It is well-known that every closed orientable 3-manifold $M^3$ is the 3-fold simple covering $M^3(K,\omega)$ of $S^3$ branched over a knot K: hence, $M^3$ may be visualized by the associated coloured knot $(K,\omega)$. On the other hand, PL-manifolds of arbitrary dimension may be represented by coloured graphs, via pseudosimplicial triangulations. The present paper produces an algorithm to construct a 4-coloured graph representing $M^3(K,\omega)$, directly 'drawn over' the coloured knot $(K,\omega)$.
Coloured knots and coloured graphs representing 3-fold simple coverings of S3 / Casali, Maria Rita. - In: DISCRETE MATHEMATICS. - ISSN 0012-365X. - STAMPA. - 137:(1995), pp. 87-98. [10.1016/0012-365X(93)E0145-T]
Coloured knots and coloured graphs representing 3-fold simple coverings of S3
CASALI, Maria Rita
1995
Abstract
It is well-known that every closed orientable 3-manifold $M^3$ is the 3-fold simple covering $M^3(K,\omega)$ of $S^3$ branched over a knot K: hence, $M^3$ may be visualized by the associated coloured knot $(K,\omega)$. On the other hand, PL-manifolds of arbitrary dimension may be represented by coloured graphs, via pseudosimplicial triangulations. The present paper produces an algorithm to construct a 4-coloured graph representing $M^3(K,\omega)$, directly 'drawn over' the coloured knot $(K,\omega)$.
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