Virtual, Welded, and Ribbon Links in Arbitrary Dimensions

Virtual, Welded, and Ribbon Links in Arbitrary Dimensions

Blake K. Winter
Abstract.

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtual links, and use such diagrams to construct a combinatorial biquandle invariant for virtual -links. In the case of -links, we also explore generalizations of Fox-Milnor movies to the virtual case. In addition, we discuss definitions extending the notion of welded links to higher dimensions. For ribbon knots in dimension or greater, we show that the knot quandle is a complete classifying invariant up to taking connected sums with a trivially knotted , and that all the isotopies involved may be taken to be generated by stable equivalences of ribbon knots.

1. Introduction

Let us begin by defining the following terms, which will be used throughout. All manifolds are taken to be smooth, unless otherwise specified. An immersed embedding, , of an -manifold into an -manifold will be called a -link in M. We will refer to this as simply a link when the dimension and ambient space are implied by context. When , it may also be termed a classical link. A link for which has only one connected component will be called a -knot (in ). Note that in some sources, the term -knot is applied only to embeddings of the form ; we use the term in the more general sense. Two links are considered equivalent if they are related by a smooth isotopy of their images. This is equivalent to requiring that their images be related by an ambient isotopy of , [35]. Note that this definition of links is also called the smooth category of links. If the requirement of immersion is replaced by requiring the embedding to be piecewise-linear and locally flat, the resulting theory is called the piecewise-linear or PL category; if we instead require the embeddings to be merely continuous and locally flat, we obtain the topological category or TOP. We will always work with the smooth category unless otherwise specified.

The concept of virtual links was introduced by Kauffman, [20], as a generalization of classical links. These virtual links were given geometric interpretations by Kamada, Kamada, and Carter et al., [18, 6] and a somewhat different geometric interpretation by Kuperberg, [23]. By using the geometric interpretation given by Kamada et al. as our starting point, we will give a generalization of the definition which applies to codimension- embeddings in any dimension. Using Roseman’s work on projections for links, [35], we define higher-dimensional generalizations of virtual link diagrams and Gauss codes. We will also discuss some methods for generalizing welded links to higher dimensions, using the work of Satoh, [37], as our starting point.

2. Link Diagrams

Every classical -link can be expressed by a link diagram, which is a combinatorial object consisting of a planar graph on a surface whose vertices are -valent. At each vertex, two of the edges of the graph are marked as the overcrossing arc and the other two are marked as the undercrossing. These markings give information on how to resolve them in three dimensions.

Let be a -link, where is a surface (possibly with boundary). Then there are canonical projections and . It is possible to isotope such that is an immersion and an embedding except for a finite number of double points called crossings. In order to recover , at each double point, we mark which strand projects under to the smaller coordinate in at that double point; this is the undercrossing. These marks allow us to recover up to ambient isotopy. Observe that is a -valent graph. The interiors of the edges of this graph will be called semi-arcs of the link diagram. Let be the set . For any connected component of , is called an arc of the link diagram.

For diagrams of -links, there exists a set of three moves, the Reidemeister moves, which satisfy the following condition: if and are ambient isotopic links in , then the diagrams of and on differ by a finite sequence of ambient isotopies of and Reidemeister moves. Thus, questions about -links may be translated into questions about equivalence classes of link diagrams, with the equivalence relation being generated by the Reidemeister moves.

Figure 1. The three Reidemeister moves for diagrams of -links.

Roseman, [33, 34, 35], has generalized the diagram presentation of -links to obtain an analogous presentation of -links. In brief, it is possible to represent an arbitrary -link in the manifold by looking at the canonical projection of to . Furthermore, it is possible to modify by an arbitrarily small isotopy such that the projection will be an immersion on an open dense subspace of . By marking the double points of the projection with over and under information, Roseman obtains a generalization of link diagrams which is applicable to any dimension. Roseman also showed that any two link diagrams of ambient isotopic links will be related by a finite sequence of ambient isotopies of and certain local changes which leave the diagram unchanged outside a disk . We will first give the full technical definitions necessary. Then we will examine some examples for -links.

