Oyuki Hayde Hermosillo-Reyes

Unidad Académica de Ciencias Básicas e Ingeniería,
Universidad Autónoma de Nayarit,
Tepic, Nay. 63155, MEXICO.

Title: The group of permutations of 3-cucas, some properties

Abstract:

A 3-cuca is a triplet $\{\Gamma, \{ P_i\}_{i=1}^3, \{\varphi_i\}_{i=1}^3\}$, where $\Gamma$ is a graph embedded in the 2-sphere with three faces, $\{P_i\}_{i=1}^3$, each one homeomorphic to a 2-disc; each face $P_i$ has an even number of nodes $\{v_{ij} \}$ in $\partial P_i$; also there is one bisector on each face; finally $\varphi_i:P_i\to P_i$ is a reflection along the bisector. We also require that

$$\bigcup \{v_{ij}\}=\{(\varphi_1\circ \varphi_2 \circ \varphi_3)^k(e_{i_r})\vert i=1,2,3,\ r=1,2,\ k\in \mathbb{Z}\}$$

where the points $e_{i_r}$ are the ends of the bisectors; this requirement means that the nodes ``moved'' by the reflections $\varphi_i$ are exactly the nodes in the faces.

The set of 3-cucas are interesting for they code all non-splittable 3-bridge links.

We will define, and give facts about diagrams and several groups associated to 3-cucas; for example, the groups generated by the three permutations of order two, $\langle \rho_1 ,\rho_2,\rho_3 \rangle$, induced by the reflections above, which contain information of how the bridges of the associated link are placed. Or, say, the number of orbits of $\langle \rho_1,\rho_2,\rho_3,\sigma_1,\sigma_2 ,\sigma_3\rangle$, give the number of components of the associated link. These different permutation groups give topological information about the link encoded by a 3-cuca.