An integral expression of the first nontrivial one-cocycle of the space of long knots in ℝ³

Keiichi Sakai

Vol. 250 (2011), No. 2, 407–419

DOI: 10.2140/pjm.2011.250.407

Abstract

Our main object of study is a certain degree-one cohomology class of the space 𝒦₃ of long knots in ℝ³. 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.

Keywords

the space of long knots, configuration space integrals, nontrivalent graphs, an action of little cubes, Gramain cycles, Casson’s knot invariant

Mathematical Subject Classification 2000

Primary: 58D10

Secondary: 55P48, 57M25, 57M27, 81Q30

Milestones

Received: 29 April 2010

Accepted: 2 March 2011

Published: 31 March 2011