Ribbon Knot Research Articles

A refined Jones polynomial for symmetric unions

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $t$ are associated to the two types of crossings, respectively on and off the symmetry axis. From sample calculations we deduce that a ribbon knot can have essentially distinct symmetric union presentations even if the partial knots are the same. If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the polynomial $W_D(s,t)$ nicely reflects the geometric properties of $K$. In particular it elucidates the connection between the Jones polynomials of $K$ and its partial knots $K_\pm$: we obtain $W_D(t,t) = V_K(t)$ and $W_D(-1,t) = V_{K_-}(t) \cdot V_{K_+}(t)$, which has the form of a symmetric product $f(t) \cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots.

BAND SURGERY ON KNOTS AND LINKS

We give some relationships of the Jones and Q polynomials between two links which are related by a band surgery. Then we consider two applications: The first one is to an evaluation of the ribbon-fusion number, the least fusion number of a ribbon knot. The second one is to DNA knot theory, helping us to understand the action of the Xer site-specific recombination at psi site.

EQUIVALENCE OF SYMMETRIC UNION DIAGRAMS

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.

An estimate of the ribbon number by the Jones polynomial

For a ribbon knot we define the notion of its ribbon number. In this paper we estimate the ribbon number for a ribbon knot by using the Jones polynomial. As a corollary we determine the ribbon number of the Kinoshita-Terasaka knot.

Structural and electrical properties of TaSe 3 ring crystals

We have investigated structural and electrical properties of “ribbon-knot” crystals of TaSe 3 (superconductor) having nontrivial topological structures. They are shaped a ring, a Möbius strip and a 2 π-twisted loop, so they can be categorized by a ribbon knot topological invariant in the mathematics field. For investigation of structural property of the ring crystals, cutting operation was performed for 45 specimens of the TaSe 3 ring crystal using focused ion beam milling. The cutting operation transformed them to arc-shape crystals having homogeneous curvatures. At low temperature, thin ring crystals of TaSe 3 showed superconductivity, however, thick rings showed nonmetallic behavior. The experimental results suggest that the ring-shape structure affect the electrical behavior in the TaSe 3 ring crystal.

Groups of ribbon knots

We prove that for each positive integer n, the V n -equivalence classes of ribbon knot types form a subgroup R n , of index two, of the free abelian group V n constructed by the author and Stanford. As a corollary, any non-ribbon knot whose Arf invariant is trivial cannot be distinguished from ribbon knots by finitely many independent Vassiliev invariants. Furthermore, except the Arf invariant, all non-trivial additive knot cobordism invariants are not of finite type. We prove a few more consequences about the relationship between knot cobordism and V n -equivalence of knots. As a by-product, we prove that the number of independent Vassiliev invariants of order n is bounded above by (n − 2)! 2 if n > 5, improving the previously known upper bound of ( n − 1)!.

Some well-disguised ribbon knots

For certain knots J in S 1 × D 2, the dual knot J ∗ in S 1 × D 2 is defined. Let J( O) be the satellite knot of the unknot O with pattern J, and K be the satellite of J( O) with pattern J ∗ . The knot K then bounds a smooth disk in a 4-ball, but is not obviously a ribbon knot. We show that K is, in fact, ribbon. We also show that the connected sum J(O) # J ∗(O) is a nonribbon knot for which all known algebraic obstructions to sliceness vanish.

Nonsimple, ribbon fibered knots

The connected sum of an arbitrary knot and its mirror image is a ribbon knot, however the converse is not necessarily true for all ribbon knots. We prove that the converse holds for any ribbon fibered knot which is a connected sum of iterated torus knots, knots with irreducible Alexander polynomials, or cables of such knots. This gives a practical method to detect nonribbon fibered knots. The proof uses a characterization of homotopically ribbon, fibered knots by their monodromies due to Casson and Gordon. We also study when cable fibered knots are ribbon and results which support the following conjecture. Conjecture. If a $(p,q)$ cable of a fibered knot $k$ is ribbon where $p(> 1)$ is the winding number of a cable in ${S^1} \times {D^2}$, then $q = \pm 1$ and $k$ is ribbon.

STABLE EQUIVALENCE OF RIBBON PRESENTATIONS

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.

Lusternik-Schnirel′mann category of ribbon knot complement

We showed in [8] that a locally flat knot is topologically unknotted if and only if the Lusternik-Schnirelmann category of the complement is one. In this paper we will show that the complement of a ribbon knot is of category two.

Ribbon knot families via stallings' twists

AbstractWe investigate the effect on the Jones polynomial of a ribbon knot when two of its bands are twisted together. We use our results to prove that each of the three S-equivalence classes of genus 2 fibered doubly slice knots in S3 can be represented by infinitely many distinct prime fibered doubly slice ribbon knots.

Relation between strings and ribbon knots

A ribbon knot can be represented as the propagation of an open string in (Euclidean) space-time. By imposing physical conditions plus an ansatz on the string scattering amplitude, we get invariant polynomials of ribbon knots which correspond to Jones and Wadatiet al. polynomials for ordinary knots. Motivated by the string scattering vertices, we derive an algebra which is a generalization of Hecke and Murakami-Birman-Wenzel (BMW) algebras of knots.

