The Fox–Hatcher cycle and a Vassiliev invariant of order three

Saki Kanou and Keiichi Sakai

Vol. 323 (2023), No. 2, 281–306

DOI: 10.2140/pjm.2023.323.281

Abstract

We show that the integration of a 1-cocycle $I(X)$ of the space of long knots in ${ℝ}^3$ over the Fox–Hatcher 1-cycles gives rise to a Vassiliev invariant of order exactly three. This result can be seen as a continuation of the previous work of the Sakai (2011), proving that the integration of $I(X)$ over the Gramain 1-cycles is the Casson invariant, the unique nontrivial Vassiliev invariant of order two (up to scalar multiplications). The result in the present paper is also analogous to part of Mortier’s result (2015). Our result differs from, but is motivated by, Mortier’s one in that the 1-cocycle $I(X)$ is given by the configuration space integrals associated with graphs while Mortier’s cocycle is obtained in a combinatorial way.

Keywords

Spaces of embeddings: Configuration space integrals: the Fox–Hatcher cycle: Vassiliev invariants

Mathematical Subject Classification

Primary: 57K16, 58D10

Secondary: 81Q30

Milestones

Received: 3 April 2022

Revised: 28 February 2023

Accepted: 13 March 2023

Published: 2 June 2023