arXiv research

Extended welded links are a generalization of Fenn, Rimányi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kam…

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.

Let $WB_n$ be the welded (or loop) braid group on n strands, $n \geq 3$. We investigate commutator subgroup of $WB_n$. We prove that the commutator subgroup $WB_n'$ is finitely generated and Hopfian. We show that $WB_n'$ is perfect if and only if $n \geq 5$. We also compute finite presentation for $FWB_n'$, the commuta…

Homotopy braid groups are shown to be 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…

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…

The paper characterizes crystallographic groups derived from virtual braid and twin groups.

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.

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…

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 $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid groups $vP_n$, as w…

Study on when the lower central series stops for various groups, including braid groups.

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 Tong-Yang-Ma representations are extended to string links and welded string links.

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…

Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitel…

Study on virtual singular braid groups with algebraic properties and homomorphisms.

Paper discusses gliding algorithm to transform tangle diagrams into a specific form.

We show that representations of the loop braid group arise from Aharonov-Bohm like effects in finite 2-group (3+1)-dimensional topological higher gauge theory. For this we introduce a minimal categorification of biracks, which we call W-bikoids (welded bikoids). Our main example of W-bikoids arises from finite 2-groups…

The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeister-type moves. We present a new class of invariants for welded…

Study of unrestricted virtual braid groups and their properties.

New invariant measures how many twists are needed to unknot welded knots.

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

The paper studies unknotting operations and numbers for plus-welded knotoids.

New knot concept extends welded knots, simplifying classification.

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. …

Combines combinatorial method to extend Milnor invariants to welded links.

Let $n$ be a positive integer. The aim of this paper is to study two local moves $V(n)$ and $V^{n}$ on welded links, which are generalizations of the crossing virtualization. We show that the $V(n)$-move is an unknotting operation on welded knots for any $n$, and give a classification of welded links up to $V(n)$-moves…

This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.

We show that any virtual or welded period of a classical knot $K$ can be realized as a classical period. A direct consequence is that a classical knot admits only finitely many virtual or welded periods.

New theory classifies knotted spheres in 4D space.

Kauffman and Lomonaco explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In the work of G. Alagic, M. Jarret, and S. Jordan it is shown that entanglement is a necessary condit…

In this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being non-unique in general, the idea is to find a topological criterion which could isolate one extension from the others. To that end, we turn to the relatio…

Using Gauss diagrams, one can define the virtual bridge number ${\rm vb}(K)$ and the welded bridge number ${\rm wb}(K),$ invariants of virtual and welded knots with ${\rm wb}(K) \leq {\rm vb}(K).$ If $K$ is a classical knot, Chernov and Manturov showed that ${\rm vb}(K) = {\rm br}(K),$ the bridge number as a classical …

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…

We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations, which can be seen as `higher order Gauss diagrams'. This Arrow calculus is used to d…

In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualizat…

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.

This note investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with…

This paper characterizes Kashiwara-Vergne groups using algebraic structures of knotted tubes.

We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers $m_{i}$ $(i=1,\ldots,n)$ and an ordered $n$-component virtual link diagram $D$, a new virtual link diagram $D(m_{1},\ldots,m_{n})$ is ob…

New diagonal move simplifies knots and links efficiently.

Paper proves link diagrams can be realized for some but not all types of links.

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. Th…

Survey of various non-classical knot theories from geometric and algebraic perspectives.

New theorem for 4D links simplifies characterisation problem.