Enlacements d'intervalles et représentation de Gassner.
Enlacements d’intervalles et torsion de Whitehead
Soit un enlacement de intervalles dans d’extérieur et soit . On utilise la propriété de la paire d’être -acyclique pour certaines représentation de l’anneau du groupe fondamental de dans un anneau pour construire des invariants de torsion à valeurs dans le groupe . Un cas particulier est le polynôme d’Alexander en variables quand est l’anneau des fractions rationnelles avec et est simplement l’abélianisation.
Erratum - Espace des représentations du groupe d'un noeud classique dans un groupe de Lie
Erratum : “Real deformations and complex topology of plane curve singularities”
Espace des représentations du groupe d'un noeud classique dans un groupe de Lie
Nous donnons, sous certaines conditions, une méthode générale de construction d’un arc de représentations non métabéliennes d’extrémité une représentation abélienne donnée du groupe d’un noeud d’une sphère d’homologie rationnelle dans un groupe de Lie complexe connexe réductif. Nous déterminons également la structure locale de la variété des représentations au voisinage de la représentation abélienne.
Essential tori admitting a standard tiling
Birman and Menasco (1994) introduced and studied a class of embedded tori in closed braid complements which admit a standard tiling. The geometric description of the tori from this class was not complete. Ng showed (1988) that each essential torus in a closed braid complement which admits a standard tiling possesses a staircase tiling pattern. In this paper, we introduce and study the so-called longitude-meridional patterns for essential tori admitting a standard tiling. A longitude-meridional...
Estimating the states of the Kauffman bracket skein module
The states of the title are a set of knot types which suffice to create a generating set for the Kauffman bracket skein module of a manifold. The minimum number of states is a topological invariant, but quite difficult to compute. In this paper we show that a set of states determines a generating set for the ring of characters of the fundamental group, which in turn provides estimates of the invariant.
Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants.
Euclidean Mahler measure and twisted links.
Every braid admits a short sigma-definite expression
A result by Dehornoy (1992) says that every nontrivial braid admits a -definite expression, defined as a braid word in which the generator with maximal index appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new...
Every knot is a billiard knot
We show that every knot can be realized as a billiard trajectory in a convex prism. This proves a conjecture of Jones and Przytycki.
Examples on Polynomial Invariants of Knots and Links.
Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot.
Exponential separation in 4-manifolds.
Extending the Dehn quandle to shears and foliations on the torus
The Dehn quandle, Q, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, 𝕋², we generalize this to show the existence of a quandle Q̂ extending Q and whose elements are measured geodesic foliations. The quandle action in Q̂ is given by applying a shear along such a foliation to another foliation. We extend some results which related Dehn quandle homology to the monodromy of Lefschetz fibrations. We apply certain quandle 2-cycles...
Families of algebraic curves with nodes
Families of polynomials with total Milnor number constant.
Fatgraph models of RNA structure
In this review paper we discuss fatgraphs as a conceptual framework for RNA structures. We discuss various notions of coarse-grained RNA structures and relate them to fatgraphs.We motivate and discuss the main intuition behind the fatgraph model and showcase its applicability to canonical as well as noncanonical base pairs. Recent discoveries regarding novel recursions of pseudoknotted (pk) configurations as well as their translation into context-free grammars for pk-structures are discussed. This...
Fibred closed braids with disc-band fibre surfaces
A classical result by Stallings provides a necessary and sufficient condition to decide whether a given embedded surface is a fibre in . In this paper it is described how to find a candidate fibre surface for a a link presented as a closed braid. Also it is described an implemented algorithm to find the main ingredients of the necessary and sufficient condition of Stallings, namely presentations of the fundamental groups of the surface and of its complement in , and an explicit expression of...
Fibred Knots and Disks with Clasps.