Narrow Results

We show that the linking signature of a closed oriented 4-manifold with infinite cyclic first homology is twice the Rochlin invariant of an exact leaf with a spin support if such a leaf exists. In particular, the linking signature of a surface-knot in the 4-sphere is twice the Rochlin invariant of an exact leaf of an associated closed spin 4-manifold with infinite cyclic first homology. As an application, we characterize a difference between the spin structures on a homology quaternion space in terms of closed oriented 4-manifolds with infinite cyclic first homology, so that we can obtain examples showing that some different punctured embeddings into S⁴ produce different Rochlin invariants for some homology quaternion spaces.

We first introduce the null-homotopically peripheral quadratic function of a surface-link to obtain a lot of pseudo-ribbon, non-ribbon surface-links, generalizing a known property of the turned spun torus-knot of a non-trivial knot. Next, we study the torsion linking of a surface-link to show that the torsion linking of every pseudo-ribbon surface-link is the zero form, generalizing a known property of a ribbon surface-link. Further, we introduce and algebraically estimate the triple point cancelling number of a surface-link.

We prove that if a surface-knot diagram is colored by a specific quandle non-trivially, then any diagram of the surface-knot consists of at least four broken sheets. In particular, the minimal number of broken sheets for the spun trefoil is shown to be exactly four.

In this paper we give a lower bound of triple point numbers of special family of 2-knots colored by the dihedral quandle of order 5.

In this paper, we prove that if a surface diagram of a surface-knot has at most two triple points and the lower decker set is connected, then the surface-knot group is isomorphic to the infinite cyclic group.

In this paper, we introduce the notion of a virtual surface-knot as a 2-dimensional generalization of a virtual knot. We define the upper and lower groups of a virtual surface-knot to prove that there are infinitely many non-classical virtual surface-knots.

In this paper, we present a construction of a family of surface-knot diagrams with cross-exchangeable curves, along which we can change the crossing information to obtain trivial diagrams. These diagrams also satisfy a kind of minimality, called $d$-minimal surface-knot diagrams.

A 2-dimensional braid over an oriented surface-knot $F$ is presented by a graph called a chart on a surface diagram of $F$. We consider 2-dimensional braids obtained by an addition of 1-handles equipped with chart loops. We introduce moves of 1-handles with chart loops, called 1-handle moves, and we investigate how much we can simplify a 2-dimensional braid by using 1-handle moves. Further, we show that an addition of 1-handles with chart loops is an unbraiding operation.

In this paper, we describe a two-dimensional rectangular-cell-complex derived from a surface-knot diagram of a surface-knot. We define a pseudo-cycle for a quandle colored surface-knot diagram. We show that the maximal number of pseudo-cycles is a surface-knot invariant.

Two-dimensional knots and links are studied in the paper. In this paper, the notion of a parity for 2D knots and links is introduced via techniques similar to the ones used by the second named author in 1D case. By using parity new invariants are constructed and some known invariants are refined.

A branched covering surface-knot over an oriented surface-knot $F$ is a surface-knot in the form of a branched covering over $F$. A branched covering surface-knot over $F$ is presented by a graph called a chart on a surface diagram of $F$. For a branched covering surface-knot, an addition of 1-handles equipped with chart loops is a simplifying operation which deforms the chart to the form of the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications.

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained.

It is known that there is no 2-knot with triple point number two. The present paper shows that there is no surface-knot of genus one with triple point number two.

Niebrzydowski and Przytycki defined a Kauffman bracket magma and constructed the invariant $P$ of framed links in $3$-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is obtained from the Kauffman bracket polynomial by the multiplication of indeterminate and it is an ambient isotopy invariant for links. In this paper, we reformulate the multiplication by using a map from the set of framed links to a Kauffman bracket magma in order that $P$ is invariant for links in $3$-space. We define a generalization of a Kauffman bracket magma, which is called a marked Kauffman bracket magma. We find the conditions to be invariant under Yoshikawa moves except the first one and use a map from the set of admissible marked graph diagrams to a marked Kauffman bracket magma to obtain the invariant for surface-links in $4$-space.

Gay and Meier asked whether or not a trisection diagram constructed by the Gluck twist on a spun or a twist spun 2-knot obtained from some method is standard. In this paper, we depict the trisection diagrams explicitly when the 2-knot is the spun $(2n+1,2)$-torus knot, where $n\ge 1$, and show that the trisection diagram is standard when $n=1$. Moreover, we introduce a notion of homologically standard for trisection diagrams and show that the trisection diagram is homologically standard for all $n$.

A diagram of a surface-knot consists of a disjoint union of compact conneted surfaces. The sheet number of a surface-knot is the minimal number of such connected surfaces among all possible diagrams of the surface-knot. This is a generalization of the crossing number of a classical knot. We give a lower bound of the sheet number by using quandle-colorings of a diagram and the cocycle invariant of a surface-knot.

We focus an interest on the torsion linking of a surface-knot. It is a knot invariant independent of the surface-knot group and its peripheral subgroup. It is identified with the torsion linking of any associated closed 4-manifold with infinite cyclic first homology. In the case of such a 4-manifold with an exact leaf, the linking of the leaf is identified with an orthogonal sum of it and a hyperbolic linking.