The linear minimal 4-chart with three crossings

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $\Gamma$ be a chart, and we denote by $Cross(\Gamma)$ the set of all the crossings of $\Gamma$, and we denote by $\Gamma_m$ the union of all the edges of label $m$. For a 4-chart $\Gamma$, if each connected component of the set $(\Gamma_1\cup \Gamma_3)-Cross(\Gamma)$ is acyclic, then $\Gamma$ is said to be {\it linear}. In this paper, we shall show that any linear minimal $4$-chart with three crossings is lor-equivalent (Label-Orientation-Reflection equivalent) to the chart describing a $2$-twist spun trefoil knot by omitting free edges and hoops.