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

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.

arXiv comment15 pages, 19 figures, Figure 4.9 corrected

References in corpus (5)

• Configuration spaces and Vassiliev classes in any dimension
• Non-trivalent graph cocycle and cohomology of the long knot space
• An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3
• Combinatorial cohomology of the space of long knots
• BV-structures on the homology of the framed long knot space