2021-07-28T00:18:52Zhttps://soar-ir.repo.nii.ac.jp/oaioai:soar-ir.repo.nii.ac.jp:000117512021-07-05T00:36:42Z1169:1170An integral expression of the first nontrivial one-cocycle of the space of long knots in R(3)Sakai, Keiichi36019the space of long knotsconfiguration space integralsnontrivalent graphsan action of little cubesGramain cyclesCasson's knot invariantOur main object of study is a certain degree-one cohomology class of the space K(3) of long knots in R(3). We describe this class in terms of graphs and configuration space integrals, showing the vanishing of some anomalous obstructions. To show that this class is not zero, we integrate it over a cycle studied by Gramain. As a corollary, we establish a relation between this class and (R-valued) Casson's knot invariant. These are R-versions of the results which were previously proved by Teiblyum, Turchin and Vassiliev over Z/2 in a different way from ours.ArticlePACIFIC JOURNAL OF MATHEMATICS. 250(2):407-419 (2011)journal articlePACIFIC JOURNAL MATHEMATICS2011-04application/pdfPACIFIC JOURNAL OF MATHEMATICS22504074190030-8730AA00767097https://soar-ir.repo.nii.ac.jp/record/11751/files/pjm-v250-n2-p08-p.pdfeng10.2140/pjm.2011.250.407http://dx.doi.org/10.2140/pjm.2011.250.407