Related papers: Ribbon 2-knot groups of Coxeter type
We introduce ribbon-moves of 2-knots, which are operations to make 2-knots into new 2-knots by local operations in B^4. (We do not assume the new knots is not equivalent to the old ones.) Let L_1 and L_2 be 2-links. Then the following hold.…
In this paper we study affine reflection subgroups in arbitrary infinite Coxeter groups of finite rank. In particular, we study the distribution of roots of Coxeter groups in the root subsystems associated with affine reflection subgroups.…
We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a…
We study the cohomology ring of the Bott--Samelson variety. We compute an explicit presentation of this ring via Soergel's result, which implies that it is a purely combinatorial invariant. We use the presentation to introduce the…
In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we…
In this work we solve the problem of providing a Morita invariant definition of Lie and Courant algebroids over Lie groupoids. By relying on supergeometry, we view these structures as instances of vector fields on graded groupoids which are…
Using a previous classification result on symmetric additive 2-cocycles, we collect a variety of facts about the Lubin-Tate cohomology of formal groups to compute the 2-primary component of the scheme of symmetric multiplicative 2-cocycles.…
In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and…
In one of his papers on the weak order of Coxeter groups, Dyer formulates several conjectures. Among these, one affirms that the extended weak order forms a lattice, while another offers an algebraic-geometric description of the join of two…
We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly,…
For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that 2-solvable knots are in particular algebraically slice and have vanishing Casson-Gordon obstructions. Similarly all…
We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of…
In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $\beta$ in its inversion set $\Phi(x)$, the set of elements $y \in W$…
We investigate restricted Lie algebras arising as analogues of (twisted) right-angled Artin groups and right-angled Coxeter groups over fields of characteristic two. These algebras are defined via quadratic relations determined by decorated…
Following the theory of principal $\infty$-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of…
We combinatorially characterize the number $\mathrm{cc}_2$ of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count…
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…
We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra…
We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…
It is well known that there are many 2-torsion elements in the classical knot concordance group. On the other hand, it is not known if there is any torsion element in the rational knot concordance group $\mathcal{C}_\mathbb{Q}$. Cha defined…