Narrow Results

Bleiler and Eudave-Muñoz showed that composite ribbon number one knots have two-bridge summands. In this paper, we show that there exists an infinite family of composite ribbon number one knots which have arbitrary large bridge numbers.

We define a local move on a ribbon 2-knot diagram, called an HC-move. We show that it is an unknotting operation for a ribbon 2-knot, and that the application of a single HC-move to a ribbon 2-knot changes the second derivative at t=1 of its normalized Alexander polynomial by either ±2 or 0. This result is applied to the calculation of the HC-unknotting numbers of ribbon 2-knots. We also consider a relation with a 1-handle unknotting operation.

We give the counterexamples for a conjecture of Fiedler concerning the ribbon genus of a link.

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of uniqueness. In order to attack this question we extend the usual Reidemeister moves to a family of moves respecting the symmetry, and consider the symmetric equivalence thus generated. This notion being in place, we discuss several situations in which a knot can have essentially distinct symmetric union representations. We exhibit an infinite family of ribbon two-bridge knots each of which allows two different symmetric union representations.

We study the ribbon disks that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number r_S(K) is introduced and compared with the classical ribbon number r(K). We show that the difference r_S(K) - r(K) can be arbitrarily large by constructing an infinite family of ribbon knots K_n such that r(K_n) = 2 and r_S(K_n) > n. The proof is based on a particularly simple description of symmetric unions in terms of certain band diagrams which leads to an upper bound for the Heegaard genus of their branched double covers.

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.

A symmetric union in the 3-space ${ℝ}^3$ is obtained from a knot in ${ℝ}^3$ and its mirror image, which is symmetric with respect to a 2-plane in ${ℝ}^3$, by connecting them with some vertical 2-string tangles with twists and a horizontal 2-string tangle with no twists along the plane. We introduce the minimal twisting number of a symmetric union and show that if a knot $K$ is a composite symmetric union with minimal twisting number one, then $K$ has a non-trivial knot and its mirror image as connected summands. As an application, we study the minimal twisting number of a symmetric union and show that there exist infinitely many symmetric unions with minimal twisting number two.

A point in the $(N,q)$-torus knot in ${ℝ}^3$ goes $q$ times along a vertical circle while this circle rotates $N$ times around the vertical axis. In the Lissajous-toric knot $K(N,q,p)$, the point goes along a vertical Lissajous curve (parametrized by $t\mapsto (sin(qt+\varphi ),cos(pt+\psi )))$ while this curve rotates $N$ times around the vertical axis. Such a knot has a natural braid representation $B_{N,q,p}$ which we investigate here. If $\mathrm{\text{gcd}}(q,p)=1$, $K(N,q,p)$ is ribbon; if $\mathrm{\text{gcd}}(q,p)=d>1$, $B_{N,q,p}$ is the $d$th power of a braid which closes in a ribbon knot. We give an upper bound for the $4$-genus of $K(N,q,p)$ in the spirit of the genus of torus knots; we also give examples of $K(N,q,p)$’s which are trivial knots.

In this paper, we show that there exists a knot which has infinitely many non-equivalent symmetric union presentations with the same partial knots. We also define an invariant of a symmetric union presentation and give alternative proofs to results of Eisermann and Lamm and a result of Collari and Lisca by using methods in 3-dimensional topology.

Satoh defined a map from virtual links to ribbon surfaces embedded in $S^4$. Herein, we generalize this map to virtual $m$-links, and use this to construct generalizations of welded and extended welded knots to higher dimensions. This also allows us to construct two new geometric pictures of virtual $m$-links, including 1-links.

We show that a two-bridge ribbon knot $K(m^2,mk\pm 1)$ with $m>k>0$ and $(m,k)=1$ admits a symmetric union presentation with partial knot which is a two-bridge knot $K(m,k)$. Similar descriptions for all the other two-bridge ribbon knots are also given.

We introduce a ribbon presentation to describe a ribbon knot, and equivalences between them. Superspun knots of classical knots are well known to be ribbon knots, and we give a construction of ribbon presentations of the knots by means of classical knot diagrams. We can then extend the classical Reidemeister moves into higher dimensions, and we prove that the presentations of a superspun knot are stably equivalent by using these moves.

A ribbon n-knot Kⁿ is constructed by attaching m bands to m + 1n-spheres in the euclidean (n + 2)-space. There are many way of attaching them; as a result, Kⁿ has many presentations which are called ribbon presentations. But concerning the case of m = 1, it was proved in the case of n = 1 by M. Scharlemann, and n ≥ 2 by Y. Marumoto that if Kⁿ is unknotted, its ribbon presentation is essentially unique. In this note, we will prove in the case of m = 1 and n ≥ 2 that there are infinitely many ribbon n-knots which has essentially different two ribbon presentations.

A ribbon n-knot Kⁿ is constructed by attaching m bands to m + 1n-spheres in the Euclidean (n + 2)-space. There are many way of attaching them; as a result, Kⁿ has many presentations which are called ribbon presentations. In this note, we will induce a notion to classify ribbon presentations for ribbon n-knots of m-fusions (m ≥ 1, n ≥ 2), and show that such classes form a totally ordered set in the case of m = 2 and a partially ordered set in the case of m ≥ 1.

We define invariants of higher dimensional theta-curves, and give a characterization of a trivial curve in the case of the classical dimension. We introduce a ribbon presentation of a knot and equivalence in ribbon presentations. Then a ribbon presentation induces a theta-curve, and invariants of theta-curves give those of ribbon presentations. Using the invariants, we can distinguish ribbon presentations of a knot.

We give an algorithm for calculating the second degree coefficient of the Conway polynomial of a ribbon 1-knot. This naturally yields a recursive calculation for the second derivative at t = 1, Δ^′′(1), of the normalized Alexander polynomial of a ribbon 2-knot in R⁴, which is the first nontrivial finite type invariant of a ribbon 2-knot defined by Habiro, Kanenobu, and Shima.

Gluck surgery is the operation of cutting out S² × D², a tubular neighborhood of a 2-knot and pasting it back in a 4-manifold. It may be expected to make a fake pair of 4-manifolds, which means a pair that are homotopy equivalent but non-diffeomorphic. We will give an alternative proof of a theorem: Gluck surgery along a banded 2-knot is independent of the bands, which was proved in P. Melvin's thesis and prove that: if a 2-knot is obtained by ribbon moves from another 2-knot, then the Gluck surgeries along them are diffeomorphic. Our method, which we call "framed links in 4-manifolds", is a (4,5)-dimensional version of the usual ((3,4)-dimensional) framed links.