Volumes for twist link cone-manifolds

Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot $4_1$ and the links $5^2_1$ and $6^2_2$, have been obtained by the second named author and his collaborators. In this paper we explicitly find the hyperbolic volume for cone-manifolds with the link $6^2_3$ as singular set. Trigonometric identities (Tangent, Sine and Cosine Rules) between complex lengths of singular components and cone angles are obtained for an infinite family of two-bridge links containing $5^2_1$ and $6^2_3$.Comment: 23 pages, 1 figure. Revised version with minor changes. Accepted for publication in the Boletin de la Sociedad Matematica Mexicana, special issue in honor of Francisco "FICO" Gonzales Acun

On Jacobian group of the Δ\Delta-graph

In the present paper we compute the Jacobian group of $\Delta$-graph $\Delta(n; k, l, m).$ The notion of $\Delta$-graph continues the list of families of $I$-, $Y$- and $H$-graphs well-known in the graph theory. In particular, graph $\Delta(n; 1, 1, 1)$ is isomorphic to discrete torus $C_3\times C_n.$ It this case, the structure of the Jacobian group will be find explicitly.Comment: arXiv admin note: text overlap with arXiv:2111.0430