Narrow Results

It has been a fundamental problem to study the crossing number and the bridge index for an 1-knot. By spinning an arbitrary 1-knot about R² in R⁴, we have a ribbon 2-knot in R⁴; therefore, for a ribbon 2-knot, we can also naturally induce notions corresponding to the crossing number and the bridge index. In this note, we will define these notions, and for a property on these notions, we will study a great contrast between 1-knots and ribbon 2-knots and in addition we will enumerate all ribbon 2-knots with the crossing number under four.

The HC-move was defined as an unknotting operation of a ribbon 2-knot. Representing a ribbon 2-knot by a virtual arc, we see that the HC-move corresponds to one of the "forbidden moves", which unknot every virtual knot. Using this, we consider the α₂-invariant of a ribbon 2-knot and a relation between the Δ-move for a 1-knot and the HC-move for its spun 2-knot.

For an arbitrary 1-knot k¹, the spun 2-knot of k¹, denoted by spun(k¹), is a ribbon 2-knot in R⁴. Hence for a ribbon 2-knot K², we can also induce a notion corresponding to the crossing number on a 1-knot, and it is said to be the crossing number of K², denoted by cr(K²). In this note, we will show that the Alexander polynomial plays an important role in determining the crossing number of a ribbon 2-knot. Lastly, we will prove the following: If k¹ is a (p,q)-torus knot, then cr(spun(k¹)) is equal to (p - 1)(q - 1).

Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's C_k-moves. C_k-moves are defined by using his claspers. In this paper we define "oriented" claspers and RC_k-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing.

A ribbon 2-knot is a 2-sphere in R⁴, which is obtained from m 2-spheres in R⁴ by connecting them with m - 1 pipes. Even in the case that a ribbon 2-knot K² is constructed by two 2-spheres and one pipe, a way of constructing K² is not always unique. That is, K² might have presentations of two or more types.

The presentation that a pipe crosses two spheres n times is said to be an n-crossing ribbon presentation, and also n be the ribbon crossing number of this presentation.

In this note, we will show that for any positive integer i, there exists a ribbon 2-knot with presentations of two distinct types such that the difference of their ribbon crossing number is i; and in addition we will also show that for any positive integer j, there exists a ribbon 2-knot with presentations of j, distinct types such that they have the same ribbon crossing number.

A symmetric union of a knot is a generalization of the operation of the connected sum of a knot and its mirror image. In this paper, we show that every ribbon 2-knot of 1-fusion has a ribbon presentation associated with a symmetric union which has either the unknot or a 2-bridge knot as a partial knot.

For a 1- or 2-dimensional knot, we give a lower bound log₂ n + 1 of the minimum number of distinct colors for all effective n-colorings. In particular, we prove that any effectively 9-colorable 1- or ribbon 2-knot is presented by a diagram where exactly five colors of nine are assigned to the arcs or sheets.

Let be a hyperbolic transformation. Let B be a new band attaching to L such that is also a hyperbolic transformation. In this paper, we will study the relationship between the realizing surfaces and . If B is a noncoherent band to both L and such that is defined, then and are ambient isotopic, where RP² is one of the standard real projective planes. We will study the triviality of because as an application, RP² can untangle some knotted sphere with suitable conditions, when it is attached to by the connected sum.

We prove that any 11-colorable knot is presented by an 11-colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially 11-colored diagrams of the knot. We also prove a similar result for any 11-colorable ribbon 2-knot.

We enumerate ribbon 2-knots presented by virtual arc diagrams with up to four classical crossings. We use a linear Gauss diagram for a virtual arc diagram.

A $2$-knot is a surface in $R^4$ that is homeomorphic to $S^2$, the standard sphere in $3$-space. A ribbon $2$-knot is a $2$-knot obtained from $m$$2$-spheres in $R^4$ by connecting them with $m-1$ pipes. Let $K^2$ be a ribbon 2-knot. The ribbon crossing number, denoted by $r$-$cr(K^2)$, is a numerical invariant of the ribbon $2$-knot $K^2$. In [T. Yasuda, Crossing and base numbers of ribbon 2-knots, J. Knot Theory Ramifications10 (2001) 999–1003] we showed that there exist just $17$ ribbon $2$-knots of the ribbon crossing number up to three. In this paper, we show that there exist no more than $111$ ribbon $2$-knots of ribbon crossing number four.

We consider classification of the oriented ribbon 2-knots presented by virtual arcs with up to four crossings. We show the difference by the 2-fold branched covering space, the Alexander polynomial, the number of representations of the knot group to SL$(2,𝔽)$, $𝔽$ a finite field, and the twisted Alexander polynomial.

We generalize Yasuda’s examples of ribbon 2-knots of 1-fusion with different ribbon presentations.

Suciu constructed infinitely many ribbon 2-knots in $S^4$ whose knot groups are isomorphic to the trefoil knot group. They are distinguished by the second homotopy groups. We classify these knots by using $\mathrm{\text{SL}}(2,ℂ)$-representations of the fundamental groups of the 2-fold branched covering spaces.

We give infinitely many pairs of ribbon 2-knots of 1-fusion in $S^4$ with isomorphic knot groups, which extend Takahashi’s examples. They are distinguished by the trace sets, which are calculated by using $\mathrm{\text{SL}}(2,ℂ)$-representations of the knot groups.

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.