Quandles and Monodromy

Quandles and Lefschetz Fibrations

We show that isotopy classes of simple closed curves in any oriented surface admit a quandle structure with operations induced by Dehn twists, the Dehn quandle of the surface. We further show that the monodromy of a Lefschetz fibration can be conveniently encoded as a quandle homomorphism from the knot quandle of the base as a manifold with a codimension 2 subspace (the set of singular values) to the Dehn quandle of the generic fibre, and discuss prospects for construction of invariants arising naturally from this description of the monodromy.

Homogeneous quandles arising from automorphisms of symmetric groups

Quandle is an algebraic system with one binary operation, but it is quite different from a group. Quandle has its origin in the knot theory and good relationships with the theory of symmetric spaces, so it is well-studied from points of view of both areas. In the present paper, we investigate a special kind of quandles, called generalized Alexander quandles $Q(G,ψ)$, which is defined by a group $G$ together with its group automorphism $ψ$. We develop the quandle invariants for generalized Alexander quandles. As a result, we prove that there is a one-to-one correspondence between generalized Al…

Automorphism groups of quandles arising from groups

Let $G$ be a group and $φ\in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=φ(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, φ)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $φ= -\id_G$, then $\Alex(G, φ)$ is the well-known Takasaki quandle. In this paper, we determine the group of automorphisms and inner automorphisms of Takasaki quandles of abelian groups with no 2-torsion, and Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. As an application, we prove that if $G\cong (\mathbb…

On the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4

A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3.

与Pin群相关的Coxeter拟阵的变体

Variants of Coxeter quandles associated with Pin groups

本文引入两类由Coxeter拟阵导出的拟阵：双覆盖拟阵和旋转$D_n$拟阵，并确定其内自同构群。

Links arising from braid monodromy factorizations

We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated and regenerated surfaces in the latter. In particular we focus on degenerations which involve intersection points of multiplicity two and three. We demonstrate when the same links arise even when the local arrangements are different.