Research of global knot theory in thickened surfaces

对于加厚曲面中的链环L，我们定义了括号（L）和链环不变量F（L，A），它们可以直接从L关于加厚曲面到曲面的投影的链环图计算。F（L，A）和F（L，A）都是由整数1生成的Laurent多项式系数的自由模的元素，且无平凡分量.如果曲面是单连通的，则F（L，A）和F（L，A）分别等价于Kauffman括号多项式和Jones多项式。非单连通加厚曲面中的纽结和链环不仅引起局部纽结-链环现象（，I。e.拓扑3球中的那些），而且是全局打结链接的现象。<L）和F（L，A）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）将三维球面上纽结和链环的已知结果推广到三维球面上，并在此基础上建立了三维球面上纽结和链环的数学模型。 ...更多信息 Ked曲面情形;（3）将（1）和（2）的结果应用于三维流形理论。我们特别感兴趣的是得到全局现象的结果（例如，支持亏格，见下面的4）。主要成果如下：1）给出了加厚曲面上链路乘积的修正F（L，A）公式，它是3-球面上链路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，并利用两端电网络的方法证明了它是链路不变量. 3）我们尝试了F（L，A）的一个推广，它的系数是反映链路分量个数的多变量Laurent多项式（参见图1）。多变量亚历山大多项式），并得到了几个结果。4）完整地证明了确定连通交错链环图的链环支撑亏格的关键引理，完成了加厚曲面上交错链环的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