Let be an -link in the manifold . If we wish to deal with links in , we may form a diagram by isotoping the link to lie in some ; two links are isotopic in this subspace iff they are isotopic in .

The crossing set of a link , denoted by , is the closure of the set . The double point set is defined as . The branch set is the subset of on which fails to be an immersion. The pure double point set is defined to be the subset of consisting of those points such that consists of exactly two points. We also define the (pure) overcrossing set to be the set containing all such that of the two , , and the (pure) undercrossing set .

Definition 1.

([34]) An immersion is said to have normal crossings if at any point of self-intersection, the pushforwards of the tangent spaces, , meet one another in general position.

Definition 2.

is in general position (with respect to ) iff the following conditions hold.

  1. is a closed dimensional submanifold of .

  2. is a union of immersed and closed -dimensional submanifolds of with normal crossings. The set of points where normal crossings occur will be labeled and called the self-crossing set of .

  3. is a submanifold of , and every has an open disk neighborhood , such that has two components , , which embeds into such that .

  4. meets transversely.

  5. The restriction of to is an immersion of with normal crossings.

  6. The crossing set of is transverse to the crossing set of .

Note that this definition is taken from [32], rather than Roseman’s original version; the two definitions are equivalent.

This somewhat technical definition is important because of Theorem 3, which says that we can always put an -link into general position. Furthermore, Roseman has given a finite collection of moves, analogous to the Reidemeister moves, which relate any two diagrams of isotopic -links, [35].

Theorem 3.

([35], [32, Theorem 6.2, Lemma 7.1])
Every -link in can be isotoped into general position with respect to the projection .

Note that in [35], Theorem 3 is proved in the case of links in only. However, by Lemma 7.1 in [32], it is true for links in as well.

Definition 4.

A link diagram for an -link , in general position with respect to , is the projection , such that at each point of crossing set whose preimage consists of exactly two points, there are markings to determine which point in the preimage has the larger coordinate in .

Note that by the above theorem every -link in can be represented by a link diagram on . It is easy to see that a link represented by is unique.

Note that in the case , this definition reduces to the usual definition of a link diagram. A -link diagram will be a surface in a -manifold with double point curves, which can meet in isolated triple points and terminate in isolated branch points. Over and under information can be given, as in the case, by breaking one of the surfaces in the neighborhood of a double point curve. A -link diagram is for this reason also called a broken surface diagram.

Roseman also shows, in [35, Prop. 2.9], that, for any dimension , two link diagrams for links which are ambient isotopic in differ by a finite sequence of ambient isotopies of and certain Roseman moves. Each Roseman move changes the link diagram only within some open disk in The set of Roseman moves for fixed is finite, although the number of moves increases with . In the case , the Roseman moves reduce to the standard Reidemeister moves, shown in Fig. 1. In the case , the Roseman moves can be given simple graphical representations in terms of broken surface diagrams. For higher , Roseman gives a local model for each Roseman move, but does not give an explicit enumeration of moves. We will return to these representations in our discussion of diagrams for virtual links.

Definition 5.

In a link diagram , given by , for , the connected components of will be termed the sheets of the diagram.

For -links, a sheet is the same as a semi-arc. In addition, sometimes the term sheet may be used to refer to the preimages in of the sheets in the diagram. When there is ambiguity, we will specify whether we are referring to a sheet in the diagram on or a sheet in .

Definition 6.

Let be in general position with respect to . The connected components of will be termed the faces of the diagram. The images of these components under may also be called faces depending upon context.

The faces of a -link diagram are its arcs.

3. Review of Virtual and Welded Links for

Virtual links were introduced by Kauffman, [20], as a generalization of classical -links. A link diagram gives rise to a combinatorial Gauss code.

Since a link diagram is inherently a graph on a surface , each link diagram gives rise to a Gauss code. There are various ways of defining Gauss codes for links. All of them specify the set of arcs in the diagram, and, for each crossing, the two arcs which terminate at that crossing are specified, together with the order in which these crossings occur on their overcrossing arcs. Concretely, a Gauss diagram for is together with a collection of triples , one for each vertex of the diagram where are preimages of under and , depending on whether is over (i.e. ) or not.

