An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3

Keiichi Sakai

doi:10.2140/pjm.2011.250.407

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arXiv · 1104.0073

An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3

Keiichi Sakai

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Abstract

Our main object of study is a certain degree-one cohomology class of the space K 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.

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Keiichi Sakai. 2011-04-01. An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3. https://doi.org/10.2140/pjm.2011.250.407

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