A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-…
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48 results for “link equivalence”
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain componen…
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.
Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band a…
Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
A pass-move and a $#$-move are local moves on oriented links defined by L.H. Kauffman and H. Murakami respectively. Two links are self pass-equivalent (resp. self $#$-equivalent) if one can be deformed into the other by pass-moves (resp. $#$-moves), where non of them can occur between distinct components of the link. T…
New equivalence relation on ribbon graphs connects to virtual links.
problem Understanding virtual links through ribbon graphs.
method Introducing a new equivalence relation on ribbon graphs.
result Correspondence between virtual links and ribbon graphs.
String links classified up to -moves and link-homotopy.
problem Classifying string links up to specific moves and link-homotopy.
method Using Milnor link-homotopy invariants and -moves.
result Equivalence classes of string links form a finite group.
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.
The paper classifies pretzel links with 2 components and gives conditions for those with 3 or more.
problem Classifying pretzel links based on their self delta-equivalence.
method Using Conway polynomials to determine self delta-equivalence for links with 2 or more components.
result Necessary and sufficient conditions for self delta-equivalence of pretzel links with 3 or more components.
Classifies Legendrian Hopf links in lens spaces.
problem Classifying Legendrian Hopf links in lens spaces.
method Classification up to coarse equivalence.
result Legendrian Hopf links in L(p,1) are classified.
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 -fold cyclic covering diagrams and proving their equivalence to original diagrams.
result Well-defined map from virtual links to mod almost classical virtual links.
Flat fully augmented links with homeomorphic complements are equivalent.
problem Determining equivalence of flat fully augmented links based on their complements.
method Careful analysis of totally geodesic surfaces and cusps in link complements and their behavior under homeomorphism.
result Complete classification of flat fully augmented link complements with multiple reflection surfaces and symmetries.
Parallelization technique for welded links preserves equivalence and yields specific decompositions.
problem Defining and proving equivalence of parallel welded link diagrams.
method Introduced a parallelization construction for welded link diagrams and showed its well-definedness.
result Parallel diagrams maintain equivalence under specific orientations and yield decompositions.
Knots in math.GT
The paper studies knots in projective space using virtual link theory.
problem Understanding knots in three-dimensional projective space.
method Associate virtual links to projective links and apply virtual knot theory techniques.
result Equivalent projective links correspond to equivalent virtual links modulo a flype move.
Researchers debunk a generalized Property R conjecture for 2-component links.
problem The generalized Property R conjecture for 2-component links in .
method Disproved a further generalization of the GPRC using specific counterexamples.
result 2-component framed links in surging to a connected sum of homology 's are not handleslide equivalent to a split link.
Recently Swatee Naik and Theodore Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called Str…
Minimal crossing virtual links have minimal supporting genus.
problem Understanding the relationship between the number of crossings and the genus of virtual links.
method Developed a new parity theory for virtual links to prove minimal crossing implies minimal genus.
result Minimal crossing virtual links have minimal supporting genus.
Study the embedding space of a Hopf link in 3D and 3-manifolds.
problem Homotopy equivalence of embedding spaces of Hopf links.
method Prove homotopy equivalence using subspace inclusions and manifold properties.
result Embedding spaces are homotopy equivalent to specific 3-manifolds.
Graphs with certain eigenvalues are linked to simply laced Dynkin diagrams.
problem Characterizing graphs with specific eigenvalues.
method Defining equivalence relation on signed graphs and using congruence over Z.
result Signed graphs with eigenvalues > -2 are linked to simply laced Dynkin diagrams.
New equivalence relation for links using cut-diagrams.
problem Classical link concordance.
method Cut-diagrams and cut-concordance.
result Nilpotent peripheral system invariant of cut-concordance.
Study on links formed by pseudocircle arrangements, focusing on three unavoidable cases.
problem Counting non-equivalent positive oriented links with pseudocircle arrangements as shadows.
method Analyzing three unavoidable arrangements of pseudocircles to estimate the number of non-equivalent links.
result Sharp estimates on the number of non-equivalent positive oriented links for the three unavoidable arrangements.
Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology f…
Study shows infinite distinct outer metric Lipschitz classes for knots in .
problem Lipschitz classification of surface singularities in .
method Outer metric Lipschitz equivalence and topological equivalence of knots.
result Infinitely many distinct outer metric Lipschitz classes for knots in .
