Narrow Results

In this article, we study the topology of the family of real analytic germs F : (ℂ³, 0) → (ℂ, 0) given by with p, q, r ∈ ℕ, p, q, r ≥ 2 and (p, q) = 1. Such a germ has an isolated singularity at 0 and gives rise to a Milnor fibration . Moreover, it is known that the link L_F is a Seifert manifold and that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of 𝕊⁵ given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ℂ³.

The notion of Gem–Matveev complexity (GM-complexity) has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper, we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via GM-complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base 𝔻² and two exceptional fibers and, therefore, for all torus knot complements.

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space $S^2$ and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot $K_{(r,s)}$. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on ${𝕋}_{p/q}$, the result of $p/q$ surgery in the left-handed trefoil knot $𝕋$, J. Knot Theory Ramifications24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

Let $X$ and $Y$ be the complementary regions of a closed hypersurface $M$ in $S^4=X{\cup }_{M}Y$. We use the Massey product structure in $H^{\ast }(M;\mathbb{Z})$ to limit the possibilities for $\chi (X)$ and $\chi (Y)$. We show also that if ${\pi }_1(X)\ne 1$ then it may be modified by a 2-knot satellite construction, while if $\chi (X)\le 1$ and ${\pi }_1(X)$ is abelian then ${\beta }_1(M)\le 4$ or ${\beta }_1(M)=6$. Finally we use TOP surgery to propose a characterization of the simplest embeddings of $F\times S^1$.

Let $t_{\alpha ,\beta }\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $\alpha$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{\alpha ,\beta }$ if $\alpha \ge 3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{\alpha ,\beta }$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.