論文

We characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is a three-variable analogue of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are 2-dimensional surfaces.

We construct efficient topological cobordisms between torus links and large connected sums of trefoil knots. As an application, we show that the signature invariant σω at ω = ζ6 takes essentially minimal values on torus links among all concordance homomorphisms with the same normalisation on the trefoil knot.

A 2-sphere embedded in the 4-sphere invariant under a circle action is called a branched twist spin. A branched twist spin is constructed from a 1-knot in the 3-sphere and a pair of coprime integers uniquely. In this paper, we study, for each pair of coprime integers, if two different 1-knots yield the same branched twist spin, and prove that such a pair of 1-knots does not exist in most cases. Fundamental groups of 3-orbifolds of cyclic type are obtained as quotient groups of the fundamental groups of the complements of branched twist spins. We use these groups for distinguishing branched twist spins.

We show, for every positive integer n, there is an alternating knot having a boundary slope with denominator n. We make use of Kabaya’s method for boundary slopes and the layered solid torus construction introduced by Jaco and Rubinstein and further developed by Howie et al.

A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form whose Reeb flow is a flow of the flow-spine. It is known by Thurston and Winkelnkemper that any open book decomposition of a closed oriented 3-manifold supports a contact structure. In this paper, we introduce a notion of positivity for flow-spines and prove that any positive flow-spine of a closed, connected, oriented 3-manifold supports a contact structure uniquely up to isotopy. The positivity condition is critical to the existence of the unique, supported contact structure, which is also proved in the paper.

In our previous paper, it is proved that for any positive flow-spine P of a closed, oriented 3-manifold M, there exists a unique contact structure supported by P up to isotopy. In particular, this defines a map from the set of isotopy classes of positive flow-spines of M to the set of isotopy classes of contact structures on M. In this paper, we show that this map is surjective. As a corollary, we show that any flow-spine can be deformed to a positive flow-spine by applying first and second regular moves successively.

Let f be a polynomial map from Rm to Rn with m > n > 0 and t0 be a regular value of f. For a small open ball Dt0 centered at t0, we show that the map f : f−1(Dt0) → Dt0 is a Serre fibration if and only if f is a Serre fibration over a finite number of certain simple arcs starting at t0. We characterize the fibration f : f−1(Dt0) → Dt0 by relative homotopy groups defined for these arcs and use it to prove the assertion.

The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form b defining a foliation on a branched simple polyhedron P satisfies db > 0, which means that the foliation is possibly a characteristic foliation of a contact form whose Reeb flow is transverse to P. In this paper, we show that if there exists a 1-form b on P with db > 0 then we can find a 1-form with the same property and additionally being S-stable. We then prove that the number of simple tangency points of an S-stable foliation on a positive or negative flow-spine is at least 2 and give a recipe for constructing a characteristic foliation of a 1-form b with db > 0 on the abalone.