arXiv subjects
Publications and source records attributed to Víctor Núñez.
We give an algorithm to calculate the minimal and maximal genus of the orientable closed surface where a graph $G$ can be embedded. For this, we construct some special branched coverings of the 2-sphere. We apply this algorithm to calculate the orientable genus and maximal genus of some Snarks graphs.
Let $t_{α,β}\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $α$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{α,β}$ if $α\geq3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{α,β}$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.
We determine the structure of the circular handle decompositions of the family of free genus one knots. Namely, if k is a free genus one knot, then the handle number h(k)= 0, 1 or 2, and, if k is not fibered (that is, if h(k)>0), then k is almost fibered. For this, we develop practical techniques to construct circular handle decompositions of knots with free Seifert surfaces in the 3-sphere (and compute handle numbers of many knots), and, also, we characterize the free genus one knots with more than one Seifert surface. These results are obtained through analysis of spines of surfaces on handlebodies. Also we show that there are infinite families of free genus one knots with either h(k)=1 or h(k)=2.