Project/Area Number  10640085 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  Tokyo Metropolitan University 
Principal Investigator 
KURATA Toshihiko Tokyo Metoropolitan University, Graduate school of Science, 理学研究科, 助手 (40311899)

CoInvestigator(Kenkyūbuntansha) 
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)
阿原 一志 明治大学, 理工学部, 専任講師 (80247147)
OKA Mutsuo Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (40011697)
荻上 紘一 東京都立大学, 総長 (10087025)
OHNITA Yoshihiro Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (90183764)
NAKAMULA Ken Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (80110849)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 2001 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥1,200,000 (Direct Cost : ¥1,200,000)

Keywords  topology / knot / energy / トポロジー / 結び目理論 / エネルギー / 低次元トポロジー / 結び目のエネルギー 
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 crossratio, we described the energy E in terms of its realpart 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 crossratio 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 recursivecalls, 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 (typefree) λcalculus 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.
