New invariant refines Milnor's triple linking number, revealing more information for complex links.
problem Indeterminacy of Milnor's triple linking number in complex link configurations.
method Introduced a new invariant called the total triple linking number, refining Milnor's original.
result The total triple linking number is non-trivial for every (n≥6)-component link, providing more information than classical triple linking numbers. Paper finds linking numbers for Montesinos links using a simple algorithm.
problem Calculating linking numbers for Montesinos links.
method Simple proof and numerical algorithm for rational links, extending to Montesinos links.
result Linking numbers found for any two components in Montesinos links.
The triple linking number of an oriented surface link was defined as an analogical notion of the linking number of a classical link. We consider a certain m-component T2-link (m≥3) determined from two commutative pure m-braids a and b. We present the triple linking number of such a T2-link, by usin…
The paper examines linking numbers in grid models and finds polynomial moments.
problem Analyzing linking numbers in grid models.
method Examined linking numbers as a random variable on isotopy classes of 2-component links, computed moments and limits.
result The uth moment of the linking number is a polynomial in the grid size with degree d≤u, and all odd moments vanish. Computed linking number of modular knots and Lorenz links.
problem Computing the linking number of modular knots in a specific space.
method Using correspondence between modular links and Lorenz links, and intersection number involving Conway topographs.
result Computed linking number of modular knots and compared to a previous formula.
New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
Delta-unlinking number measures how to unlink algebraically split links.
problem Measuring unlinking complexity of algebraically split links.
method Defining delta-unlinking number as minimum delta-moves to unlink, proving bounds and calculating specific values.
result Precise delta-unlinking numbers for algebraically split prime links up to 9 crossings, and 4-genus values for most.
The paper extends CF-moves to classify virtual links of any number of components.
problem Classifying virtual links using CF-moves.
method Extending CF-moves to classify virtual links of arbitrary number of components using the virtual linking number and invariants.
result Classification of 3-component even virtual links up to CF-moves.
The paper generalizes linking number properties for complete graphs.
problem Understanding linking numbers in spatial complete graphs.
method Analyzing the sum of square linking numbers and triangle-triangle links.
result Explicit formula for the sum of square linking numbers in large complete graphs.
New bounds on clasp number for 3-component links.
problem Measuring how close a 3-component link is to being a boundary link.
method Using C-complexes and minimizing the area of word curves.
result Constructs links achieving new lower bounds for clasp number.
Link crossing number problem is NP-hard.
problem Determining the crossing number of a link.
method NP-hardness proof.
result Link crossing number problem is NP-hard.
Paper defines linking numbers for periodic tangles.
problem Defining invariant linking numbers for periodic tangles.
method Defined linking numbers using motifs and showed invariance.
result Linking numbers are well-defined and invariant.
This paper improves bounds on how many Delta-moves are needed to trivialize a link.
problem Counting the minimum number of Delta-moves to make a link homotopy trivial.
method Classification of link homotopy and extremal graph theory.
result Quadratic and cubic upper bounds on the homotopy trivializing numbers of links.
Characterizes adequate links using Jones polynomial and crossing number.
problem Characterizing adequate links.
method Using Jones polynomial and crossing number, proving links are adequate.
result Links with specific polynomial properties are adequate.
The paper studies the minimal number of clasps in C-complexes for links.
problem Determining the minimal number of clasps in C-complexes for links.
method Examining C-complexes bounded by links, focusing on the number of clasps and their relation to link invariants.
result For 2-component links with nonzero linking number, the linking number determines the minimal number of clasps.
Links with minimum tunnel number have one less component than their number of parts.
problem Finding the minimum tunnel number for complex links.
method Combinatorial argument in link diagrams.
result Links with minimum tunnel number have one less component than their number of parts.
Enhanced Alexander module detects linking numbers in links.
problem Detecting linking numbers in links using Alexander modules.
method Defining and singling out meridians and longitudes in reduced Alexander modules.
result The enhanced Alexander module determines all linking numbers.
The paper identifies all link projections with isolate-region number one.
problem Determining link projections with a specific isolate-region number.
method Analyzing link projections to find isolated regions and their cardinality.
result All link projections with isolate-region number one are identified.
The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link. In this paper, we show that the minimal coloring number of any non-splittable Z-colorable links is four. As an example, we consider the link obtained by…
In this paper we investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinkin…
Defines linking number and Milnor invariants, their properties.
problem None explicitly stated; definitions and properties are discussed.
method Definitions and properties of linking number and Milnor invariants are discussed.
result Definitions and properties of linking number and Milnor invariants are established.