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link d…
Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Typesmath.GT
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent s…
We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.
Polynomial algorithm found for alternating link equivalence.
problem Link equivalence of alternating links in 3-space.
method Tait flyping conjectures, observations from graph theory, and topological graph theory.
result Alternating link equivalence has a polynomial algorithm.
The paper verifies stable handleslide triviality of some R-links and shows many are stably equivalent.
problem Stable handleslide triviality of R-links as potential counterexamples to the generalized property R conjecture.
method Implemented an algorithm to construct all R-links explicitly and verified their stable handleslide triviality.
result Many R-links are stably handleslide equivalent.
For an -component link , the Milnor's isotopy invariant is defined for each multi-index $I=i_1i_2...i_m (i_j\in\n)$. Here is called the length. Let denote the maximam number of times that any index appears. It is known that Milnor invariants with are link-homotopy invariant. N. Habegger and X. S.…
We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
We use an action, of 2l-component string links on l-component string links, defined by the first author and Xiao-Song Lin, to lift the indeterminacy of finite type link invariants. The set of links up to this new indeterminacy is in bijection with the orbit space of the restriction of this action to the stabilizer of t…
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
In the context of finite type invariants, Stanford introduced a family of equivalence relations on knots defined by the lower central series of the pure braid groups and characterized the finite type invariants in terms of the structure of the braid groups. It is known that this equivalence and Ohyama's equivalence def…
The paper studies equivalence classes of links using k-moves and analyzes the Hopf crossing number.
problem Equivalence classes of links and Hopf crossing number.
method Defining k-moves on arrow diagrams of links and studying equivalence classes up to k-moves.
result There are infinitely many classes of knots modulo k-moves for odd k ≥ 5.
A Laundry Surface for Braidsmath.GT
Laundry surfaces for closed braid diagrams are presented. It is shown that braid diagrams are characterized by linking matrices obtained by lifting cycles from these surfaces. Oriented link types are then characterized by equivalence classes of linking matrices. Similar equivalence classes can be composed of Gordon and…
The paper studies right-angled links on higher genus surfaces.
problem Classifying and understanding right-angled links on surfaces of higher genus.
method Defining and proving equivalence of properties for RGCR links, using diagram restrictions and polygonal checkerboard surfaces.
result Classification of RGCR links and bounds on their number for a given genus.
A knot (or link) diagram is said to be everywhere equivalent if all the diagrams obtained by switching one crossing represent the same knot (or link). We classify such diagrams of a closed 3-braid.
Legendrian Lavrentiev linksmath.SG
Legendrian Lavrentiev links are shown to be equivalent to smooth links.
problem Understanding equivalence of Legendrian and smooth links.
method Definition and analysis of Legendrian isotopies.
result Equivalence classes of Legendrian Lavrentiev links coincide with smooth links.
For any link and for any modulus we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtaine…
Links and dynamicsmath.DS
This paper classifies links in 3D dynamical systems.
problem Classifying links in 3D dynamical systems.
method Analyzing diffeomorphisms and link complements in .
result Countable number of equivalence classes of tame links in .
We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and l…
The paper extends Alexander invariant and medial quandle equivalence to links.
problem Understanding stronger invariants for links than Alexander modules.
method Extending Joyce's observation to multiple medial quandles and reduced Alexander modules.
result Medial quandles provide stronger invariants than reduced Alexander modules for links.
This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.
problem Finite type invariants for ribbon knotted surfaces and their relation to welded string links.
method Developed a theory of finite type invariants for welded string links up to -equivalence, studied algebraic structures, and showed characterizations.
result Characterizes the information contained by finite type invariants in low degrees for welded string links.
Defines concordance for spatial graphs and proves sliceness equivalence.
problem Understanding concordance and sliceness for spatial graphs.
method Smooth definitions and linking number conditions.
result Sliceness of a spatial graph is equivalent to a condition on linking numbers and a link.
Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual homological notion of linking by taking into account the fundamental group of the ambient …
Two Seifert surfaces of links in are said to be twist equivalent if one can be obtained from the other, up to isotopy, by repeatedly performing operations consisting of cutting along an embedded arc, applying a full twist near one copy of the arc, and re-gluing. By using bridge spheres for their boundary links, w…