Composite ribbon number one knots have two-bridge summands

A composite ribbon knot which can be sliced with a single band move has a two-bridge summand.

A Ribbon Knot Group which has no Free Base

A Ribbon Knot Group which has no Free Base

A ribbon knot group which has no free base

We consider the following problem: If a group G G satisfies the conditions (1) G G has a finite presentation with r + 1 r + 1 generators and r r relators, and (2) there exists an element x x of G G such that G = ⟨ ⟨ x ⟩ ⟩ G G = {\left \langle {\left \langle x \right \rangle } \right \rangle ^G} where ⟨ ⟨ x ⟩ ⟩ G {\left \langle {\left \langle x \right \rangle } \right \rangle ^G} is the normal closure of x x in G G , then is G G an HNN (Higman-Neumann-Neumann) extension of a free group of finite rank? In this paper, we give a negative answer to the problem. Thus it follows that there exists a ribbon n n -knot group ( n ≥ 2 ) (n \geq 2) which has no free base.

Infinitely many ribbon knots with the same fundamental group

We work in the DIFF category. A knot K = (Sn+2, Sn) is a ribbon knot if Sn bounds an immersed disc Dn+1 →Sn+2 with no triple points and such that the components of the singular set are n-discs whose boundary (n – l)-spheres either lie on Sn or are disjoint from Sn. Pushing Dn+1 into Dn+3 produces a ribbon disc pair D = (Dn+3, Dn+1), with the ribbon knot (Sn+2, Sn) on its boundary. The double of a ribbon (n+1)-disc pair is an (n + l)-ribbon knot. Every (n+l)-ribbon knot is obtained in this manner.

A loop theorem for duality spaces and fibred ribbon knots

In this paper we prove a version of the loop theorem for surfaces in the boundary of a 3-dimensional duality space, i.e. a space which resembles a 3manifold only in that it satisfies the appropriate form of Poincar6-Lefschetz duality over some field of untwisted coefficients. Our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance, and as the main application of our theorem we show that if a fibred knot in the 3-sphere is a ribbon knot, then its monodromy extends over a handlebody. We approach the loop theorem via the study of planar coverings of a surface, as in the original paper of Papakyriakopoulos [11] and the subsequent work of Maskit [9]. w contains a simple geometric treatment of these matters. The main result of w is that a duality space actually satisfies duality with (twisted) coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal. In w 4, the results of w167 2 and 3, together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring, are used to prove the loop theorem for 3-dimensional duality spaces. (For the reader's convenience a proof of the algebraic temma is included as an appendix.) w contains the application to fibred ribbon knots mentioned above. In w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4-manifolds. In w we apply our methods to another aspect of knot concordance, and show that for any concordance with a rationally anisotropic fibred knot (see [7]) at one end, the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups. For torus knots, this question was raised by Scharlemann [14].

On certain non-trivial ribbon knots

Any ribbon knot K in S3 can be obtained from the unlink of r + 1 components {U0, U1,…, Ur}, for some positive integer r, by the simultaneous performance of r band operations (fusions), using disjoint bands. These bands may be assumed to run from U0 to U1, U2, …, Ur respectively, and labelled b1, b2…, br accordingly. This paper investigates the question of what conditions on these bands guarantee that the knot K so constructed is non-trivial. Besides being of independent interest (see 1·2 (A) of (2), and the example below), this problem is related to the ‘slice implies ribbon’ conjecture (1·33 of (2)), to the triviality question for 2-spheres in the 4-sphere (section 10 of (4)), and other unknotting questions for surfaces in 4-space (see, for example, 4·30 of (2)). The main result proved here is that K is non-trivial if the union of the bands ‘geometrically links’ (in a sense made precise below) every Ui at most once, and at least one U0 exactly once. The linking of the bands with U0 does not affect the argument, so may be completely arbitrary.

Is every slice knot a ribbon knot

Is every slice knot a ribbon knot

The Minkowski units of ribbon knots

2. Ribbon projections. Let k be a ribbon knot in R3, let 12 be the unit square, and let f: 12 -R3 be a ribbon whose boundary is k. By definition of a ribbon, the singular set S= {xflff(x) xx} of f consists of an even number 2n of pairwise disjoint arcs. Half of these arcs, say C1, * * , C., have the property that both their endpoints lie on 2, but they are otherwise disjoint from J2. For each Ci, i= 1, * , n there is a unique arc C in S such that f(Ci) =f(Ci) and C! fI2 The arcs C* separate f2 into n +1 components Xo, * * , Xn, and the endpoints of the arcs Ci separate 12 into a number of component arcs. At least two of these component arcs, say E1, * * *, Em, have the property that they join the two endpoints of some arc Ci. For each i=1, = * , n, let pi be an interior point of Ci and let p be the unique point of C' such that f(p) =f(pi). Also let qi be an interior point of Ei for each i= 1, * , m. Then each component Xi of I2-Ufo1 Ci has a certain number r of pi's and qi's designated on its boundary and a certain number s of p!'s designated on its interior.