Mathematical Sciences: Knotting in 3-Manifolds

三名研究人员参与了这一低维拓扑的研究项目。(I)Scharlemann打算将新的研究工具“薄位置”、曲面相交的组合学和缝合流形理论应用于三维空间中的纽结和图的理论以及三维流形的Heegaard分裂的几个问题。特别令人感兴趣的是确定和刻画3-流形中的图的纽结的方法，用于将这些工具应用于在3-流形的Heegaard分裂中自然产生的图，以及在图由简单的闭合曲线组成的相对简单的情况下，理解在称为Dehn手术的过程下拓扑如何变化。(Ii)Long计划研究低维流形、辫子群和表示簇理论中几个相互关联的问题。具体内容涉及3-流形的基本群的表示及其在边界坡度和性质P中的应用，以及辫子群及其表示。(Iii)Cooper打算研究低维拓扑与双曲几何之间的联系。纽结补中的不可压缩曲面与SL2(C)纽结多项式的牛顿多边形之间的联系提供了一种获得纽结补拓扑的深层信息的方法。纽结补的双曲粘合方程提供了处理相同问题的另一种方法。霍奇森关于双曲体积形式的公式提供了由经线和子午线上的多项式给出的任何曲线上的闭合的1-形式。从结点产生的曲线上的这种体积形式的精确度为多项式以这种方式产生提供了很强的条件。根据这些线索，库珀希望找到具有不同SL2(C)多项式的突变结，即使不能显式计算这些多项式。结是相当基本的几何对象，其真正有趣的属性是拓扑。我们的意思是，如果两个几何纽结中的一个可以在不切断或解开它的情况下被变换成与另一个一样的样子，那么两个几何纽结就没有什么有趣的区别，只需推动它的线来重新排列交叉点。然而，当一个复杂的几何节点在拓扑上与另一个不同时，认识到这不是一件微不足道的事情，而不仅仅是不同的几何实现。这个问题可以通过计算某些数字或多项式来解决，这些数字或多项式被称为“拓扑不变量”，这意味着对于同一拓扑结点的不同几何实现，它们总是具有相同的值。如果有一个不变量对不同的拓扑节点的几何实现总是有不同的值，那么这个问题就会简化为纯代数，但生活并不那么简单--没有一个单一的不变量达到这个理想，甚至所有已知的不变量加在一起也不是。因此，研究新的不变量是有价值的，其中一些最有用的是近年来受到量子物理思想启发的那些不变量。特别是，纽结理论在DNA长链生物学中的应用利用了这些较新的不变量的知识。

Three investigators are engaged in this research project in low-dimensional topology. (i) Scharlemann intends to apply the new research tools of "thin position," combinatorics of surface intersections, and sutured manifold theory to several problems in the theory of knots and graphs in 3-space and of Heegaard splittings of 3-manifolds. Of particular interest are methods for determining and characterizing knotting of graphs in 3-manifolds, for applying these tools to the graphs which arise naturally in Heegaard splittings of 3-manifolds, and, in the relatively simple case in which the graphs consist of simple closed curves, understanding how the topology changes under a process called Dehn surgery. (ii) Long plans to examine several interrelated questions in the theory of low-dimensional manifolds, braid groups, and representation varieties. The specifics concern representations of the fundamental groups of 3-manifolds and applications of these to boundary slopes and Property P, as well as the braid groups and their representations. (iii) Cooper intends to study ties of low- dimensional topology to hyperbolic geometry. The connection between incompressible surfaces in knot complements and the Newton polygons of the SL2(C) knot polynomial provides a method of getting at deep information about the topology of knot complements. The hyperbolic gluing equations for a knot complement provide another means to approach the same problem. Hodgson's formula for the hyperbolic volume form provides a closed 1-form on any curve given by a polynomial in the longitude and meridian eigenvalues. The exactness of this volume form on a curve arising from a knot puts strong conditions on which polynomials arise this way. Following these leads, Cooper expects to find mutant knots with distinct SL2(C) polynomials, even without being able to calculate these polynomials explicitly. Knots are rather elementary geometric objects whose really interesting properties are topological. By this we mean that two geometric knots do not differ in an interesting way if one of them can be transformed to look just like the other without cutting or untying it, just by pushing its string about to rearrange the crossings. Nevertheless, it is not a trivial matter to recognize when one complicated geometric knot is topologically different from another, rather than just a different geometric realization. This problem can be addressed by computing certain numbers or polynomials which are called "topological invariants," meaning that they always have the same value for different geometric realizations of the same topological knot. The problem would be reduced to pure algebra if there were one invariant which also always had different values for geometric realizations of different topological knots, but life is not so simple -- no single invariant achieves this ideal, nor even all the known invariants taken together. It is therefore valuable to investigate new invariants, some of the most useful being those inspired in recent years by ideas from quantum physics. In particular, applications of knot theory to the biology of long strands of DNA have drawn upon knowledge of these newer invariants.