### $3$-dimensional Euclidean manifolds represented by locally regular coloured graphs

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We describe a combinatorial algorithm for constructing all orientable 3-manifolds with a given standard bidimensional spine by making use of the idea of bijoin (Bandieri and Gagliardi (1982), Graselli (1985)) over a suitable pseudosimplicial triangulation of the spine.

We formulate a conjectural formula for Khovanov's invariants of alternating knots in terms of the Jones polynomial and the signature of the knot.

An approach to construction of topological invariants of the Reshetikhin-Turaev-Witten type of 3- and 4-dimensional manifolds in the framework of SU(2) Chern-Simons gauge theory and its hidden (quantum) gauge symmetry is presented.

Heegaard splittings and Heegaard diagrams of a closed 3-manifold $M$ are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on $M$. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when $dimM\>2$. The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.