arXiv research

Computes a specific homology for a type of braid.

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.

The paper studies Coxeter quotients of surface braid groups.

This paper introduces cluster exchange groupoids for Coxeter-Dynkin diagrams and finds their fundamental groups are braid groups.

Triangulates permutahedra for Coxeter groups, revealing braid group connections.

We interpret Coxeter's truncated braid groups in terms of Platonic solids.

Study cobordism distances between 3-braid links and trefoil knots.

New reflection groups derived from torus knots with finite meridians.

A Coxeter link is a closure of a product of two braids, one being a quasi-Coxeter element and the other being a product of partial full twists. This class of links includes torus knots \(T_{n,k}\) and torus links \(T_{n,nk}\). We identify the knot homology of a Coxeter link with the space of sections of a particular li…

In the present paper, we prove that the group $G_{n}^{2}$ of free $n$-strand braids is isomorphic to a subgroup of a semidirect product of some Coxeter group that we denote by $C(n,2)$ and the symmetric group $S_{n}$.

The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.

We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use…

Virtual twin groups map to symmetric groups, revealing automorphism structure.

Research on knots, braids, and their invariants.

Study extends symmetries of sphere points to surface mapping classes.

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical …

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A_n, B_n=C_n and D_n and the affine diagrams tilde{A}_n, tilde{B}_n, tilde{C}_n and tilde{D}_n as subgroups of the braid gr…

Study on twin groups with algorithm and properties.

We conjecture that the complex of Soergel bimodules associated with the full twist braid is categorically diagonalizable, for any finite Coxeter group. This utilizes the theory of categorical diagonalization introduced earlier by the authors. We prove our conjecture in type $A$, and as a result we obtain a categorifica…

The study characterizes subgroups of braid groups and their finite quotients.

We prove the $K(π,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol'd, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on sever…

We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on th…

Reduces conjecture for Artin groups to simpler cases.

The study explores continuous noncrossing partitions and their relation to weighted circular factorizations.

The final result of this article gives the order of the extension $$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] & 1}$$ as an element of the cohomology group $H^2(W,P/[P,P])$ (where $B$ and $P$ stands for the braid group and the pure braid group associated to the complex reflection group $W$)…

Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös-Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about t…

A classical result of H. S. M. Coxeter asserts that a certain quotient $B(m,n)$ of the braid group $B(m)$ on $m$ strands is finite if and only if $(m,n)$ corresponds to the type of one of the five Platonic solids. If ${\bf k}$ is a knot or virtual knot, one can study similar quotients $G({\bf k}, n)$ for the correspond…

Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.

New proof shows not all Salem numbers are growth rates of Coxeter groups.

Let $Γ$ be a Coxeter graph, let $(W,S)$ be its associated Coxeter system, and let $(A,Σ$) be its associated Artin-Tits system. We regard $W$ as a reflection group acting on a real vector space $V$. Let $I$ be the Tits cone, and let $E_Γ$ be the complement in $I +iV$ of the reflecting hyperplanes. Recall that Charney, D…

Survey on Coxeter groups for Lie group examples.

Study on connectivity of Morse boundaries of Coxeter groups.

New proofs for growth series of Coxeter groups using complex structures.

Introduces Coxeter polyhedra in various geometries.

Characterizes Coxeter groups with convex cocompact representations in projective space.

The paper characterizes Conway-Coxeter friezes using rational links.

New groups found in hyperbolic 4D and 5D space have minimal growth rate.

We prove that two angle-compatible Coxeter generating sets of a given finitely generated Coxeter group are conjugate provided one of them does not admit any elementary twist. This confirms a basic case of a general conjecture which describes a potential solution to the isomorphism problem for Coxeter groups.

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cu…

Flat coordinates found for algebraic Frobenius manifolds in low dimensions.

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-…

We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of symmetries. The periodic mapping classes can in turn be used to construct sequences of pse…

New classification of hyperbolic Coxeter prisms.

Surprising circles found in Coxeter group boundaries.

The paper calculates the Saito determinant for Coxeter discriminant strata.

New noncompact Coxeter polytopes found in various dimensions.

We consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for c…

The paper explores growth rates and Perron numbers in Coxeter systems with low-dimensional Davis complexes.