Study on commutator subgroups of welded braid groups, proving their finiteness and perfection.
problem Investigating the structure of commutator subgroups in welded braid groups.
method Proved finiteness and Hopfian property, showed perfection for n≥5, computed finite presentations. result Commutator subgroups of welded braid groups are finitely generated, Hopfian, and perfect for n≥5. Paper proves Markov's theorem for extended welded braids and links.
problem Generalizing welded links to extended welded links and braids.
method Following Kamada's approach, proving Alexander and Markov's theorems for extended welded braids and links.
result Proves versions of Alexander and Markov's theorems for extended welded braids and links.
The study of commutator subgroups of virtual and welded braid groups.
problem Characterizing and generating commutator subgroups of virtual and welded braid groups.
method Obtained generators and relations for commutator subgroups, proved finiteness and perfection.
result Finitely generated commutator subgroups for VBn and WBn for specific n values. Homotopy braid groups are shown to be torsion-free.
problem The existence of non-abelian torsion-free quotients of the braid group.
method Diagrammatic theory of welded braids and Artin representation.
result Homotopy braid groups are torsion-free.
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups Pn, the welded pure braid groups wPn, the virtual pure braid groups vPn, as w…
Study on when the lower central series stops for various groups, including braid groups.
problem Understanding when the lower central series stops for different groups.
method Various techniques applied to braid groups and related groups.
result Complete computation of the lower central series for most groups studied.
In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathem…
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from VSGn to Sn and obtained corresponding semi-direct product decompositions. New representations of loop braid group from higher gauge theory effects.
problem Understanding representations of loop braid group in higher gauge theory.
method Introducing W-bikoids and constructing representations from groupoid algebras.
result Candidate for higher quantum group and flux metamorphosis.
The Tong-Yang-Ma representations are extended to string links and welded string links.
problem Extending Tong-Yang-Ma representations to string links and welded string links.
method Using the method of Silver and Williams, the authors extend the Tong-Yang-Ma representations.
result The kernels of the extended representations can be described using linking numbers.
In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. Thes…
Study of unrestricted virtual braid groups and their properties.
problem Characterize and describe homomorphisms of unrestricted virtual braid groups.
method Analyzing homomorphisms to symmetric groups and finite groups, characterizing images, proving characteristic subgroups, determining automorphism groups, and studying residual properties.
result Complete description of homomorphisms from UVBn to Sn for n≥5. In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. Thes…
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.
We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. Th…
New theory classifies knotted spheres in 4D space.
problem Classifying knotted punctured spheres in 4D space.
method Diagrammatic theory of welded graphs, Tube map extension, Milnor invariants.
result Complete link-homotopy classification of knotted punctured spheres.
In this survey paper we present the L--moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3--manifolds and in handlebodies, as well as for virtual knots, …
The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.
problem Proving the non-triviality of welded knots and ribbon torus-knots.
method By generating examples and determining the fundamental group of the concerned welded knot.
result Non-triviality of welded knots and ribbon torus-knots is demonstrated.
We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.
The paper studies generalized virtualization moves on welded links and classifies their equivalence.
problem Classifying equivalence of welded links under specific moves.
method Introduced and analyzed two new moves, V(n) and Vn, on welded links. result The V(n)-move is an unknotting operation for any n, while Vn is not for neq1. This paper characterizes Kashiwara-Vergne groups using algebraic structures of knotted tubes.
problem Characterizing Kashiwara-Vergne groups using algebraic structures.
method Using algebraic structures of welded foams and arrow diagrams, the paper describes the Kashiwara-Vergne groups and their associated graded circuit algebras.
result The paper provides a description of the graded Grothendieck-Teichmüller group as automorphisms of arrow diagrams.
Using Gauss diagrams, one can define the virtual bridge number vb(K) and the welded bridge number wb(K), invariants of virtual and welded knots with wb(K)≤vb(K). If K is a classical knot, Chernov and Manturov showed that vb(K)=br(K), the bridge number as a classical …
Paper discusses gliding algorithm to transform tangle diagrams into a specific form.
problem Transforming tangle diagrams into a specific form.
method Gliding algorithm to bring tangle diagrams to Over-then-Under (OU) form.
result Obtained a braid classification result and extended it to virtual braids.
