A triangulation $\Delta$ of $S^{3}$ defines uniquely a number $m\leq4,$ a subgraph $\Gamma$ of $\Delta$ and a representation $\omega(\Delta)$ of $\pi_{1}(S^{3}\backslash\Gamma)$ into $\Sigma_{m.}$ It is shown that every $(K,\omega)$, where $K$ is a knot or link in $S^{3}$ and $\omega$ is transitive representation of $\pi_{1}(S^{3}\backslash K)$ in $\Sigma_{m},$ $2\leq m\leq3,$ equals $\omega(\Delta)$, for some $\Delta$. From this, a representation of closed, orientable 3-manifolds by triangulations of $S^{3}$ is obtained. This is a theorem of Izmestiev and Joswig, but, in contrast with their proof, the methods in this paper are constructive. Some generalizations are given. The method involves a new representation of knots and links, which is called a butterfly representation.