Virtual singular braids and links
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.
Key words and phrases:braids, knots, -moves, Markov-type moves, singular knots, virtual knots
2010 Mathematics Subject Classification:57M25, 57M27; 20F36
J.W. Alexander  showed that any oriented classical link can be represented as the closure of a braid. Moreover, it is well-known that two braids have isotopic closures if and only if they are related by braid isotopy and a finite sequence of the so-called Markov’s moves (see [13, 16]). The first complete proof of this result was given by J. Birman . Other proofs have been provided by D. Bennequin , H. Morton , P. Traczyk , and S. Lambropoulou .
Analogous theorems for the virtual braid group have been proven by L.H. Kauffman and S. Lambropoulou  using the, so-called, -equivalence and by S. Kamada  using Gauss data. Moreover, J. Birman  proved an Alexander-type theorem for the singular braid monoid and singular links and B. Gemein  provided a Markov-type theorem for singular braids. Further, S. Lambropoulou  derived the -move analogue for singular braids via -move methods, recovering the result of Gemein.
In this paper we consider oriented virtual singular links and prove Alexander-and Markov-type theorems for this class of links. These theorems are crucial in understanding the structure of virtual singular knots and links. We first define the virtual singular braid monoid using generators and relations. This definition reveals that the virtual singular braid monoid on strands is an extension of the singular braid monoid on strands by the symmetric group on letters. Various braiding algorithms can be used to prove that the Alexander theorem extends to the class of virtual singular braids. For our purpose, we borrow the braiding algorithm described in  and extend it to include singular crossings. We then show that the -moves used in  for the class of virtual braids and links can be extended to the class of virtual singular braids and links. In the presence of singular crossings and additional relations describing the virtual singular braid monoid, we need to introduce a new type of -moves, namely a new type of ‘threaded -moves’ involving classical, singular, and virtual crossings. We state and prove first an -move Markov-type theorem for virtual singular braids and then use it to provide an algebraic Markov-type theorem for virtual singular braids.
During our study of this problem, we found that we were able to modify the arguments of  and take the same diagrammatic geometry so as to prove our main results. Consequently, several figures in this paper are similar or exactly the same as certain figures in . For example, if the reader would examine in this paper Figures 6 through 11 and compare with Figures 7, 9, 11, 12, and 13 in , they would see the precise analogy of our arguments and the arguments of .
Motivated by L.H. Kauffman and S. Lambropoulou’s work in [8, Section 3], we also prove that the virtual singular braid monoid on strands admits a reduced presentation using fewer generators, namely three braiding elements together with the generators of the symmetric group on letters.
2. Virtual singular links
A virtual singular link diagram is a decorated immersion of (finitely many) disjoint copies of into , with finitely many transverse double points each of which has information of over/under, singular, and virtual crossings as in Figure 1. The over/under markings are the classical crossings, which we will refer to as real crossings. Virtual crossings are represented by placing a small circle around the point where the two arcs meet transversely. A filled in circle is used to represent a singular crossing. We assume that virtual singular link diagrams are the same if they are isotopic in .
Note that the set of classical link diagrams, or singular link diagrams, or virtual link diagrams comprise subsets of the set of virtual singular link diagrams.
Two virtual singular link diagrams are said to be equivalent if they are related by a finite sequence of the extended virtual Reidemeister moves depicted in Figure 2 (where only one possible choice of crossings is indicated in the diagrams). A virtual singular link (or a virtual singular link type) is the equivalence class of a virtual singular link diagram.
Note that the moves involving virtual crossings can be considered as special cases of the detour move depicted in Figure 3 ([7, 8, 9]). This move is the representation of the principle that the virtual crossings are not really there but that are rather byproducts of the projection. To understand the detour move, suppose an arc is free of real (classical) and singular crossings, and which may contain a consecutive sequence of virtual crossings. Then that arc can be arbitrarily moved, keeping its endpoints fixed, to any new location and placed transversally to the rest of the diagram, adding virtual crossings whenever these intersections occur. (In Figure 3, the grey box represents an arbitrary virtual singular tangle diagram; a braid representation of the detour move is given in Figure 28.)
