Paper finds knots with stair-step bridge spectra but are not high distance.
problem Understanding the relationship between bridge spectra and knot distance.
method Computed bridge spectra and distances of generalized Montesinos knots, including pretzel and Montesinos knots.
result First example of knots with stair-step bridge spectra but not high distance.
Study bridge spectra of 2-bridge knots and their cables.
problem Computing bridge spectra for 2-bridge knots and their cables.
method Computed bridge spectra of cables of 2-bridge knots.
result Results on bridge spectra and distance of Montesinos knots.
Alternative proof for 2-bridge knots' bridge positions.
problem Proving genus one 1-bridge positions for 2-bridge knots.
method Alternative proof using disk complexes.
result Every genus one 1-bridge position is a stabilization.
For every bridge number, critical spheres exist for links.
problem Finding critical bridge spheres for links with arbitrary bridge numbers.
method Constructing infinite families of links with square bridge spheres.
result Existence of critical bridge spheres for links with arbitrary bridge numbers.
Study on knot diagrams showing bridge number can differ from crossing number.
problem Incompatibility between crossing number and bridge number for knot diagrams.
method Defined and compared various bridge number computations for knot diagrams, studied minimizing diagrams, and constructed families of minimal crossing diagrams.
result Found examples where bridge number differs from crossing number, and demonstrated this difference can grow infinitely.
New examples of knots with special bridge positions found.
problem Understanding special types of bridge positions for knots.
method Derived examples of knots with unperturbed weakly reducible non-minimal bridge positions.
result Connected sum of unperturbed bridge positions is unperturbed (a new conjecture).
Reduction of knots in bridge position results in a lower bridge number if certain conditions are met.
problem Understanding the reduction of knots in bridge position and its implications.
method Analyzing the reduction of knots along bridge disks and considering the existence of cancelling pairs.
result If a reduction of a knot yields an (n-1)-bridge position, the knot bounds a disk containing the bridge disk and intersecting the sphere in n arcs.
The paper provides examples of keen weakly reducible bridge spheres for links in b-bridge position.
problem Characterizing and finding examples of keen weakly reducible bridge spheres.
method Analyzing bridge spheres and their properties in terms of compressing disks and width complex.
result Infinitely many examples of keen weakly reducible bridge spheres for links in b-bridge position.
Any 2-bridge knot in the 3-sphere has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.
Lower bounds on knot distortion related to bridge distance and number.
problem Finding lower bounds on knot distortion in 3D space.
method Extending Pardon's techniques to relate distortion to bridge distance and number.
result Lower bound on knot distortion proportional to minimum of bridge distance and bridge number.
New method finds infinitely many surface knots with specific bridge numbers.
problem Finding numerical invariants for surface links.
method Colorings of surface links by keis to prove bridge number existence.
result Existence of infinitely many surface knots with bridge number n for n ≥ 4.
Researchers found infinite links with specific bridge positions.
problem Finding links with minimal bridge positions.
method Applying Takao et al.'s criterion to create links with locally minimal n-bridge and globally minimal m-bridge positions. result Provided an infinite family of links with specific bridge positions.
Character varieties of 2-bridge knots and links explained using Farey recursion.
problem Understanding character varieties of 2-bridge knots and links.
method Using Farey recursion to define polynomials for character varieties.
result Character varieties described in terms of polynomials defined by Farey recursion.
We show that if K is a knot in S3 and Σ is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by Σ via an isotopy is n. We also construct a knot with two different bridge spheres with 2n and 2n−1 bridges respecti…
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.
Paper explores link and plat presentations, showing equivalence under bridge isotopy.
problem Relationship between links in bridge position and plat presentations.
method Established equivalence between Hilden double coset classes of plat presentations and bridge positions up to isotopy.
result Revealed equivalence of Hilden double coset classes for n-bridge unknot and torus knots in plat position.
We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.
We give a complete characterization of those essential simple loops on 2-bridge spheres of 2-bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridian-pair-preserving epimorphisms between 2-bridge link groups.
Paper refines generating function for 2-bridge knot groups.
problem Determining the number of epimorphisms between 2-bridge knot groups.
method Refined generating function considering genus and crossing number.
result Improved formula for epimorphisms between 2-bridge knot groups.
Bridge positions of handlebody-knots are equivalent when stable.
problem Equivalence of bridge positions in handlebody-knots.
method Demonstrated stability of bridge positions.
result Bridge positions of handlebody-knots are stably equivalent.
The study calculates braid indices for two-bridge knots and proves inequalities.
problem Calculating the braid index of two-bridge knots.
method Proved inequalities and provided average braid index for knots of a given crossing number.
result Average braid index for two-bridge knots with a given crossing number.
We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.
For any given number of crossings c, there exists a formula to determine the number of 2-bridge knots of c crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of whether a given (prime) knot is a 2-bridge knot remains a nontrivial exercise, and …
New unbiased methods for generating stochastic bridges with given extrema.
problem Generating unbiased stochastic bridges with a specified extremum.
method Comparison and generalization of two algorithms for Brownian bridges to other diffusions, and application to Ornstein-Uhlenbeck and unconstrained processes.
result Generalization of unbiased generation methods to other diffusions and application to various processes.
