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

小行星9401139 使用大大改善了计算机的能力，现在可在工作站上，调查人员最近分类结多达14个过境点。 他将继续通过15或16个过境点汇编结。 这项计算工作正在与J. Hoste（Pitzer学院）联合进行。 研究者将奋进回答分类过程中自然产生的各种理论问题。 特别是，他将寻求完成计算的映射类组的一个交替链接对，他将探讨有关问题素的结和问题的同态图像的结群或子群的结群。 他将寻找第二个证明泰特猜想和可能的扩展这一猜想。 这些推广可以用来回答有关交错纽结的渐近密度的问题。 Thistlethwaite的研究涉及节点和链接的拓扑结构。 结是一个简单的闭合曲线，人们可以把它看作是一个闭合的弦环，位于空间中;链接是一种更一般的对象，由一个或多个独立的环组成。 数学的兴趣不在于内在循环，而在于它在三维空间中的位置，即它可能被“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节点移动到另一个节点（或链接），则两个节点（或链接）是等效的;不允许切割。 纽结理论的一个迷人的方面是，它包含了许多问题，这些问题很容易陈述，也很容易为外行人所理解，但这些问题始终抵制所有解决方案的尝试。 去年，纽约州立大学布法罗分校的威廉·梅纳斯科（William Menasco）和西斯尔思韦特（Thistlethwaite）解决了一个这样的问题，那就是关于交替链的有百年历史的泰特猜想。 Thistlethwaite和Hoste目前正在编制的最多16个渡口的综合结表将用于测试和制定其他方案。 此外，Thistlethwaite将研究一个交替结的解结次数，这是通过一个结使其变得微不足道所需的最小次数。 他的搜索替代证明泰特猜想将涉及“非欧几里德”几何。 ***

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. ***