies not so much in the intrinsic loop, but rather in the way in which it is situated in 3-dimensional space, i.e. in the way in which it might be "knotted." Topologists consider that two knots (or links) are equivalent if it is possible to maneuver one to the other by bending or stretching; no cutting is allowed. One of the fascinating aspects of knot theory is that it contains many problems which are easy to state and easy for the layman to understand, yet which persistently resist all attempts at solution. One such problem, which William Menasco (SUNY Buffalo) and Thistlethwaite solved last year, was the hundred-year old Tait conjecture on alternating links. The comprehensive table of knots up to 16 crossings which Thistlethwaite and Hoste are currently producin g will be used to test and formulate other conjectures. In addition, Thistlethwaite will investigate the unknotting number of an alternating knot, this being the minimal number of times it is necessary to pass a knot through itself to render it trivial. His search for an alternative proof of the Tait conjecture will involve "non-Euclidean" geometry. ***

9401139利用现在工作站上可用的极大改进的计算机能力，研究人员最近对多达14个交叉点进行了分类。他将继续编制15或16个交叉口的结点。这项计算工作是与J.Hoste(皮策学院)共同完成的。调查者将努力回答分类过程中自然产生的各种理论问题。特别是，他将寻求完成交错连杆对的映射类群的计算，并将研究关于纽结的素性的问题和关于纽结群或纽结群的子群的同态像的问题。他将寻找泰特猜想的第二个证明，并寻找该猜想的可能扩展。这些推广可以用来回答有关交错纽结的渐近密度的猜想。希斯尔斯韦特的研究涉及到结和链的拓扑结构。结是一条简单的闭合曲线，人们可能会认为它是一个封闭的线环，以某种方式位于空间中；链接是一种更一般的对象，由一个或多个单独的环组成。数学上的兴趣不在于内在的循环，而在于它在三维空间中的位置，也就是它可能“打结”的方式。拓扑学家认为，如果可以通过弯曲或拉伸将一个结(或链节)移动到另一个结(或链节)，则两个结(或环)是等价的；不允许切割。纽结理论的一个吸引人的方面是，它包含了许多容易陈述和外行人容易理解的问题，但这些问题始终抵制一切试图解决的尝试。威廉·梅纳斯科(纽约州立大学水牛城分校)和希斯尔斯韦特去年解决了一个这样的问题，那就是百年来交替联系的泰特猜想。希斯尔斯韦特和霍斯特目前正在制作的多达16个交叉点的综合表将用于测试和表述其他猜想。此外，希斯尔斯韦特将研究一个交替的结的解开次数，这是一个结通过自身使其变得微不足道所需的最小次数。他寻找泰特猜想的另一种证明将涉及“非欧几里得”几何。***