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Arnaud Mortier

Publications and source records attributed to Arnaud Mortier.

10 recordsLinked to original sources
arXiv2025

Somatosensory prediction in premature neonates: iatrogenic pain experience increases repetition suppression and deviance detection of innocuous stimuli in a tactile oddball protocol

Sensory prediction (SP) is a fundamental mechanism of perception that supports cognitive development. Atypical SP has been reported across multiple neurodevelopmental disorders (ND), suggesting it may constitute an early cross-syndromic marker. Premature birth is a major risk factor for ND, with risk increasing as gestational age (GA) at birth decreases. However, how perinatal risk factors shape the development of SP remains poorly understood. We do not know if untimely birth itself, or exposure to iatrogenic pain during neonatal intensive care, cause neurodevelopmental impairments. In this study, we first assessed whether SP can be detected in the brains of premature neonates at 35 weeks corrected GA using a tactile oddballomission paradigm with EEG. We then examined the effects of the degree of prematurity and of the exposure to painful care procedures on neural indices of SP. Results demonstrate the presence of repetition suppression (RS) and a mismatch response (MMR) to deviant stimuli in the contralateral somatosensory cortex of premature neonates. The amplitude of these SP proxies was significantly affected by the number of painful procedures experienced since birth, independently of the effect of GA at birth. Contrary to our initial hypothesis that greater neurodevelopmental risk would be associated with less mature SP, infants with higher exposure to pain exhibited more robust indices of SP. These findings suggest that increased ex utero experience, even painful, is associated with accelerated maturation of predictive somatosensory processing. Longitudinal follow-up of participants at age 2 will explore how these early markers relate to developmental outcomes.

q-bio.NC↗
arXiv2022

A Kontsevich integral of order 1

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--Zamolodchikov connection. We show that the well-known close relationship between the Kontsevich integral and Vassiliev invariants (via the algebra of chord diagrams and 1T-4T relations) is preserved between our integral and Vassiliev 1-cocycles, via a change of variable similar to the one that led Birman--Lin to discover the 4T relations. We explain how this construction is related to Cirio--Faria Martins' categorification of the Knizhnik--Zamolodchikov connection.

math.GT↗
arXiv2021

Abelian quandles and quandles with abelian structure group

Sets with a self-distributive operation (in the sense of $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c))$, in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study of the Yang-Baxter equation, and other areas. An important invariant of quandles is their structure group. The structure group of a finite quandle is known to be either "boring" (free abelian), or "interesting" (non-abelian with torsion). In this paper we explicitly describe all finite quandles with abelian structure group. To achieve this, we show that such quandles are abelian (i.e., satisfy $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft b)$; present the structure group of any abelian quandle as a central extension of a free abelian group by an explicit finite abelian group; and determine when the latter is trivial. In the second part of the paper, we relate the structure group of any quandle to its 2nd homology group $H_2$. We use this to prove that the $H_2$ of a finite quandle with abelian structure group is torsion-free, but general abelian quandles may exhibit torsion. Torsion in $H_2$ is important for constructing knot invariants and pointed Hopf algebras.

math.GR↗
arXiv2017

A translation of A.A.Vinogradov's "On the free product of ordered groups"

This is a translation from Russian of the article by A.A.Vinogradov: On the free product of ordered groups, Mat. Sb. (N.S.), 1949, Volume 25(67), Number 1, 163-168. The main result is that the free product of two ordered groups is orderable. Minor corrections were made and highlighted in footnotes.

math.GR↗
arXiv2014

Combinatorial cohomology of the space of long knots

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 achieve this goal by defining a combinatorial graded cochain complex, such that the elements of an explicit submodule in the cohomology define algebraic intersections with some "geometrically simple" strata in the space of knots. Such strata are endowed with explicit co-orientations, that are canonical in some sense. The combinatorial tools involved are natural generalisations (degeneracies) of usual methods using arrow diagrams.

math.GT↗
arXiv2014

Virtual knot theory on a group

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 knots on a given arbitrary surface. This theory contains all of the earlier attempts to decorate Gauss diagrams, in a way that is made precise via symmetry-preserving maps. These maps become crucial when one makes use of decorated Gauss diagrams to describe finite-type invariants. In particular they allow us to generalize Grishanov-Vassiliev's formulas and to show that they define invariants of virtual knots.

math.GT↗
arXiv2014

Finite-type 1-cocycles of knots given by Polyak-Viro formulas

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 these formulas always represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to coincide over Z/2 with the Teiblum-Turchin cocycle v_3^1.

math.GT↗
arXiv2012

Gauss diagrams of real and virtual knots in the solid torus

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 diagram can be fully recovered from its decorated Gauss diagram, and we use it to establish a characterization of the decorated Gauss diagrams of closed braids.

math.GT↗
arXiv2010

On homotopies with triple points of classical knots

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant $v_2(K)$ by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the homotopy.

math.GT↗