Ribbon Knot Research Articles

Trace embeddings from zero surgery homeomorphisms

AbstractManolescu and Piccirillo (2023) recently initiated a program to construct an exotic or by using zero surgery homeomorphisms and Rasmussen's ‐invariant. They find five knots that if any were slice, one could construct an exotic and disprove the Smooth 4‐dimensional Poincaré conjecture. We rule out this exciting possibility and show that these knots are not slice. To do this, we use a zero surgery homeomorphism to relate slice properties of two knots stably after a connected sum with some 4‐manifold. Furthermore, we show that our techniques will extend to the entire infinite family of zero surgery homeomorphisms constructed by Manolescu and Piccirillo. However, our methods do not completely rule out the possibility of constructing an exotic or as Manolescu and Piccirillo proposed. We explain the limits of these methods hoping this will inform and invite new attempts to construct an exotic or . We also show that a family of homotopy spheres constructed by Manolescu and Piccirillo using annulus twists of a ribbon knot are all standard.

Ribbonness on Classical Link

It is shown that if a link in 3-space bounds a proper oriented surface (without closed component) in the upper half 4-space, then the link bounds a proper oriented ribbon surface in the upper half 4-space which is a renewal embedding of the original surface. In particular, every slice knot is a ribbon knot, answering an old question by R. H. Fox affirmatively

On the symmetry of composite ribbon knots

The connected-sum of a knot and its mirror image with orientation reversed is a ribbon knot. Conversely, we show that if K0 and K1 are knots which are minimal with respect to ribbon concordance and the connected sum of K0 and K1 is a ribbon number one knot, then K0 is equivalent to the mirror image of K1 with orientation reversed. We also show that the connected sum of two torus knots is a slice knot, then it is a connected sum of a torus knot and its mirror image.

Linking number and folded ribbon unknots

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for [Formula: see text]-stick unknots with ribbon linking numbers [Formula: see text] and [Formula: see text], and we prove that the minimum folded ribbonlength for [Formula: see text]-gons with obtuse interior angles is achieved when the [Formula: see text]-gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number [Formula: see text] is bounded from above by [Formula: see text].

Ribbonlength of families of folded ribbon knots

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, $(2,q)$ torus, twist, and pretzel knots, and these upper bounds turn out to be linear in the crossing number. We give a new way to fold $(p,q)$ torus knots and show that their folded ribbonlength is bounded above by $2p$. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 6. We then show that any $(p,q)$ torus knot $K$ with $p\geq q>2$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $c\cdot Cr(K)^{1/2}$. This provides an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.

A short proof of a theorem of Tanaka on composite knots with symmetric union presentations

We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twist region, then it has a non-trivial knot and its mirror image as connected summands.

Extending fibrations of knot complements to ribbon disk complements

We show that if $K$ is a fibered ribbon knot in $S^3=\partial B^4$ bounding a ribbon disk $D$, then given an extra transversality condition the fibration on $S^3\setminus\nu(K)$ extends to a fibration of $B^4\setminus\nu(D)$. This partially answers a question of Casson and Gordon. In particular, we show the fibration always extends when $D$ has exactly two local minima. More generally, we construct movies of singular fibrations on $4$-manifolds and describe a sufficient property of a movie to imply the underlying $4$-manifold is fibered over $S^1$.

Ribbonlength and crossing number for folded ribbon knots

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants [Formula: see text] such that the ribbonlength is bounded above by [Formula: see text], and also by [Formula: see text]. We use a different method for each bound. The constant [Formula: see text] is quite small in comparison to [Formula: see text], and the first bound is lower than the second for knots and links with [Formula: see text] 12,748.

A lower bound for the double slice genus

In this paper, we develop a lower bound for the double slice genus of a knot in the 3-sphere using Casson-Gordon invariants. As an application, we show that the difference between the slice genus and the double slice genus can be arbitrarily large. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound. In particular, through the study of superslice genus we show that the degree of the Alexander polynomial is an upper bound for the topological double slice genus of a ribbon knot.

