Research of global knot theory in thickened surfaces

对于加厚曲面上的连杆L，我们定义了括号<L)和连杆不变量F (L, a)，它们可以从L的连杆图中直接计算出从加厚曲面到该曲面的投影。<L)和F （L， A）都被求值为自由模的元素，该自由模的系数是由整数1和表面上无平凡成分的所有低维1墨水的同位素类生成的劳伦多项式系数。如果（加厚）曲面是单连通的，<L)和F （L, a）分别等价于Kauffman的托架多项式和Jones多项式。在非单连通的加厚表面上的结和连接不仅会引起局部结连现象（即拓扑3球中的结连现象），还会引起全局结连现象。<L)和F （L， A）都很好地反映了这两种现象。本课题的主要目的是：(1)研究<L)和F （L， A）的性质；(2)利用(1)的结果将3球面上结和连杆的已知结果推广到更复杂的曲面情况；(3)将(1)和(2)的结果应用到3流形理论。我们特别感兴趣的是得到全局现象的结果（例如，支持属，见下面的4）。本课题的主要成果如下：1)我们得到了加厚曲面中连杆乘积的修正F （L， A）公式，这是对3球中连杆乘积的Jones多项式公式的推广。2) Kauffman-goldman定义了加厚2穿孔平面上特殊隧道链路的电导，并通过基于双终端电网的方法证明了电导是一个链路不变量。基于<D)的性质，我们给出了另一种证明，并推广了加厚多穿孔平面上的链路，这对应于具有多终端的电网的电导。3)我们尝试对F （L, a）进行概化，F （L, a）的系数是反映环节成分数量的多变量Laurent多项式（cf.多变量Alexander多项式），得到了几个结果。4)完整证明了用连通的交替连杆图确定连杆支持格的关键引理，完成了加厚曲面上交替连杆的Tait型定理的证明。少

For a link L in a thickened surface, we had defined the bracket <L) and the link-invariant F (L, A) which are computable directly from a link diagram for L with respect to the projection from the thickened surface to the surface. Both <L) and F (L, A) are evalued as elements of the free module with Laurent polynomial coefficients generated by the integer 1 and all isotopy classes of lcodimension 1 inks in the surface without trivial component. If a (thickened) surface is simply connected, <L) and F (L, A) are equivalent to Kauffman's bracket polynomial and Jones polynomial respectively. Knots and links in non simply connected thickened surfaces cause not only locally knotted-linked phenomena (, I. e. those in a topological 3-ball) but also globally knotted-linked phenomena. <L) and F (L, A) reflect well both such phenomena. The main purposes of this Project are (1) research of properties of <L) and F (L, A), (2) generalization of know results for knots and links in the 3-sphere to thic … More ked surface case by using the results of (1), and (3) application of the results of (1) and (2) to 3-manifold theory. Our special interest is to get results on global phenomena (for example, supporting genus, see 4 below). Main results of this Project are the followings:1) We have the formulae on modified F (L, A) corresponding to product of links in thickened surfaces which is a generalization of the formulae on Jones polynomial corresponding to those of links in the 3-sphere.2) Kauffman-goldman defined conductance for special tunnel links in the thickened 2-punctured plane and showed that it is is a link invariant by the method based on electric network with two terminals. We had an alternative proof based on the property of <D) and a generalization for links in thickened multi-punctured planes which corresponds to conductance of electric networks with multi terminals.3) We tried a generalization of F (L, A) whose coefficients are multi-variable Laurent polynomials reflecting the number of components of links (cf. Multivariable Alexander polynomial) and had several results.4) We had a complete proof of the key lemma to decide supporting genus of links with connected alternating link diagram and completed the proof of Tait type Theorem for alternating links in thickened surfaces. Less