The crossing number of the generalized Petersen graph $P(3k,k)$ in the projective plane

The crossing number of a graph $G$ in a surface $\mathrm{\Sigma }$, denoted by $c{r}_{\mathrm{\Sigma }}(G)$, is the minimum number of pairwise intersections of edges in a drawing of $G$ in $\mathrm{\Sigma }$. Let $k$ be an integer satisfying $k\ge 3$, the generalized Petersen graph $P(3k,k)$ is the graph with vertex set $V(P(3k,k))=\{u_i,v_i\mathrm{\mid }i=1,2,\cdots ,3k\}$ and edge set $E(P(3k,k))=\{u_i{u}_{i+1},u_i{v}_i,v_i{v}_{k+i}\mathrm{\mid }i=1,2,\cdots ,3k\},$ the subscripts are read modulo $3k\mathrm{.}$ This paper investigates the crossing number of $P(3k,k)$ in the projective plane. We determine the exact value of $c{r}_{N_1}(P(3k,k))$ is $k-2$ when $3\le k\le 7,$ moreover, for $k\ge 8,$ we get that $k-2\le c{r}_{N_1}(P(3k,k))\le k-1.$