Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
This paper confirms a bound for folded ribbonlength of 2-bridge knots.
problem Bounding the folded ribbonlength of 2-bridge knots.
method Investigated the folded ribbonlength of 2-bridge knots and proved a linear upper bound.
result The folded ribbonlength of a 2-bridge knot K is bounded above by 2c(K)+2. Study on ribbonlength and crossing number for folded ribbon knots.
problem Understanding the relationship between ribbonlength and crossing number for folded ribbon knots.
method Used two different methods to establish upper bounds on ribbonlength in terms of crossing number.
result Found constants c1 and c2 such that ribbonlength is bounded above by c1⋅Cr(K)2 and c2⋅Cr(K)3/2, respectively. Upper bounds on ribbonlength of various knots, showing linear and sub-linear behavior.
problem Estimating the ribbonlength of different types of knots.
method Using Kauffman's model of folded ribbon knots, we derive upper bounds on ribbonlength for specific knot types.
result Upper bounds on ribbonlength are linear in crossing number for some knots and sub-linear for others.
Study on the ribbonlength of knots and links, improving upper bounds.
problem Finding the infimal folded ribbonlength of knots and links.
method Creating new folded ribbon knots and applying them to improve upper bounds.
result Improved upper bounds for (2,q)-torus links, twist knots, and pretzel links. This survey reviews 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 ribbon, and the ribbonlength problem asks to minimize the ribbonlength for a given knot type. We give a summary of known results. For the mos…
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 ribbon, and it turns out that the way the ribbon is folded influences the ribbonlength. We give an upper bound of ncot(π/n) for the ribbonlength of $…
Paper proves ribbonlength grows linearly with knot complexity.
problem Proving ribbonlength grows linearly with knot complexity.
method Binary grid diagrams and bisected vertex leveling techniques.
result Ribbonlength is bounded by a linear function of the crossing number.
This paper finds bounds on the ribbonlength of knots and links with up to 9 crossings.
problem Finding the infimal folded ribbonlength of knots and links.
method Analyzing pretzel links and constructing knots to find bounds on ribbonlength.
result Uniform bounds on infimal folded ribbonlength for certain link families.
Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An important problem in knot theory is to study the minimal ratio of …
Proves bounds on ribbonlength for various knot types.
problem Finding bounds on the infimal folded ribbonlength of knot types.
method Applied techniques to multi-twist Möbius bands and knot constructions.
result Established bounds for (2,q) torus knots and twist knots. We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert s…
Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or equivalently, embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a ch…
In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings is a simple-ribbon knot. In this paper, we give a formula for th…
Study knots that divide ribbon knotted surfaces, computing their half ribbon genus and fusion number.
problem Understanding knots that divide ribbon knotted surfaces and their properties.
method Defining half ribbon knots, computing half ribbon genus and fusion number, and comparing with Levine-Tristram signatures.
result Computed half ribbon genus and fusion number for various knots, including new computations of doubly slice genus.
Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral 3-fold cover of S3 branched along a…
The paper calculates ribbon numbers for 12-crossing knots using Alexander polynomials.
problem Determining the minimum number of ribbon singularities for knots.
method Using Alexander polynomials and systematic treatment of knot invariants.
result Computed ribbon numbers for many 12-crossing knots.
Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.
problem Determining Alexander polynomials for ribbon and virtual knots.
method Using ribbon's intrinsic singularity information, defining half Alexander polynomial, and developing simplified formulas.
result New formulas for Alexander polynomials of general knots and virtual knots in terms of Gauss diagrams.
Study shows not all ribbon knots can be symmetric unions.
problem Whether every ribbon knot can be a symmetric union.
method Exhibited a specific ribbon Montesinos knot that cannot be a symmetric union.
result Found a ribbon knot that is not a symmetric union.
Two-bridge ribbon knots have symmetric union presentations.
problem Characterizing two-bridge ribbon knots.
method Symmetric union presentations and partial knot analysis.
result Symmetric union presentations for various two-bridge ribbon knots.
Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.
problem Understanding the concordance properties of knots and their cables.
method Using Seifert genus and concordance invariant γ0 from bordered Heegaard Floer homology.
result Connected sums of γ0-sharp fibered knots are ribbon only if they are of a specific form, or the slice-ribbon conjecture fails.
A new approach to Morse theory using folded ribbon trees.
problem Applying Morse theory on symmetric products of surfaces.
method Introducing an A-infinity category with objects as κ-tuples of Morse functions, and showing conditions for the endomorphism to be a Hecke algebra.
result The endomorphism of a specific type of κ-tuple of Morse functions on T*R^2 is the Hecke algebra associated to the symmetric group.
