Invariants for links in thickened surfaces and its applications

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Tait conjectures on crossing numbers of alternating links in the 3-dimensional space means the following two conjectures :TC1) If two connected, reduced and alternating link diagrams represent equivalent (ie, ambient isotopic) links, their crossing numbers will be the same.TC2) The crossing number of a connected, reduced and alternating link diagram D will be minimal among all crossing numbers of link diagrams which represent links equivalent to that D does.These conjectures had been unsolved for more than 100 years untill they were positively solved in 1987 after Jones polynomial appeared. In 1996, N.Kamada made a pioneer-work to establish anologous results for alternating links in thickened surfaces. She showed an anologous result to TC1) under a certain assumption and conjectured that the assumption can be removed in her paper. In this research project, we solved her conjecture positively . And, by applying the idea of the proof of her conjecture, we established an anologous result … More to TC2), which is a final goal to seeking anologous results to Tait conjectures in knot-link theory in thickened surfaces because TC2) implies TC1. Our these results was showed for more general both surfaces and link diagrames than those in her paper. In the proofs of our these results, we use the following two fundamental lemmta :ENGULFING LEMMA : For two link diagrams on a surface, one of their regular neiborhoods in the the surface can contain the other by isotopic deformation in the surface under a certain condition,DUAL STATE LEMMA (thickened surface version) : For a pair of dual states of a link diagram D on a surface, the sum of numbers of connected components of two (no crossing) diagrams obtained from D by resolving all crossings according to each state does not exceed to the number of boundary components of the regular neighhood of D in the surface under a certain condition.These lemmata, themselves, are impotant results in our resaerch. Each invesigator got intereting own results as the fundamental research related to this project. Less