Quantum entanglement is linked to topological braiding through Yang-Baxter equations.
problem Understanding the relationship between quantum entanglement and topological braiding.
method Viewing unitary entangling operators as braiding operators and using Yang-Baxter equations.
result Quantum entanglement is necessary for forming invariants of knots, as shown by solutions to the Yang-Baxter Equation.
New invariants for welded and extended welded knots found using multi-skein relations.
problem Classifying and distinguishing welded and extended welded knots and links.
method Developed a multi-skein relation approach to find invariants for these knots and links.
result Found sufficient conditions on coefficients for invariance under extended Reidemeister moves.
Characterizes Milnor invariants with limited repetitions.
problem Understanding Milnor invariants with constraints on repetition.
method Algebraic and diagrammatic approaches using k-reduced free groups and welded knot theory. result Algebraic and diagrammatic characterizations of Milnor invariants.
Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball~B4, "bounding" 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant A of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group G. Th…
This paper classifies virtual string links up to cobordism using group theory.
problem Classifying virtual string links up to cobordism.
method Using group theory, specifically the group Zn(n−1). result Virtual string links up to cobordism are classified by elements of Zn(n−1). New invariant measures how many twists are needed to unknot welded knots.
problem Measuring complexity of welded knots.
method Local twist move, Alexander quandle coloring, Gordian distance.
result Established lower bound on twist number and related it to other knot invariants.
Virtual and welded periods of knots are equivalent to classical periods.
problem Understanding the equivalence of virtual and welded periods to classical periods.
method Direct demonstration of equivalence.
result Classical knots have only finitely many virtual or welded periods.
Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-link…
The paper studies unknotting operations and numbers for plus-welded knotoids.
problem Understanding unknotting operations and numbers for plus-welded knotoids.
method The paper proves transformations and introduces new operations to calculate unknotting numbers.
result Upper bounds for unknotting numbers of plus-welded knotoids are found.
In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. …
New knot concept extends welded knots, simplifying classification.
problem Classifying welded knots and their complements.
method Introducing 'wen knots', proving subset relationships, characterizing complements.
result Extended welded knots can be fully characterized by the parity of wens.
Combines combinatorial method to extend Milnor invariants to welded links.
problem Extending Milnor invariants to welded links.
method Combinatorial approach.
result Invariance of extended Milnor invariants for welded links.
Classifies welded string links up to specific moves.
problem Classifying welded string links up to moves.
method Using Milnor invariants, classify welded string links up to 2n-move and Vn-move. result Classifies welded string links up to 2n-move and Vn-move. 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 wk-equivalence, studied algebraic structures, and showed characterizations. result Characterizes the information contained by finite type invariants in low degrees for welded string links.
The paper explores meridional ranks of knotted surfaces and welded knots, proving equalities and relationships.
problem Investigating the Meridional Rank Conjecture for knotted surfaces and welded knots.
method Constructing knots with specific properties, establishing equalities, and using Tube map.
result Established the equality of bridge number and meridional rank for certain knots and knotted spheres.
The paper explores extensions of local moves on string links and their relation to ribbon surfaces.
problem Finding a unique extension of classical local moves on string links.
method Relating local moves to surgeries on ribbon surfaces in R^4.
result There can be at most one unique welded extension of a classical move that is a ribbon residue.
Develops a calculus for knotted objects, including classical links.
problem Characterizing and classifying knotted objects, including classical links.
method Introduces Arrow presentations and w-tree presentations to encode knot diagrams, uses clasper theory for welded knots.
result Characterizes finite type invariants of welded knots and long knots.
We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.
We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and con…
The generalized Alexander polynomial vanishes for virtually slice knots.
problem Identifying knots that are concordant to homologically trivial knots.
method Using Bar-Natan Zh-correspondence and concordance properties of maps between links and tori.
result The generalized Alexander polynomial vanishes on virtually slice knots.
Classic braids embed in virtual braids.
problem Embedding classic braids in virtual braids.
method Elementary proof using generalizations of braid groups.
result Classical braid group injects into virtual braid group.