Introduces Schubert normal form for 3-bridge links and presents their groups.
problem Representing and understanding 3-bridge link groups.
method Develops Schubert normal form and uses it to present 3-bridge link groups.
result Two presentations of 3-bridge link groups based on Schubert normal form.
In this paper and its prequel, we give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a 2-bridge sphere in an even …
In this paper and its sequel, we give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a 2-bridge sphere in an even H…
The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when…
Perturbing non-minimal bridge positions of a knot ensures similar behavior for its cable links.
problem Ensuring similar non-minimal bridge positions for cable links.
method Perturbing non-minimal bridge positions of a knot K and analyzing (2,2q)-cable links L. result Every non-minimal bridge position of a (2,2q)-cable link L is perturbed. The author, in her previous paper, constructed an infinite family of 3-bridge links each of which admits infinitely many 3-bridge spheres up to isotopy. In this paper, we prove that if a prime, unsplittable link L in S3 admits infinitely many 3-bridge spheres up to isotopy then L belongs to the family.
Random links defined by bridge splitting are hyperbolic with high probability.
problem Understanding the hyperbolic nature of random links.
method Random bridge splitting to define links and probabilistic analysis.
result Random links are hyperbolic with asymptotic probability 1.
We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-spher…
Short note shows epimorphism in 2-bridge knots using Riley polynomials.
problem Classifying 2-bridge knots using Riley polynomials. method Elementary argument using Riley polynomials.
result Existence of epimorphism between groups of 2-bridge knots. The paper calculates bridge numbers for knots using machine learning.
problem Determining the bridge number for virtual knots with multiple definitions.
method Employed computational techniques and machine learning models to classify knots based on their bridge numbers.
result Demonstrated that the bridge number for virtual knots can differ significantly.
Formula found for braid index of n-bridge braids.
problem Finding a formula for the braid index of n-bridge braids. method Elementary, effective, self-contained proof.
result Closed form formula for braid index of n-bridge braids. This is the second of a series of papers which give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. The first paper of the series treated the case of the 2-bridge torus links. In this paper, we treat the case …
We show that any non-minimal bridge decomposition of a torus knot is stabilized and that n-bridge decompositions of a torus knot are unique for any integer n. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…
Paper introduces danceability index as a new bridge index definition.
problem Defining the bridge index in various mathematical contexts.
method Proves danceability index as equivalent to bridge index, extends to virtual knots.
result Danceability index is a new equivalent definition of the bridge index.
The study of epimorphisms between 2-bridge knot groups and their crossing numbers.
problem Understanding the relationship between the crossing numbers of 2-bridge knots and their epimorphisms.
method Analyzing epimorphisms and their impact on crossing numbers.
result The crossing number of a 2-bridge knot is at least three times the crossing number of the target knot.
Algorithm improves blockchain bridge efficiency.
problem Efficient cross-chain wealth management.
method Dynamic algorithm to optimize bridge capacities.
result Optimized fund transfers across networks.
The volume and genus g bridge numbers of knots are independent.
problem The relationship between hyperbolic volume and genus g bridge numbers of knots. method Constructing explicit sequences of knots to show independence of volume and genus g bridge numbers. result Hyperbolic volume and genus g bridge numbers are completely independent. Proves integration formula for Brownian bridge measure on manifolds.
problem None explicitly stated in the abstract.
method Integration by parts formula for semi-classical Riemannian Brownian bridge measure.
result Proof of integration by parts formula.
Explicit formula for Reidemeister torsion of two-bridge knots.
problem Calculating Reidemeister torsion for two-bridge knots.
method Provided an explicit formula and proved vanishing identities.
result Adjoint Reidemeister torsion satisfies vanishing identities.
Normal distribution found for 2-bridge knots signatures.
problem Distribution of signatures for 2-bridge knots.
method Calculated average signature, introduced s(c,σ), used limit theorem. result Distribution of signatures approaches normal as knot complexity increases.
A new method estimates Schrödinger bridges without iterative simulations or neural networks.
problem Estimating the time-dependent drift between two probability distributions.
method Solving the static entropic optimal transport problem and modifying the potentials.
result The Sinkhorn bridge method provably estimates Schrödinger bridges with a rate of convergence dependent on the target measure's intrinsic dimensionality.
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.
We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern L that lies in a solid torus T in which it has index k, then the bridge numbers satisfy the following: 1) The bridge number of K is greater than or equal to the product of k and the bridge number of J; …
Extends martingale Schrödinger bridge to arbitrary dimensions and characterizes it.
problem Tackles the martingale Schrödinger bridge in arbitrary dimensions.
method Identifies continuous-time counterpart and relates to variational problems.
result Continuous martingale Schrödinger bridge coincides with Föllmer martingale in irreducible case.