arXiv research

New algebras model Ozsváth-Szabó's Kauffman-states.

Isomorphism found between Floer homology and contact geometry.

This paper explores A-infinity structures in contact categories and strand algebras.

We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured in 1998 by Broué, Malle and Rouquier and has for consequence that this algebra is a flat deformation of the group algebra of the complex reflection group $G_{32}$, of order 1…

New findings on algebraic structure of hyperbolic graph braid groups.

New homological results for bordered Floer algebras derived from hypertoric categories.

The paper introduces an algebraic structure for the stable Khovanov homology of torus links.

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Math…

We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph …

We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Broué, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequence…

The paper defines a new algebra and provides a basis for it.

Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giv…

In this paper we introduce the tied links, i.e. ordinary links provided with some ties between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpret…

Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.

Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[π]$ of the fundamental group $π$ of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the seco…

Homologies of Jones and partition algebras match cyclic and symmetric groups.

Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…

New Alexander-type invariant for tangles defined and studied.

Researchers define new algebraic structures for knot Floer homology.

Researchers use Mellin-Barnes integrals to study trinomial equations and their braids.

Algorithm finds positive braids with specific crossing matrices.

The paper constructs representations of flat virtual braids by free group automorphisms.

We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the c…

Virtual singular braids embed in a group with normal form.

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

Novel braid groups on surfaces with colored strands intersecting according to graphs.

Braid group proof shows 7-strand is CAT(0).

Characterizes topological sliceness of 3-strand pretzel knots.

Characterizes the OU matrix for up to 5 strands in braids.

New method calculates knot and link biquandle brackets using trace diagrams.

Extends pillowcase homology for immersed curves in a 3-ball.

Let $G$ be one of the Artin groups of finite type ${\mathbf B}_n={\mathbf C}_n$, and affine type $\tilde{\mathbf A}_{n-1}$ and $\tilde{\mathbf C}_{n-1}$. In this paper, we show that if $α$ and $β$ are elements of $G$ such that $α^k=β^k$ for some nonzero integer $k$, then $α$ and $β$ are conjugate in $G$. For the Artin …

Sharp signature bound for 3-strand torus knots proved.

Let $B_n$ denote the classical braid group on $n$ strands and let the {\em mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\cup_nB_{m,n}$. We will describe explicit algebraic moves on $B_{m,\infty}$ such that equivale…

Computes a specific homology for a type of braid.

The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology…

Decategorifies higher actions in Heegaard Floer homology.

Classifies homomorphisms between specific braid groups.

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and the minimal number of conways. The continued fraction associated to rational kno…

Proves knots in handlebodies can be represented as plats of braids.

A new simple method for representing Artin braid groups.

We use a variation on the commutator collection process to characterize those pure braids which become trivial when any one strand is deleted, or, more generally, those pure braids which become trivial when all the strands in any one of a list of sets of strands is deleted.

Just as the Temperley-Lieb algebra is a good place to compute the Jones polynomial, the Kauffman bracket skein algebra of a disk with $2k$ colored points on the boundary, each with color $n$, is a good place to compute the $n^{th}$ colored Jones polynomial. Here, this colored skein algebra is shown to be a cellular alg…

Paper proves homotopy braid group properties over integers and three strands.

Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.

The total homology of the loop space of the configuration space of ordered distinct n points in R^m has a structure of a Hopf algebra defined by the 4-term relations if m>2. We describe a relation of between the cohomology of this loop space and the set of finite type invariants for the pure braid group with n strands.…

New groups from strand diagrams show polycyclic subgroups are virtually abelian and undistorted.

A pretzel knot $K$ is called $odd$ if all its twist parameters are odd, and $mutant$ $ribbon$ if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5-stranded pretzel knots are $mutant$ $ribbon$. We d…