Study Milnor invariants for covering links to distinguish certain links.
problem Distinguish links using Milnor invariants for covering links.
method Generalize Hartley and Murasugi's covering linkage invariants and use cobordism invariants.
result First non-vanishing Milnor invariants of a Brunnian link modulo 2 equals a sum of linking numbers of covering links.
We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering T2-link is equivalent to the split union of spun T2-links and turned spun T2-links. We show th…
Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the o…
We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T2-knots and turned spun T2-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…
The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.
problem Understanding and distinguishing twisted links.
method Twisted intersection colorings and double coverings.
result There exist infinitely many pairs of twisted links with equivalent double coverings.
Special covers of alternating links have finite index subgroups in certain groups.
problem Understanding the structure of alternating link complements and their subgroups.
method Constructing special covers with bounded degree and embedding into specific groups.
result Explicit bounds on the index of subgroups in right-angled Artin and Coxeter groups.
We consider surface links in the 4-space which are presented by the form of simple branched coverings over the standard torus, which we call torus-covering links. In this paper, we study unknotting numbers of torus-covering links. In some cases, we can determine the unknotting numbers.
Study involutions on 3D small covers, proving quotient spaces are linked 2-spheres.
problem Understanding quotient spaces of 3D small covers by involutions.
method Analyzing quotient spaces and using topological properties.
result Quotient spaces of 3D small covers by involutions are linked 2-spheres.
Researchers study chirality in a specific type of torus-covering link.
problem Determining chirality in a specific type of torus-covering link of degree 3.
method Investigates invariants like triple linking numbers, Fox p-colorings, and quandle cocycle invariants.
result Determines the quandle cocycle invariant for S3(a,b) associated with tri-colorings. Paper constructs cyclic coverings of virtual link diagrams, proving equivalence of certain maps.
problem Constructing and proving equivalence of cyclic coverings of virtual link diagrams.
method Introducing m-fold cyclic covering diagrams and proving their equivalence to original diagrams. result Well-defined map from virtual links to mod m almost classical virtual links. The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused bou…
Theory developed for p-adic Mahler measure applied to Z-covers of links.
problem Study of Z-covers of rational homology 3-spheres branched over links. method Developed a theory of p-adic Mahler measure and applied it to Z-covers of links. result Obtained a p-adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among coefficients and invariants. The article proves properties of Seifert links and their cyclic branched covers.
problem Properties of Seifert links and their cyclic branched covers.
method Left-orderability, co-oriented taut foliations, and L-space properties. result Proves the ADE link conjecture for Seifert links. Study calculates non-trivial link cohomologies and applies to branched double covers.
problem Calculating non-trivial link cohomologies.
method Uses Baldwin-Ozsváth-Szabó cohomology and Heegaard-Floer homology.
result Demonstrates applications of cohomology to branched double covers.
We use virtual knot theory to detect the non-invertibility of some classical links in S3. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot υ to a knot K in a 3-manifold N, under certain hypotheses on K and N. Virtual covers of links in …
The study of universal links in 3-manifolds and their properties.
problem Existence and characterization of universal links in 3-manifolds.
method Analyzing branched coverings and distinguishing between universal and complement universal links.
result Closed spherical 3-manifolds are the only ones admitting universal links.
New link types derived from old links.
problem Understanding the structure of modular links.
method Analyzing the complements of arithmetic modular links and comparing them to augmented chainlinks.
result Arithmetic modular links are homeomorphic to augmented chainlinks.
New transverse links solve contact manifold problems.
problem Contact manifold structures.
method Transverse links in 3-sphere contact structures.
result All contact 3-manifolds are contact branched covers over a transverse link.
Study finds symplectic fillings' properties for specific contact covers.
problem Determining symplectic fillings' Euler characteristics and signatures.
method Analyzing exact symplectic fillings of contact branched covers.
result Identified links' symplectic fillings' properties.
We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, w…
We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in Q and in Q(Z[t,t−1]) respectively, where Q(Z[t,t−1]) denotes the quotient field of Z[t,t−1]. It is known that the modulo-Z …
Study definite strongly quasipositive links and their L-space branched covers.
problem Characterize strongly quasipositive links with definite Seifert forms and L-space branched covers.
method Investigate definite strongly quasipositive links, apply previous results, and use Garside elements and braid closures.
result If a strongly quasipositive braid closure is definite, it must be one of specific links or has an L-space branched cover.
This paper studies isotopies of periodic tangles in 3-manifolds using finite covers.
problem Understanding equivalence relations of periodic tangles in 3-manifolds.
method Employing techniques from 3-manifold topology to study complements of links.
result Isotopies between different finite covers of isotopic links are preserved.
Finite n-quandles linked to finite branched covers.
problem Characterizing finite n-quandles of links. method Proving conjecture about n-quandles and branched covers. result Finite n-quandles correspond to finite branched covers. New class of links with specific homology properties and instanton computations.
problem Understanding homology properties of links and their instanton invariants.
method Introduced a new class of links, computed instanton homology, and discussed spectral sequences.
result Computed framed instanton homology for double branched covers of new links.
