論文

We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topological study of microphase separation of a block copolymer melt.

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.

Nozaki et al. gave a homotopy classification of the knotted defects of ordered media in three-dimensional space by considering continuous maps from complements of spatial graphs to the order parameter space modulo a certain equivalence relation. We extend their result by giving a classification scheme for ordered media in handlebodies, where defects are allowed to reach the boundary. Through monodromies around meridional loops, global defects are described in terms of planar diagrams whose edges are colored by elements of the fundamental group of the order parameter space. We exhibit examples of this classification in octahedral frame fields and biaxial nematic liquid crystals.

We give a homotopy classification of the global defects in ordered media and explain it via the example of biaxial nematic liquid crystals, that is, systems where the order parameter space is the quotient of the three-sphere S 3 by the quaternion group Q. As our mathematical model, we consider continuous maps from complements of spatial graphs to the space S 3 / Q modulo a certain equivalence relation and find that the equivalence classes are enumerated by the six subgroups of Q. Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of Q; once we pass to planar diagrams, these labels can be refined to elements of Q associated with each arc. The same classification scheme applies not only in the case of Q but also to arbitrary groups.

Quantum enhancement polynomials are invariants for oriented links, defined in association with an algebraic structure called a tribracket. In this paper, we focus on the particular case of the canonical two-element tribracket. We prove that, in that case, the quantum enhancement polynomials can be recovered by five specific polynomials, which we refer to as the universal quantum enhancement polynomials. After presenting several notable properties of these polynomials, we show that they are strictly stronger than the Jones polynomial. Furthermore, we provide computational results for links with up to 10 crossings.

A stable map of a closed orientable 3-manifold into the real plane is called a stable map of a link in the manifold if the link is contained in the set of definite fold points. We give a complete characterization of the hyperbolic links in the 3-sphere that admit stable maps into the real plane with exactly one (connected component of a) fiber having two singular points.

A (g,n)-decomposition of a link L in a closed orientable 3-manifold M is a decomposition of M by a closed orientable surface of genus g into two handebodies each intersecting the link L in n trivial arcs. The Goeritz group of that decomposition is then defined to be the group of isotopy classes of orientation-preserving homeomorphisms of the pair (M,L) that preserve the decomposition. We compute the Goeritz groups of all (1, 1)-decompositions.

We prove that if the distance of a bridge decomposition of a link with respect to a Heegaard splitting of a 3-manifold is at least 6, then the Goeritz group is a finite group.

For a bridge decomposition of a link in the 3-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere that preserve each of the bridge sphere and link setwise. After describing basic properties of this group, we discuss the asymptotic behavior of the minimal pseudo-Anosov entropies. We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings of the 3-sphere and the real projective space.

Any knot K in genus-1 1-bridge position can be moved by isotopy to lie in a union of n parallel tori tubed by n - 1 tubes so that K intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal n for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the (1, 1)-length. We show that the (1, 1)-length equals the level number. We then find braid descriptions for (1,1)-positions of all 2-bridge knots providing upper bounds for their level numbers and also show that the (-2, 3, 7)-pretzel knot has level number two.

We introduce the concept of a homotopy motion of a subset in a manifold and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behavior of simple loops on a Heegaard surface and monodromies of virtual branched covering surface bundles associated with a Heegaard splitting.