Narrow Results

Any ribbon torus-knot in 4-space is naturally associated with a virtual knot. We investigate a relationship between ribbon torus-knots and virtual knots from this viewpoint. We also give a new example of a non-classical virtual knots which can not be detected by the group and the Z-polynomial.

Given a tangle diagram of a classical knot K, we construct a surface diagram of any twist-spun 2-knot of K. From the crossings of the tangle diagram, we can get information of the corresponding triple points of the surface diagram, which are used to compute cocycle invariants of twist-spun 2-knots.

We calculate the dihedral quandle cocycle invariants of twist-spins of alternating odd pretzel knots. The calculation leads us to the conclusion that there exist non-ribbon 2-knots which admit a non-trivial coloring by the dihedral quandle R_p and all of whose cocycle invariants derived from ℤ_p-valued 3-cocycles on R_p take value in ℤ ⊂ ℤ[ℤ_p] for any odd prime integer p.

The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.

In this paper, we develop a method to change the label of a ring in a chart by C-moves and a stabilization where a ring is a simple closed curve consisting of edges of the same label which does not contain any white vertices.

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.

We show that a 2-knot group discovered in the course of a census of 4-manifolds with small triangulations is an HNN extension with finite base and proper associated subgroups, and has the smallest base among such knot groups.

If $X$ is an orientable, strongly minimal $P{D}_4$-complex and ${\pi }_1(X)$ has one end, then it has no nontrivial locally finite normal subgroup. Hence, if $\pi$ is a 2-knot group, then (a) if $\pi$ is virtually solvable, then either $\pi$ has two ends or $\pi \cong \mathrm{\Phi }$, with presentation $〈a,t|ta=a^{2}t〉$, or $\pi$ is torsion-free and polycyclic of Hirsch length 4 (b) either $\pi$ has two ends, or $\pi$ has one end and the center $\zeta \pi$ is torsion-free, or $\pi$ has infinitely many ends and $\zeta \pi$ is finite, and (c) the Hirsch–Plotkin radical $\sqrt{\pi }$ is nilpotent.

Ng constructed an invariant of knots in ${ℝ}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${ℝ}^4$ using marked graph diagrams.

Ng constructed an invariant of knots in ${ℝ}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${ℝ}^4$ using diagrams in ${ℝ}^3$.

We prove that the Gaussian parity on free two-dimensional knots is universal.

We consider a locally flat 2-sphere embedded in S² × S², which is referred to as a 2-knot in S² × S², and we study the problem whether a 2-knot in S² × S² is determined by its exterior. In this paper, we show that for almost homology classes ξ, 2-knots in S² × S² representing ξ are determined by their exteriors.

Any closed oriented surface embedded in R⁴ is described as a closed 2-dimensional braid and its braid index is defined. We study 2-dimensional braids through a method to describe them using graphs on a 2-disk and show that braid index two surfaces in R⁴ are unknotted and braid index three surfaces are ribbon.

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an infinite cyclic fundamental group.

The weak unknotting number of a 2-knot K in a 4-sphere is the minimum number of elements g₁, g₂,…, g_n of the knot group πK of K such that πK becomes the infinite cyclic group by adding the relations g_ix=xg_i, where x is a meridian. We give an example of the nonadditivity of the weak unknotting number under connected sum of 2-knots.