Ribbon knots, cabling, and handle decompositions

The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the (p,1)-cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number p. Our main tools are Juhasz-Miller-Zemke's bound on fusion number coming from the torsion order of knot Floer homology and Hanselman-Watson's cabling formula for immersed curves.

Knot cobordisms, bridge index, and torsion in Floer homology

Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for T p , q and T p , q # T ¯ p , q , respectively. We also show that the bridge index of T p , q is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show that knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot.

A Fox–Milnor theorem for the Alexander polynomial of knotted 2-spheres in $S^4$

For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon 2-knot in $S^4$ is not even symmetric in general. Via an alternative notion of ribbon 2-knots, we give a topological condition on a 2-knot that implies the factorization of the Alexander polynomial.

Lissajous-toric knots

A point in the [Formula: see text]-torus knot in [Formula: see text] goes [Formula: see text] times along a vertical circle while this circle rotates [Formula: see text] times around the vertical axis. In the Lissajous-toric knot [Formula: see text], the point goes along a vertical Lissajous curve (parametrized by [Formula: see text] while this curve rotates [Formula: see text] times around the vertical axis. Such a knot has a natural braid representation [Formula: see text] which we investigate here. If [Formula: see text], [Formula: see text] is ribbon; if [Formula: see text], [Formula: see text] is the [Formula: see text]th power of a braid which closes in a ribbon knot. We give an upper bound for the [Formula: see text]-genus of [Formula: see text] in the spirit of the genus of torus knots; we also give examples of [Formula: see text]’s which are trivial knots.

Symmetric Union Diagrams and Refined Spin Models

AbstractAn open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that non-equivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many Reidemeister equivalent but symmetrically non-equivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.

A tabulation of ribbon knots in tangle form

It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the closed tangle is equivalent to the unlink. For each of these 21 ribbon knots, we present a 4-strand tangle that satisfies these properties. We provide diagrams of these tangles and also express them in planar diagram notation.

Slice implies mutant ribbon for odd 5–stranded pretzel knots

A pretzel knot $K$ is called $odd$ if all its twist parameters are odd, and $mutant$ $ribbon$ if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5-stranded pretzel knots are $mutant$ $ribbon$. We do this in stages by first showing that 5-stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group, and thus in the smooth knot concordance group as well. Next, we show that any odd, 5-stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.

Dehn surgery on ribbon knots

In this paper, we show that if 0-surgery along a ribbon knot [Formula: see text] and 0-surgery along another knot [Formula: see text] give diffeomorphic 3-manifolds then [Formula: see text] has to be a slice knot. Moreover, they have diffeomorphic slice disk exteriors [Formula: see text] for some ribbon disk [Formula: see text] and slice disk [Formula: see text], where [Formula: see text] and [Formula: see text] are tubular neighborhoods of [Formula: see text] and [Formula: see text], respectively.

Band-passes and long virtual knot concordance

Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group [Formula: see text]. It is shown that for every concordance class [Formula: see text], there is a [Formula: see text] that is not band-pass equivalent to [Formula: see text] and an [Formula: see text] that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that [Formula: see text] is a band-pass invariant but not a concordance invariant of long virtual knots, where [Formula: see text] and [Formula: see text] generate the degree two Polyak group for long virtual knots.

Symmetric ribbon disks

We study the ribbon disks that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number rS(K) is introduced and compared with the classical ribbon number r(K). We show that the difference rS(K) - r(K) can be arbitrarily large by constructing an infinite family of ribbon knots Kn such that r(Kn) = 2 and rS(Kn) > 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.

The Kanenobu knots and Khovanov-Rozansky homology

Kanenobu has given infinite families of knots with the same HOMFLYPT polynomials. We show that these knots also have the same s l ( n ) sl(n) and HOMFLYPT homologies, thus giving the first example of an infinite family of knots indistinguishable by these invariants. This is a consequence of a structure theorem about the homologies of knots obtained by twisting up the ribbon of a ribbon knot with one ribbon.