Research of global knot theory in thickened surfaces

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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