Vassiliev invariants for flows via Chern–Simons perturbation theory

The perturbative expansion of Chern–Simons gauge theory leads to invariants of knots and links, the so-called finite type invariants or Vassiliev invariants. It has been proved that at any order in perturbation theory the superposition of certain amplitudes is an invariant of that order. Bott–Taubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological framework. One of the consequences of this formalism is that the resulting amplitudes are rewritten in cohomological terms in configuration spaces. This cohomological structure can be used to translate Bott–Taubes integrals into Chern–Simons perturbative amplitudes and vice versa. In this paper, this program is performed up to third order in the coupling constant. This expands some work previously worked out by Thurston. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the 3-manifold. The Bott–Taubes integrals obtained are used for constructing higher-order average asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.

Crossing change on Khovanov homology and a categorified Vassiliev skein relation

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the “genus-one” operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

Finite-type invariants of order one of long and framed virtual knots

We generalize three invariants, first discovered by Henrich, to the long and/or framed virtual knot case. These invariants are all finite-type invariants of order one, and include a universal invariant. The generalization will require us to extend the notion of a based matrix of a virtual string, first introduced by Turaev and later generalized by Henrich, to the long and framed cases.

Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of [Formula: see text]-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.