Narrow Results

There is a nonribbon 2-link all of whose components are trivial 2-knots and one of whose band-sums is a nonribbon 2-knot.

We discuss the ribbon-move for 2-knots, which is a local move. Let K and K' be 2-knots. Then we have the following: Suppose that K and K' are ribbon-move equivalent.

(1) Let (respectively, ) be the ℤ-torsion submodule of the Alexander module (respectively, ). Then is isomorphic to not only as ℤ-modules but also as ℤ[t, t^-1]-modules.

(2) The Farber–Levine pairing for K is equivalent to that for K'.

(3) The set of the values of the ℚ/ℤ-valued invariants for K is equivalent to that for K'.

Intersection homology enables the definition of a twisted signature of a stratified pseudomanifold with coefficients in a local system that is typically only given on the top stratum. If the local system extends to the entire space, then the twisted signature can be computed by a version of a characteristic class formula first observed by Atiyah for smooth fiber bundles. In the first part of this paper, we construct examples of singular spaces, equipped with local systems that do not extend, such that the above characteristic class formula fails. In the second part, we consider smooth codimension two embeddings of manifolds. We view the target as a stratified space with bottom stratum the image of the embedding and top stratum the complement. When the embedded manifold is a sphere, we establish various formulae that compute the twisted signature even when the local system does not extend from the top stratum to the entire space.

We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families of distinct homotopy-ribbon disks with homotopy equivalent exteriors, with potential relevance to the Slice–Ribbon Conjecture. We show that any fibered ribbon 2-knot can be obtained by doubling infinitely many different slice disks (sometimes in different contractible 4-manifolds). Finally, we illustrate these ideas for the examples arising from spinning fibered 1-knots.

An $(m,n)$-branched twist spin is a fibered $2$-knot in $S^4$ which is determined by a $1$-knot $K$ and coprime integers $m$ and $n$. For a $1$-knot, Nagasato proved that the number of conjugacy classes of irreducible $SL(2,ℂ)$-metabelian representations of the knot group of a $1$-knot is determined by the knot determinant of the $1$-knot. In this paper, we prove that the number of irreducible $SL(2,ℂ)$-metabelian representations of the knot group of an $(m,n)$-branched twist spin is determined up to conjugation by the determinant of the associated $1$-knot in the orbit space by comparing a presentation of the knot group of the branched twist spin with the Lin’s presentation of the knot group of the $1$-knot.

In this paper, we prove that given two cubical links of dimension two in ${ℝ}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots (links) of dimensions one and two, respectively.

Spun-knots (respectively, spinning tori) in $S^4$ made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in $S^4$. Virtual 1-knots are generalizations of classical 1-knots. We generalize these constructions to the virtual 1-knot case by using what we call, in this paper, the spinning construction of submanifolds. The construction proceeds as follows: For a virtual 1-knot $K$, take an embedded circle $C$ contained in (a closed oriented surface $F$)×(a closed interval $[0,1]$), where $F$ is called a representing surface in virtual 1-knot theory. Embed $F$ in $S^4$ by an embedding map $f$, and let $F$ stand for $f(F).$ Regard the tubular neighborhood of $F$ in $S^4$ as the result of rotating $F\times [0,1]$ around $F$. Rotate $C$ together then with $F\times [0,1]$. When $C\cap (F\times \{0\})=\varphi$, we obtain an embedded torus $Q\subset S^4$. We prove the following: The embedding type $Q$ in $S^4$ depends only on $K$, and does not depend on $f$. Furthermore, the submanifolds, $Q$ and “the embedded torus made from $K$ by using Satoh’s method”, of $S^4$ are isotopic.

Fiberwise equivalence of diagrams refers to fiberwise equivalence of tori in 4-space that lie over the diagrams. We prove that two virtual 1-knot diagrams $\alpha$ and $\beta$ are fiberwise equivalent if and only if $\alpha$ and $\beta$ are rotational welded equivalent (see the body of the paper for this definition).

We generalize the construction in the virtual 1-knot case written in the first paragraph, and we also succeed to make a consistent construction of one-dimensional-higher submanifolds from any virtual two-dimensional knot. Note that Satoh’s method says nothing about the virtual 2-knot case. Rourke’s interpretation of Satoh’s method is that one puts “fiber-circles” on each point of each virtual 1-knot diagram. If there is no virtual branch point in a virtual 2-knot diagram, our way gives such fiber-circles to each point of the virtual 2-knot diagram. Furthermore we prove the following: If a virtual 2-knot diagram $\alpha$ has a virtual branch point, $\alpha$ cannot be covered by such fiber-circles. Hence Rourke’s method cannot be generalized to the virtual 2-knot case. Only the spinning construction introduced in this paper works for now.

We construct a family of trivial $2$-knots $k^i$ in ${ℝ}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $k^i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $k^i$.

Here, we can either construct $k^i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots $k^i$, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat $2$-simplices in its triangulation.

These results contrast with the situation of classical knots in ${ℝ}^3$, where every unknot can be untied through knots of complexity that is only polynomially higher than the complexity of the initial knot.

For a surface diagram, there is a cell-complex associated with the surface diagram. The index of a surface diagram is the rank of the second homology group of the cell-complex. We present that if a 2-knot has the triple point number four, its t-minimal surface diagram has four branch points and its cell-complex has index 1, then the knot group of the surface-knot for the surface diagram has a presentation of the knot group of the double twist spun trefoil.