The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.
problem Investigating exotic surface links in 4-ball.
method Two constructions of Brunnian exotic surface links, using satellite operations and covering links.
result Provided constructions of exotic Brunnian surface links with varied properties.
Let L⊂S3 be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S3, branched along L. When L is an alternating link, $\HFa$ of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a …
Method converts virtual link diagrams to normal ones.
problem Convert virtual link diagrams to normal ones.
method Double covering technique and generalized Reidemeister moves, Kauffman flypes.
result Normal virtual link diagrams obtained from equivalent virtual link diagrams are related by moves.
Links with isotopic preimages in S3 are isotopic in RP3.
problem Characterizing isotopy in RP3 from S3 preimages. method Analysis of isotopy invariants under double covering map.
result Isotopic preimages in S3 imply isotopic links in RP3. Formula calculates linking numbers in knot theory.
problem Obtaining obstructions for satellite operations.
method Explicit formula for linking numbers in branched covers.
result Evaluation of obstructions in various cases.
Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…
Study p-torsion growth in covers of 3-manifolds, proving Iwasawa formulas.
problem Investigate p-torsion growth in compatible systems of covers of 3-manifolds. method Establish analogues of Iwasawa's class number formula for branched covers of links.
result Prove Cuoco--Monsky type formula for branched covers of links.
This paper introduces a new method to create mod m almost classical links from virtual knots.
problem Creating mod m almost classical links from virtual knots.
method Using knots in S_g × S^1 to construct m-fold coverings.
result A new family of m-fold coverings over virtual knots are mod m almost classical.
Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted l…
Study fractional twist behavior in branched covers, with applications to 3-manifolds.
problem Behavior of fractional Dehn twist coefficients in branched coverings.
method Analyzes fully ramified branched coverings and their effects on 3-manifolds.
result Non-right-veering braids represent virtually loose transverse links.
We develop a method to show the fundamental group of the double branched covering of a link is not left-orderable by introducing the notion of the coarse presentation. As in the usual group presentations, a coarse presentation is given by a set of generators and relations, but inequalities are allowed as relations. By …
New covering moves for 3-manifolds up to degree 4.
problem Relating colored link diagrams in 3-manifolds.
method Complete set of covering moves on braids in fixed degree d≥4. result Two local tangle replacements are sufficient after stabilization to the same degree at least 4.
We view closed orientable 3-manifolds as covers of S^3 branched over hyperbolic links. For a p-fold cover M \to S^3, branched over a hyperbolic link L, we assign the complexity p Vol(S^3 minus L) (where Vol is the hyperbolic volume). We define an invariant of 3-manifolds, called the link volume and denoted LV, that ass…
We prove that any knot or link in any 3-manifold can be nicely decomposed (splitted) by a filling Dehn sphere. This has interesting consequences in the study of branched coverings over knots and links. We give an algorithm for computing Johansson diagrams of filling Dehn surfaces out from coverings of 3-manifolds branc…
The paper presents a chain complex for 3-manifold covers, including surface bundles and surgeries.
problem Calculating linking forms and Dijkgraaf-Witten invariants for specific 3-manifold covers.
method Presentation of cellular chain complexes for universal covers of 3-manifolds in a specified class.
result Established a formula for linking forms and developed procedures for Dijkgraaf-Witten invariants.
Algorithm calculates linking numbers in three-manifolds.
problem Computing linking numbers in three-manifolds.
method Algorithm based on dihedral covers of knots.
result Explicit algorithm for all linking numbers.
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
This paper extends link invariants using functors on nanophrases.
problem Link invariants are less informative in certain categories.
method Introduces functors from nanophrases to virtual strings, pseudolinks, quasilinks, and free links.
result Extends the Jones pseudolink polynomial to general nanophrases.
Study shows pretzel links are not slice even with ribbon mutants.
problem Determining which pretzel links are slice.
method 3-fold branched covers and Donaldson's diagonalization theorem.
result Slice-ribbon conjecture proven for 4-stranded 2-component pretzel links.
Study on pretzel knots showing cyclic branched covers are L-spaces.
problem Understanding cyclic branched covers of pretzel knots and their properties.
method Analyzing pretzel knots Kk and their n-fold cyclic branched covers for all n≥1. result The n-fold cyclic branched covers of pretzel knots Kk are L-spaces for all n≥1. Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational homology 3-sphere. Using elementary principles, we provide a similar calculation f…
New findings on branched covers of quasipositive links and their L-space properties.
problem Understanding the conditions under which branched covers of quasipositive links are L-spaces.
method Analyzing Alexander polynomials and using properties of cyclic covers.
result Conditions for the L-space property of branched covers of quasipositive links, including specific cases for strongly quasipositive and quasipositive links.
Dihedral linking invariant uses knot colorings to distinguish knots.
problem Distinguishing knots using knot colorings and linking numbers.
method Algorithm for computing linking numbers in dihedral branched covers.
result The dihedral linking invariant distinguishes more than 98% of prime knot pairs.