Conversely, the detour move can be obtained by a finite sequence of the moves shown in Figure 2 that involve virtual crossings. Consequently, the virtual singular equivalence is generated by the Reidemeister-type moves for singular link diagrams (that is, the classical Reidemeister moves together with the moves and ) and the detour move.
When working with equivalent virtual singular link diagrams, it is important to avoid the moves depicted in Figure 4. Although these moves are similar to some of the extended virtual Reidemeister moves, the diagrams of the two sides of a forbidden move do not represent equivalent virtual singular links. For this reason, we refer to these as the forbidden moves for virtual singular link diagrams.
Recall that a singular link is an immersion of a disjoint union of circles in three-dimensional space, which has finitely many singularities (namely singular crossings) that are all transverse double points. Equivalently, a singular link is an embedding in three-dimensional space of a 4-valent graph with rigid vertices (where these vertices are the singular crossings). These type of embedding are also called rigid vertex knotted graphs.
Similar to the case of virtual knot theory, there is a useful topological interpretation for virtual singular knot theory in terms of embeddings of singular links (or equivalently, of rigid vertex knotted graphs) in thickened surfaces. For this, interpret each virtual crossing as a detour of one of the arcs in the crossings through a 1-handle that has been attached to the 2-sphere of the original diagram. We obtain an embedding of a collection of immersed circles into a thickened surface , where is the unit interval, is a compact oriented surface of genus , and is the number of virtual crossings in the original diagram. Then singular knot theory in is represented by diagrams drawn on taken up to the Reidemeister-type moves for singular link diagrams transferred to diagrams on . Recall that the Reidemeister-type moves for singular link diagrams contain the classical Reidemeister moves and together with the moves and shown in Figure 2.
3. Alexander- and Markov-type theorems
A virtual singular braid on strands is a braid in the classical sense, which may contain real, singular, and virtual crossings as ‘interactions’ among the strands of the braid. By connecting the top endpoints with the corresponding bottom endpoints of a virtual singular braid using parallel arcs without introducing new crossings we obtain a virtual singular link diagram, called the closure of the braid.
Similar to the case of classical braids, virtual singular braids are composed using vertical concatenation. For two -stranded virtual singular braids and , the braid is obtained by placing on top of and connecting their endpoints. The set of isotopy classes of virtual singular braids on strands forms a monoid, which we denote by . The monoid operation is the composition of braids, and the identity element, denoted by , is the braid with vertical strands.
3.1. The virtual singular braid monoid
The virtual singular braid monoid on strands, , is the monoid generated by the virtual singular braids and , for , depicted below:
and subject to the following relations:
These relations taken together define the isotopies for virtual singular braids. Each relation in is a braided version of a virtual singular link isotopy. That is, two equivalent virtual singular braids have isotopic closures. Note that the type 1 moves and are not reflected in the defining relations for , because these moves cannot be represented using braids. Note also that only the generators are not invertible in .
3.2. A braiding algorithm
In this section we present a method for transforming any virtual singular link diagram into the closure of a virtual singular braid. For that, we borrow the braiding algorithm introduced in  and extend it to our set-up where we add singular crossings, to prove a theorem for virtual singular links analogous to the Alexander theorem for classical braids and links.
We will work in the piecewise linear category, which gives rise to the operation of the subdivision of an arc (in a virtual singular link diagram) into smaller arcs, by marking it with a point. Note that local minima and maxima are subdivision points of a diagram.
We fix a height function in the plane of the diagram, and use the following conventions necessary for our braiding algorithm: First it is understood that only one crossing (real, singular or virtual) can occur at each level (with respect to the height function) in a virtual singular link diagram. Likewise, we arrange our diagram so that no crossings or subdivision points are vertically aligned, so as to avoid triple points when new pairs of braid strands are created with the same endpoints (this will be made more clear later as we explain our braiding algorithm). In addition, a crossing must not coincide with a local maximum or minimum. Lastly, a diagram should not have any horizontal arcs (it will only have up-arcs and down-arcs). If a virtual singular link is arranged so that it satisfies each of these conventions, we say that the diagram is in general position.