However, not every Gauss code corresponds to a link diagram on . A Gauss code which corresponds to a link diagram on is termed realizable. One method for defining virtual links is to consider arbitrary Gauss codes, modulo local changes corresponding to Reidemeister moves. These moves for Gauss codes are illustrated in Fig. 2.

Figure 2. The effects of the three Reidemeister moves on a Gauss code. Here denotes the sign of the crossing and denotes a crossing of the opposite sign.

Such Gauss codes may also be represented using virtual link diagrams. A virtual link diagram is a link diagram where we allow arcs to cross one another in virtual crossings, indicated in the diagram by a crossing marked with a little circle. All the Reidemeister moves are permitted on such diagrams, and any arc segment with only virtual crossings may be replaced by a different arc segment with only virtual crossings, provided the endpoints remain the same. This is equivalent to allowing the additional moves shown in Fig. 3.

Figure 3. The additional virtual Reidemeister moves. Virtual crossings are indicated by crossings with small circles placed on them. The unmarked crossing in the third move can be either a virtual crossing or an arbitrary classical crossing.

Virtual links share many similarities with classical ones. For example, one can define virtual link groups and peripheral subgroups of virtual knot groups, cf. Sec. 7. Furthermore, polynomial invariants of links, like the Alexander, Jones, and Homfly-pt polynomials, extend to virtual links.

Since the group and peripheral structure are invariants of virtual knots, by a result of Gorden and Luecke, [16] along with a theorem of Waldhausen, [39], two classical knots which are virtually equivalent must be classically equivalent as well. It can be shown, however, that not every virtual link is virtually equivalent to a classical link. We will discuss such an example in Section 5.

Welded links were first studied in the context of braid groups and automorphisms of quandles by Fenn, Rimanyi, and Rourke, [12]. For the case, welded links are virtual links considered modulo the so-called “forbidden move” (as it is a forbidden move for virtual links), shown in Fig. 4. The link group, and the peripheral structure, can be extended to invariants of welded links. In particular the link group is computed from a welded link diagram using the same algorithm as for virtual or classical link diagrams. Consequently, just as in the virtual case, any two classical knots which are welded equivalent must be classically equivalent as well. There are examples of welded knots which are not welded equivalent to classical links, however. For example, there are welded knots whose fundamental groups are not infinite cyclic, but whose longitudes are trivial. Examples of such welded knots are easily constructed using a consequence of the work of Satoh, [37]. Satoh has shown that every ribbon embedding of a torus in can be represented by a welded knot, whose fundamental group and peripheral subgroup are isomorphic to the fundamental group and peripheral subgroup of the torus. Given any nontrivial ribbon embedding of into , we may take the connected sum of this surface knot with a torus forming the boundary of a genus- handlebody in . The resulting knotted torus will be ribbon and have a trivial longitude. It follows from Satoh’s correspondence between welded knots and ribbon embeddings of tori in that a welded knot which represents this knotted torus will be nontrivial, but will have a trivial longitude. Diagrams of such welded knots will also provide examples of virtual knots which are not virtually equivalent to classical knots.

Figure 4. The “forbidden move,” which is not allowed for virtual -links but is permitted for welded -links.

As we will see, the fact that every welded -link diagram is also a virtual -link diagram is unique to the case ; for larger , the geometrically-motivated definition of a welded link, which we will give, does not reduce to the notion of a virtual link with an additional move permitted.

4. Ribbon Links

Because we will be making reference to ribbon links and their ribbon presentations in further sections, we give here a brief review of these concepts. See [7] for further discussion of ribbon links in arbitrary dimensions.

Remark 7.

Throughout, we will use the notation to represent a closed -ball.

