  An orientation of a graph is semi-transitive if it is acyclic, and for any
directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either
there is no edge between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an edge
for all $0\leq i<j\leq k$. An undirected graph is semi-transitive if it admits
a semi-transitive orientation. Semi-transitive graphs include several important
classes of graphs such as 3-colorable graphs, comparability graphs, and circle
graphs, and they are precisely the class of word-representable graphs studied
extensively in the literature.
  In this paper, we study semi-transitive orientability of the celebrated
Kneser graph $K(n,k)$, which is the graph whose vertices correspond to the
$k$-element subsets of a set of $n$ elements, and where two vertices are
adjacent if and only if the two corresponding sets are disjoint. We show that
for $n\geq 15k-24$, $K(n,k)$ is not semi-transitive, while for $k\leq n\leq
2k+1$, $K(n,k)$ is semi-transitive. Also, we show computationally that a
subgraph $S$ on 16 vertices and 36 edges of $K(8,3)$, and thus $K(8,3)$ itself
on 56 vertices and 280 edges, is non-semi-transitive. $S$ and $K(8,3)$ are the
first explicit examples of triangle-free non-semi-transitive graphs, whose
existence was established via Erd\H{o}s' theorem by Halld\'{o}rsson et al. in
2011. Moreover, we show that the complement graph $\overline{K(n,k)}$ of
$K(n,k)$ is semi-transitive if and only if $n\geq 2k$.