It is easy to see that by applying small planar shifts, if necessary, any virtual singular link diagram can be transformed into a diagram in general position.
When converting a virtual singular link diagram to a diagram in general position, we make certain choices which result in local shifts (which are called direction sensitive moves in ) of crossings and subdivision points with respect to the horizontal or vertical direction. The swing moves given in Figure 5 are the most interesting direction sensitive moves; these moves are necessary so that we avoid the coincidence between a crossing (real, singular, or virtual) and a maximum or minimum in a diagram.
Two isotopic virtual singular link diagrams in general position differ by the extended virtual Reidemeister moves (provided in Figure 2) and the direction sensitive moves. For the remainder of this section, we will work with virtual singular link diagrams in general position.
We describe now the braiding algorithm for transforming an oriented virtual singular link diagram (assumed in general position) into the closure of a virtual singular braid.
After placing the subdivision points using the conventions explained above, we apply the braiding algorithm locally, by eliminating each up-arc in the diagram (which can be either an up-arc in a crossing or a free up-arc), one at a time.
We first braid the crossings containing one or two up-arcs. If a crossing has no up-arcs we leave it as it is. We place each crossing that needs to be braided in a narrow rectangular box, called the braiding box, with the arcs of the crossing serving as diagonals of the box. A braiding box would have to be sufficiently narrow, so that the region it defines does not intersect the braiding box of another crossing. We braid each crossing, one at a time, according to the braiding chart given in Figure 6 (see also [9, Figure 7]). Any new crossing created between the new braid strands and the rest of the diagram outside the braiding box will be assumed to be virtual; this is indicated abstractly by putting virtual crossings at the ends of the new pair of braid strands.
Note that, locally speaking, for each crossing that was braided, connecting the corresponding pair of braid strands (outside of the resulting diagram) yields a virtual singular tangle diagram (with four endpoints) which is isotopic to the starting one (the tangle represented by the crossing in the braiding box).
The free up-arcs are arcs joining braiding boxes. Once all crossings have been braided, we braid each of the free up-arcs using the basic braiding move depicted in Figure 7 (see [9, Figure 9]). During this move, we first cut a free up-arc and then extend the upper end upward and the lower end downward, such that the new pair of strands are vertically aligned and such that they cross only virtually any other arcs in the original diagram (which is represented by an abstract virtual crossing on the ends of the new braid strands), as shown in Figure 7. As in the case of braiding a crossing, by connecting the pair of the new braid strands outside of the original diagram results in a local virtual singular tangle diagram (with two endpoints) which is isotopic to the local tangle before the braiding.
The braiding algorithm given above will braid any virtual singular link diagram, creating a virtual singular braid whose closure is isotopic to the original diagram. Indeed, for all braiding moves, even for those that do not contain singular crossings, it is important to observe that there may be singular crossings in the rest of the braid and that upon closure these are detoured freely by the virtual crossings of the new braid strands. Therefore, we have proved the following statement.
Theorem 1 (Alexander-type theorem for virtual singular links).
Every oriented virtual singular link can be represented as the closure of a virtual singular braid.
3.3. L-moves and Markov-type theorems for virtual singular braids
Two virtual singular braids may have isotopic closures, and thus we would like to describe virtual singular braids that result in isotopic virtual singular link diagrams via the closure operation. Therefore, we are interested in Markov-type theorems for virtual singular braids and links. For this purpose, we need to introduce the singular -moves for virtual singular braids. These moves enlarge the set of the -moves for virtual braids, described in . Here, the subscript stands for ‘virtual’.
We remind the reader that the classical -moves were introduced by S. Lambropoulou in  to provide a one-move Markov-type theorem for classical braids and links. We also refer the reader to , where the -move equivalence for classical braids is established.
We recall from  that a basic -move involves cutting a braid strand and pulling the upper endpoint of the cut downward and the lower endpoint upward, and in doing so, creating a pair of new braid strands which cross virtually all of the other strands in the diagram; this is abstractly denoted by a pair of virtual crossings at the points where the two new braid strands cross the box in which the -move is applied (see Figure 8).
