A triangulation $\Delta $ of $S^{3}$ defines uniquely a number $m\leq
4,$ 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 $\Sigma _{m},$ $2\leq m\leq 3,$ equals
$\omega (\Delta )$, for some $\Delta $. From this, a representation of
closed, orientable 3-manifolds by triangulations of $S^{3}$ will be
obtained. This is a theorem of Izmestiev and Joswig, but, in contrast
with their proof, our methods are constructive. Some generalizations are
given. The method involves a new representation of knots and links,
which is called a butterfly representation.
The talk will present a constructive proof of a recent theorem of
Izmestiev and Joswig. A triangulation $\Delta $ of $S^{3}$ defines
uniquely a number $m\leq 4,$ a subgraph $\Gamma $ of $\Delta $ and a
representation $\omega (\Delta )$ of $\pi _{1}(S^{3}\backslash \Gamma )$
into $\Sigma _{m.}$ . It will be 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 $\Sigma _{m},$ $2\leq
m\leq 3,$ equals $\omega (\Delta )$, for some $\Delta $. From this, the
theorem of Izmestiev and Joswig will be obtained, that is, a
representation of closed, orientable 3-manifolds by triangulations of
$S^{3}$.