Study of knots and links in 2-complexes, defining linking numbers and polynomials.
problem Understanding knots and links in 2-dimensional complexes.
method Definition of linking numbers and Kauffman-type bracket polynomials for links in 2-complexes.
result Established relationships between 2-complexes and knots/links in 3-manifolds.
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the sp…
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.
The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.
problem Computing the total number of alternating pretzel links for a given crossing number.
method Derived a closed formula to compute the total number of alternating pretzel links, P(c), for any given crossing number c. result The number of alternating pretzel links grows exponentially with the crossing number.
New measure shows how links can be untangled as twists increase.
problem Understanding how links can be simplified through repeated twists.
method Introduced the stable unknotting number to analyze links in a twist family.
result The stable unknotting number depends only on the winding number of the link, not the wrapping number.
The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting nu…
We define the crosscap number of a 2-component link as the minimum of the first Betti numbers of connected, non-orientable surfaces bounding the link. We discuss some properties of the crosscap numbers of 2-component links.
New proof for minimizing tunnel systems in satellite chain links.
problem Minimizing the tunnel number of satellite chain links.
method Proving the tunnel number is minimized for links with a specific number of components and bridge number.
result The result is sharp for satellite chain links over a 2-bridge knot.
Study sharpens unlinking number bounds for special alternating links.
problem Determining the exact unlinking number for special alternating links.
method Analyzes links in the 3-sphere, focusing on special alternating links and their crossing changes.
result Sharp lower bounds for unlinking number realized by crossing changes in alternating diagrams.
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link.…
Estimates how many times a star appears due to gravitational lensing.
problem Estimating the number of times an observer sees a star due to gravitational lensing.
method Use affine linking numbers to estimate the number of times an observer sees a star.
result Estimates the number of times an observer sees a star due to gravitational lensing.
Study on multiple linking numbers, extending Gauss diagram formulas.
problem Extending link invariants to more complex diagrams.
method Investigate second Gauss diagram formula involving two arrows.
result Discover two types of multiple linking numbers, one Vassiliev invariant, the other sensitive to Reidemeister moves.
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its charact…
Triple linking numbers were defined for 3-component oriented surface-links in 4-space using signed triple points on projections in 3-space. In this paper we give an algebraic formulation using intersections of homology classes (or cup products on cohomology groups). We prove that spherical links have trivial triple lin…
Proves links can be simplified to trivial form in few changes, limiting Milnor's invariants.
problem Limits of Milnor's invariants in understanding unlinking number.
method Sequence of links, crossing changes, and geometric filtration.
result Crossing changes to Ck-trivial link grows quadratically with number of components. New method calculates number of components in twisted torus links.
problem Determining the number of components of twisted torus links.
method Euclidean algorithm-like procedure based on parameters.
result Number of components is a multiple of gcd(p, q, r, s).
As a generalization of the linking number, we construct a set of invariant numbers for two-component handlebody-links. These numbers are elementary divisors associated with the natural homomorphism from the first homology group of a component to that of the complement of another component.
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related knots.
Study Heegaard Floer homology for two-component L-space links with zero linking number.
problem Characterize and calculate invariants for two-component L-space links with zero linking number.
method Use Heegaard Floer homology, Sato-Levine invariant, and Casson invariant to describe the relationship between these invariants and surgeries on links.
result Explicitly describe the relationship between the h-function, Sato-Levine invariant, and Casson invariant for two-component L-space links with zero linking number. A flat plumbing basket is a surface consisting a disk and finitely many bands which are contained in distinct pages of the trivial open book decomposition of S3. In this paper, we construct a Legendrian link from a flat plumbing basket, and we describe a relation among the self-linking number, the Thursto…
Calculates lower bounds for type III Reidemeister moves in link diagrams.
problem Determining the minimum number of Reidemeister III moves between link diagrams.
method Introducing non-self OU sequence and OU number for link diagrams.
result Provides a lower bound for the number of Reidemeister III moves.
We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems. Moreover these two conditions …
Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…
Paper introduces new link homotopy invariants and applies them to 3-bouquet graphs.
problem Milnor's triple linking number and its applications in link homotopy.
method Developed new integer-valued link homotopy invariants and applied them to 3-bouquet graphs.
result Found new integer-valued invariants derived from four terms summing to Milnor's triple linking number.
K. Ichihara and E. Matsudo introduced the notions of Z-colorable links and the minimal coloring number for Z-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable Z-colorable link is 4. In this paper, we sh…