Invariants of -type links via an extension of the Kauffman bracket.
Involutory Hopf group-coalgebras and flat bundles over 3-manifolds
Given a group π, we use involutory Hopf π-coalgebras to define a scalar invariant of flat π-bundles over 3-manifolds. When π = 1, this invariant equals the one for 3-manifolds constructed by Kuperberg from involutory Hopf algebras. We give examples which show that this invariant is non-trivial.
Kashaev's conjecture and the Chern-Simons invariants of knots and links.
Khovanov homology, its definitions and ramifications
Mikhail Khovanov defined, for a diagram of an oriented classical link, a collection of groups labelled by pairs of integers. These groups were constructed as the homology groups of certain chain complexes. The Euler characteristics of these complexes are the coefficients of the Jones polynomial of the link. The original construction is overloaded with algebraic details. Most of the specialists use adaptations of it stripped off the details. The goal of this paper is to overview these adaptations...
Khovanov-Rozansky homology for embedded graphs
For any positive integer n, Khovanov and Rozansky constructed a bigraded link homology from which you can recover the 𝔰𝔩ₙ link polynomial invariants. We generalize the Khovanov-Rozansky construction in the case of finite 4-valent graphs embedded in a ball B³ ⊂ ℝ³. More precisely, we prove that the homology associated to a diagram of a 4-valent graph embedded in B³ ⊂ ℝ³ is invariant under the graph moves introduced by Kauffman.
Khovanov's homology for tangles and cobordisms.
Knot and braid invariants from contact homology. I.
Knot and braid invariants from contact homology. II.
Knot Floer homology and the four-ball genus.
Knot theory with the Lorentz group
We analyse perturbative expansions of the invariants defined from unitary representations of the Quantum Lorentz Group in two different ways, namely using the Kontsevich Integral and weight systems, and the R-matrix in the Quantum Lorentz Group defined by Buffenoir and Roche. The two formulations are proved to be equivalent; and they both yield ℂ[[h]]h-valued knot invariants related with the Melvin-Morton expansion of the Coloured Jones Polynomial.
Large embedded balls and Heegaard genus in negative curvature.
Legendrian and transverse twist knots
In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot with crossing number . In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that has exactly Legendrian representatives with maximal Thurston–Bennequin...
Legendrian knots and monopoles.
Les invariants des 3-variétés périodiques
Soit un entier . Une 3-variété est dite -périodique si et seulement si le groupe cyclique agit semi-librement sur avec un cercle comme l’ensemble des points fixes. Dans cet article, nous utilisons les invariants quantiques pour établir des conditions nécessaires pour qu’une 3-variété soit périodique.
Levelling an unknotting tunnel.
Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers
Link homology and Frobenius extensions
We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.
Link invariants from finite biracks
A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, (t,s)-racks, Alexander biquandles and Silver-Williams switches, known as (τ,σ,ρ)-biracks. We consider enhancements of the counting invariant using writhe vectors,...
Link invariants from finite racks
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
Links and cycles II: families of commutation relations, parametrized by knots, in the mapping class groups and beyond
We generate families of commutation relations in various groups, by examining quandle colorings of knots and their quandle 2-cycles. The colorings are via quandles associated to the given groups.