On the sharpness of some lower bounds for the crossing number of links in thickened surfaces

We prove the sharpness of the two lower bounds for the crossing number of oriented links in thickened surfaces previously which were obtained by the author. The first bound $\operatorname{cr}(\ell) \geq 2{\operatorname{vg}}(\ell) +1$ concerns with homologically trivial links. The second bound $\operatorname{cr}(\ell) \geq {\operatorname{h}}(\ell) +2{\operatorname{vg}}(\ell) -2$ relates to the links that are not homologically trivial. Here $\operatorname{cr}(\ell)$, ${\operatorname{vg}}(\ell)$, and ${\operatorname{h}}(\ell)$ denote the crossing number, the virtual genus, and the homological multiplicity of an oriented link $\ell$. The homological multiplicity of $\ell$ is the greatest common divisor of the nonzero coefficients in the expansion of $[\ell]$ in an arbitrary basis of the first homology group of the thickened surface. By the sharpness of these bounds we mean that, for each nonnegative value of the homological multiplicity and for each positive value of the virtual genus, there is an oriented link for which the corresponding inequality becomes an equality. We prove the existence of these links by providing the explicit description of their minimal diagrams; i.e., we explicitly describe some infinite two-parameter family of oriented links in thickened surfaces for which the exact value of the crossing number is computed.