Note that an -move may introduce a crossings, which may be real or virtual, as shown in Figure 9 (see [9, Figure 11]). To stress the existence of the real or virtual crossing, these moves are called the real -move or virtual -move, respectively (abbreviated to - or -move, respectively), and there are two versions of them, namely left or right (depending whether the new crossing is on the left or on the right of the arc that was cut during the move). Figure 9 displays right virtual and left real -moves.
Note that by connecting the pair of the newly created braid strands (outside of the diagram) we obtain a tangle diagram which is isotopic to the tangle diagram we started with (the detoured loop contracts to a kink which involves either a virtual crossing or a real crossing). This is explained in Figure 10.
A threaded -move is an -move with a virtual crossing in which, before stretching the arc of the kink, we perform a classical type 2 Reidemeister move using another strand of the braid, called the thread. Depending whether we pull the kink over or under the thread, we have an over-threaded -move or an under-threaded -move; both of these moves come with the left and right versions. (We refer the reader to the analogous definition in [9, Definition 4].)
Figure 11 shows under-threaded -moves, both left and right versions. Due to the forbidden moves, a threaded -move cannot be simplified on the braid level; that is, the move does not involve isotopic braids but isotopic closures of braids.
In addition, we can create a multi-threaded -move by performing two or more classical type 2 Reidemeister moves before pulling open the arc of the kink. (See [9, Figure 14].)
When singular crossings are present, there is another type of threaded move in which the thread ‘crosses’ the detoured loop in a pair of a singular crossing and a real crossing. We call such a move an -threaded -move; this move also comes in two variants, namely left and right. Figure 12 exemplifies such a move, with only one of the two versions for the real crossing involved in the move. An -threaded -move cannot be applied (simplified) in the braid. However, it is not hard to see that the closures of the two sides of an -threaded -move are isotopic diagrams (via an move), as explained in Figure 13.
Finally, we define the notion of conjugation and commuting in the virtual singular braid monoid, . Given a virtual singular braid , we say that the braids , for , are related by real conjugation. Similarly, we say that the braids are related by virtual conjugation, where . Note that since is its own inverse in , virtual conjugation is equivalent to . Similarly, real conjugation can be rewritten in the form or . Finally, we say that are related by singular commuting (note that is not invertible in ).
We say that two virtual singular braids are singular -equivalent if they differ by virtual singular braid isotopy and a finite sequence of the following moves or their inverses:
Real conjugation and singular commuting
Right virtual and right real -moves
Left and right under-threaded -moves
Left and right -threaded -moves.
We remark that the singular -equivalence on virtual singular braids contains as a subset the -equivalence on virtual braids defined in [9, Definition 6]. We remind the reader that the -equivalence for virtual braids comprises the real conjugation, the right real and right virtual -moves, the left and right under-threaded -moves, and the virtual braid isotopy.
It was proved in  that the virtual conjugation, basic -moves, left real and left virtual -moves, over-threaded -moves, and multi-threaded -moves follow from the -equivalence. Therefore, these moves also follow from the singular -equivalence, and thus we do not need to include them in our -move Markov-type theorem for virtual singular braids, which we are now ready to state and prove.
Theorem 2 (-move Markov-type theorem for virtual singular braids).
Two virtual singular braids have isotopic closures if and only if they are singular -equivalent.
It is easy to see that singular -equivalent virtual singular braids have isotopic closures.
We will now work on the converse. First, we need to show that different choices made in the braiding process result in braids that are singular -equivalent. The choices made during the braiding process are the subdivision points and the order of the braiding moves. The subdivision points are needed for marking the braiding boxes and the up-arcs. Using a similar argument as in [9, Corollary 2], it is not hard to see that given two subdivisions of a virtual singular diagram, the resulting virtual singular braids obtained by our braiding algorithm are singular -equivalent. Due to the narrow condition for the braiding boxes, the braidings of the crossings are local and independent, so the order in which we braid the crossings has no effect on the final output. Moreover, the order in which we braid the free up-arcs is also irrelevant. Due to the braid detour moves, we can in fact braid first the free up-arcs (or just some of them) and then braid the crossings (and any remaining free up-arcs).
