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

TL;DR

This paper demonstrates that integrating a specific 1-cocycle over Fox-Hatcher cycles yields a Vassiliev invariant of order three, extending previous work on invariants of lower order and connecting configuration space integrals with knot invariants.

Contribution

It introduces a new Vassiliev invariant of order three derived from configuration space integrals, expanding the understanding of knot invariants beyond order two.

Findings

Integration over Fox-Hatcher cycles produces a third-order Vassiliev invariant.

The result generalizes previous work on the Casson invariant and Gramain cycles.

It links configuration space integrals with Vassiliev invariants of higher order.

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 second named author, 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. 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.