Embeddability of right-angled Artin groups on complements of trees

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for finite simplicial graphs $\Gamma_1$ and $\Gamma_2$ if $\Gamma_1$ embeds into $\Gamma_2^e$ as an induced subgraph then $A(\Gamma_1)$ embeds into $A(\Gamma_2)$. In this article we show that the converse of this theorem does not hold for the case $\Gamma_1$ is the complement of a tree and for the case $\Gamma_2$ is the complement of a path graph.Comment: published version in International Journal of Algebra and Computatio

Unknotting number and genus of 3-braid knots

Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.Comment: published versio

Some power of an element in a Garside group is conjugate to a periodically geodesic element

We show that for each element $g$ of a Garside group, there exists a positive integer $m$ such that $g^m$ is conjugate to a periodically geodesic element $h$, an element with |h^n|_\D=|n|\cdot|h|_\D for all integers $n$, where |g|_\D denotes the shortest word length of $g$ with respect to the set \D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside group, its stable super summit set.Comment: Subj-class of this paper should be Geometric Topology; Version published by BLM

Welfare Effects of Intellectual Property Rights Under Asymmetric Spillovers

We develop a model with one innovating northern firm and several heterogeneous Southern firms that compete in a final product market. We assume the southern firms differ in their ability to adapt technology and use this heterogeneity to study the differing incentives of southern governments to protect intellectual property rights. We find that governments representing more efficient firms have greater incentive to protect IPR than do those representing less efficient firms. However, efficiency considerations imply that, given policies resulting in the same overall innovation rate, it would be better to have weaker IPR protection for the more efficient southern firms.innovation; imperfect competition; commercial policy; intellectual property rights protection; trade

Braid groups of imprimitive complex reflection groups

We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group $B(de,e,r)$ is a semidirect product of the braid group of affine type $\widetilde A_{r-1}$ and an infinite cyclic group. Elements of $B(de,e,r)$ are visualized as geometric braids on $r+1$ strings whose first string is pure and whose winding number is a multiple of $e$. We classify periodic elements, and show that the roots are unique up to conjugacy and that the braid group $B(de,e,r)$ is strongly translation discrete.Comment: published versio