9 citations · 30 across the 9 of their papers we have counts for
7 papers · 1 filter
A Kontsevich integral of order 1
Arnaud Mortier
We define a 1-cocycle in the space of long knots that is a natural generalization of the Kontsevich integral seen as a 0-cocycle. It involves a 2-form that generalizes the Knizhnik…
Combinatorial cohomology of the space of long knots
Arnaud Mortier
The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We a…
Virtual knot theory on a group
Arnaud Mortier
Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual kn…
Finite-type 1-cocycles of knots given by Polyak-Viro formulas
Arnaud Mortier
We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that…
Polyak type equations for virtual arrow diagram formulas in the annulus
Arnaud Mortier
We describe the space of arrow diagram formulas for virtual knot diagrams in the annulus as the kernel of a linear map, inspired from a conjecture due to M. Polyak. As a main appli…
Gauss diagrams of real and virtual knots in the solid torus
Arnaud Mortier
We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagr…
On homotopies with triple points of classical knots
Thomas Fiedler, Arnaud Mortier
We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point of the cylinder is called {\em coherent} if all three branches intersect at pairwise with the…