Second, we need to show that, different choices in bringing a virtual singular diagram to general position result in braids (obtained by our braiding algorithm) that are singular -equivalent. Using a similar argument as in [9, Lemma 7], it is easily seen that planar isotopy moves applied away from any of the crossings in a virtual singular link diagram result in braids that are related by braid isotopy and the basic -move. Indeed, the addition of singular crossings in the setting does not change the situation. It was also shown in [9, Lemma 7] that, by applying the braiding algorithm to diagrams that differ by a swing move involving a virtual crossing or a real crossing results in braids that are -equivalent. Therefore, for our case of virtual singular link diagrams, it remains to verify the swing moves containing a singular crossing. These swing moves can be verified in the same manner as the swing moves involving a real crossing, by merely replacing the real crossing in [9, Figures 26, 27]) with a singular crossing.
Finally, we need to show that two virtual singular braids with isotopic closures are related by singular -equivalence. For that, we need to prove that virtual singular link diagrams (in general position) that differ by the extended virtual Reidemeister moves (recall Figure 2) correspond to closures of virtual singular braids that are singular -equivalent. By [9, Theorem 2], we know that the isotopy moves involving only real and virtual crossings follow from the -equivalence for virtual braids (and thus from singular -equivalence). Therefore, we only need to consider the extended virtual Reidemeister moves involving singular crossings, and these moves need to be considered with any given orientation of the strands.
Note that if all strands involved are oriented downward, the statement follows directly from the relations defined on . We consider all cases of each isotopy move involving singular crossings. We consider diagrams that are identical, except in a small region where they differ as shown in the figures; that is, the isotopy move is applied in that small region.
We start with the move and allow one strand to be oriented upward. If we apply the braiding algorithm to the diagrams on both sides of the move (followed by braiding isotopy), the resulting diagrams differ by a left -threaded -move, as explained in Figure 14. Note that if we reverse the orientations of the two strands in the move, the corresponding braids differ by a right -threaded -move, as explained in Figure 15 (besides reversing the orientations of the two strands, we also changed the sign of the classical crossings from positive to negative, for more variety).
Figure 16 shows that if we take the isotopy move with both strands oriented upward and braid the diagrams on each side of the move, we find that the two corresponding braids are related by a series of conjugations and the braid-type move.
Observe that in Figure 15 we used virtual conjugation, which is not a move of the singular -equivalence. However, recall that the virtual conjugation follows from the -equivalence and hence from the singular -equivalence (see [9, Figures 17, 18]).
We will now take a slightly different approach to prove that the moves and hold with any possible orientations on the strands. Again, the case where all strands are oriented downward follow from braid equivalence. We start by considering the move with one strand oriented upward and apply an move to create locally three downward oriented strands. After a couple of moves, we apply an move in braid form (see Figure 17).
Similarly, if we start with two strands oriented upward, we apply an move to reduce to the case with one strand oriented upward, as exemplified in Figure 18.
In Figure 19 we consider an move with all three strands oriented upward and show that it can be reduced to the previous case with two strands oriented upward.
The proof for the moves with various orientations on the strands is done similarly as for the moves, and therefore they are omitted to avoid repetition. This completes the proof. ∎
In the following theorem we will use to represent an arbitrary virtual singular braid in . We also regard as an element of by adding a strand on the right of . (We will not use an extra notation when we regard as an element in .) Using this operation (of adding a single identity strand on the right of a braid) the monoid embeds in , and we define . In what follows, we also allow adding an identity strand at the left of and we denote by the braid in obtained in this way.
Theorem 3 (Algebraic Markov-type theorem for virtual singular braids).
Two virtual singular braids have isotopic closures if and only if they differ by a finite sequence of braid relations in together with the following moves or their inverses:
Real and virtual conjugation, and singular commuting (see Figure 20):
Right real and right virtual stabilization (see Figure 21) :
Right and left algebraic under-threading (see Figure 22):
Right and left algebraic -threading (see Figure 23):
where and .