We consider ribbon n-knots for n\geq 2. For such knots we define a set of moves on ribbon disks, and show that any two ribbon disks for isotopic knots are related by a finite sequence of such moves and ambient isotopies. Using this we are able to prove that there is a natural geometric correspondence between ribbon n-k…
This paper investigates symmetric ribbon numbers of low-complexity knots.
problem Determining the minimum number of ribbon singularities in symmetric ribbon disks for knots with up to 12 crossings.
method Systematic investigation using knot polynomials and determinants.
result Novel lower bounds for symmetric ribbon numbers of knots with up to 12 crossings.
New knots bound multiple non-isotopic ribbon disks.
problem Finding knots that bound multiple non-isotopic ribbon disks.
method Classification of fibered, homotopy-ribbon disks for generalized square knots.
result Infinitely many knots bound infinitely many pairwise non-isotopic ribbon disks.
New knot quandles distinguish ribbon knots with isomorphic groups.
problem Distinguishing knots with isomorphic fundamental groups.
method Examined knot quandles of Suciu's ribbon knots and computed their types.
result Knot quandles of Suciu's ribbon knots are mutually non-isomorphic.
Study ribbon concordance and minimal compressions, proving new results about fibered knots.
problem Understanding ribbon concordance and minimal compressions of surface homeomorphisms.
method Proving monotonicity of simplicial volume and dilatation under ribbon concordance, algorithmic enumeration of minimal compressions.
result Every fibered knot has only finitely many predecessors in the ribbon-concordance partial order.
New bounds found for ribbon numbers of knots and links.
problem Finding bounds for the ribbon number of knots and links.
method Using determinant and Jones polynomial, we find new lower bounds and prove the finiteness of certain sets.
result We determine the set of Jones polynomials for ribbon knots with 11 or fewer crossings.
The study limits the number of ribbon concordant fibered knots.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Combining Floer homology results with fixed points of monodromy and volume inequalities.
result There are only finitely many hyperbolic fibered knots ribbon concordant to any given knot.
Knots can be ordered by ribbon concordance, solving a long-standing question.
problem Ordering knots by ribbon concordance.
method Representation varieties of knot groups to SO(N) and relations induced by ribbon concordance. result Ribbon concordance forms a partial ordering on the set of knots.
New insights into knot fusion numbers via cabling.
problem Understanding fusion numbers of ribbon knots and their behavior under cabling.
method Utilizing knot Floer homology and cabling formulas to analyze fusion numbers.
result The fusion number and strong homotopy fusion number of (p,1)-cable knots are preserved.
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…
Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.
problem Prove strong ribbon concordance induces a partial order on links.
method Use results from knot Floer homology to certify minimality under the ribbon partial order.
result Certify minimality for a handful of knots and find minimal ribbon minimal knots.
Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.
problem Finding ribbon disks for alternating knots to determine sliceness.
method Algorithm based on Donaldson's diagonalization theorem, computer implementation.
result Successfully finds ribbon disks for many prime knots, resolves sliceness for most.
We show that every Z-torsion free knot module is realized by a ribbon 2-knot with group of geometric dimension at most 2, and give some partial results on the characterization of the knot modules of fibred ribbon 2-knots.
We present the results of Axel Seeliger's tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and…
Ribbonness proven for slice knots, solving an old question.
problem Proving ribbonness for slice knots.
method Showing a link bounds a ribbon surface that is a renewal embedding of a bounding surface.
result Every slice knot is a ribbon knot.
Classifies fibered ribbon pretzels, except for a few cases.
problem Classifying fibered ribbon pretzel knots up to mutation.
method Combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks.
result Complete classification except for a few cases.
New theorem connects handle-ribbon knots to slice derivatives.
problem Characterizing knots that bound handle-ribbon disks.
method Proves equivalence between handle-ribbon and slice derivatives.
result Handle-ribbon knots are equivalent to slice derivatives.
We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger obstructions are based on the Ozsvath-Szabo correction term in Heegaard-Floer homology, al…
Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.
problem Understanding homotopy ribbon concordance for knots.
method Using Blanchfield pairings and twisted Alexander polynomials.
result Existence of infinite families of knots with same Blanchfield form but not homotopy ribbon concordant.
Khovanov homology shows (4,5) torus knot is a summand in its concordance class.
problem Understanding the global ribbon minima in knot concordance classes.
method Application of Khovanov homology to ribbon concordance.
result Khovanov homology of (4,5) torus knot is a summand in any knot's homology.
This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let K and K′ be 2-knots. Let K and K′ are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimen…
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.
Machine learning finds knots that bound ribbon disks.
problem Detecting ribbon knots in topology.
method Bayesian optimization and reinforcement learning.
result Successfully detected many ribbon knots up to 70 crossings.
The twisting number of a ribbon knot is at least as large as its doubly slice genus.
problem Proving a lower bound for the twisting number of ribbon knots in terms of their doubly slice genus.
method Analyzing symmetric unions and tangle replacements to establish the bound.
result Ribbon knots have arbitrarily high twisting numbers, matching their doubly slice genus.
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 d…