| Co-Investigator(Kenkyū-buntansha) |
OKA Mutsuo Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (40011697)
OHNITA Yoshihiro Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (90183764)
IMAI Jun Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 助教授 (70221132)
DEGUCHI Tetsuo Ochanomizu university, Faculty of Science, 理学部, 助教授 (70227544)
NAKAMULA Ken Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (80110849)
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| Research Abstract |
Roughly speaking, energy of knots is a functional on the space of knots which fulfills the property that it explodes whenever a knot degenerates to a singular knot with double points. During the period of the project, we have studied this concept of energy functional together with Prof. R. Langevin of Universit de Bourgogne mainly to clarify the correspondence between our introduced functional E and the one due to Prof. Langevin which is contrived from the viewpoint of integral geometry. As a result, considering the notion of infinitesimal cross-ratio, we described the energy E in terms of its real-part and its absolute value, and also obtained a formula which describes Langevin's introduced quantity. Furthermore, we newly introduced several conformally invariant functional from a discussion independent of the approach of infinitesimal cross-ratio and integral geometry.
Besides the observation above, we also direct our attention to a strong analogy between the assignment of algebraic invariant of knots and the assignment of denotation of programs containing recursive-calls, and especially study the general mechanism of denotational semantics of cyclic structures appearing in the syntax of programs. In consequence, we obtained an algebraic structure which enables us to model not only the computational paradigm of (type-free) λ-calculus but the internal structure of functional body. Our construction can be regarded as enough simple in comparison with some known results, and so we expect to find a neat relationship to the discussion of energy of knots through a general framework, such as the theory of traced monoidal categories.
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