We show that the integration of a 1-cocycle $I(\Gamma)$ 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 speaker (2011), proving that the integration of $I(\Gamma)$ over the Gramain 1-cycles is Casson's knot invariant. The result 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 $I(\Gamma)$ is given by the configuration space integrals associated with graphs, while Mortier's cocycle is obtained in a combinatorial way. This talk is based on joint work with Saki Kanou.