Vol. 250, No. 2, 2011

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An integral expression of the first nontrivial one-cocycle of the space of long knots in 3

Keiichi Sakai

Vol. 250 (2011), No. 2, 407–419
Abstract

Our main object of study is a certain degree-one cohomology class of the space 𝒦3 of long knots in 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 (-valued) Casson’s knot invariant. These are -versions of the results which were previously proved by Teiblyum, Turchin and Vassiliev over 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
Authors
Keiichi Sakai
Department of Mathematical Sciences
Shinshu University
3-1-1 Asahi, Matsumoto 390-8621
Japan
http://math.shinshu-u.ac.jp/~ksakai/index.html