Let be an -manifold. Consider a pair , consisting of a set of disjoint oriented -disks and a collection of oriented -handles , with embedded in , such that the orientations agree, and such that for each , the components of are a finite collection of -disks embedded in , such that each component separates from . This implies that is contained in the interior of the image of ; such intersections between a handle and a base are called ribbon intersections or ribbon singularities. A handle can intersect any number of bases in ribbon intersections, and it can intersect the same base multiple times (or zero). We will in general identify the maps and with their images. The disks in are called bases, while the elements of are called (fusion) handles or bands. Let be the link defined as . Then is called a ribbon link, and the pair are a ribbon presentation for . We identify two ribbon presentations, and , of if there is a path between them in the space of ribbon presentations of This is equivalent to requiring that , , and that there exists an ambient isotopy of , , such that is the identity, and for all , , and .

The subspace is called a ribbon solid for the ribbon presentation or for the link. Note that a given may be a ribbon solid for multiple ribbon presentations. If an -link has a ribbon presentation then it is called a ribbon link.

Observe that since we require that the handles be glued to the bases in an orientation-preserving manner, will be orientable. Note also that for ribbon -links, the components of a ribbon link may not be ribbon knots. However, for -links with , each component of a ribbon link will automatically be a ribbon knot.

Given a ribbon link , there are in general multiple ribbon presentations for , making it convenient to consider notions of equivalence for ribbon presentations. We will, for the moment, restrict our attention to knots; the generalization to links is straightforward. Let be a ribbon knot with ribbon presentations and . These presentations are called simply equivalent if , , and there exists an isotopy , , of , such that is the identity, and for all , , and .

Note that simple equivalence may create new intersections of handles with the interiors of bases, as illustrated in Fig. 5.

Figure 5. Two simply equivalent ribbon presentations with distinct ribbon intersections along a single handle. Here, the bases are indicated with shading; ribbon intersections are shown by thickened lines.
Lemma 8.

For a ribbon presentation in a simply-connected manifold is determined uniquely up to simple equivalence by specifying the following ribbon data:
(a) the cardinality of and,
(b) for each , a sequence of bases, through which the handle passes through, up to reversal of that sequence.

Proof.

Clearly there is a ribbon presentation fulfilling every possible ribbon data as above. We need to show that such ribbon presentation is unique, up to simple equivalence. Note that for for each , the sequence determines the homotopy class of the handle in uniquely. Since and the core of the handle is -dimensional, the homotopy class of the handle determines its isotopy class as well. ∎

In this is of course not the case, since the homotopy class of the handle does not determine its isotopy class in . In addition, one must specify a cyclic ordering of handles glued to a base for .

There is a broader notion of equivalence between ribbon presentations called stable equivalence, defined by Nakanishi, [29]. This is the equivalence relation generated by simple equivalence together with the following three moves and their inverses, which are illustrated in Fig. 6:

  1. Add a new base with a handle starting on and terminating on any other base, without passing through any bases.

  2. Handle pass: Isotope through , leaving fixed.

  3. Handle slide: If connects with , and has an end on , we may slide that end along over to .

Figure 6. The three moves permitted for ribbon presentations under stable equivalence.

Observe that for a knotted -sphere, by Euler characteristic arguments. It is easy to check that for , a ribbon -knot will have the homeomorphism type of an -sphere with -handles attached.

Given a ribbon -knot , it is an open question whether any two ribbon presentations for are related by stable equivalence for . For the case , Nakanishi and Nakagawa provide a family of knots, each of which admits multiple ribbon presentations which are not stably equivalent, [30, 29].

5. Quandles and Biquandles

Quandles and biquandles are nonassociative algebraic structures which provide invariants for classical and virtual links. As we will show, they also provide invariants for our generalization of virtual links to higher dimensions. Since we will be referring to these structures repeatedly, we list their definitions here for reference.

A quandle, [17], (also called a distributive groupoid in [28]) consists of a set and a binary operation on such that for all , the following equalities hold.

  1. .

  2. There is a unique such that .

  3. .

When the operation is implied, we may denote the quandle by , suppressing the operation.

Example 9.

For any group , is a quandle, for (i.e. ). We call the group quandle of the group

It is easily checked that this defines a forgetful functor from the category of quandles to the category of groups.

In general, it is common to denote the operation in any quandle by using the conjugation notation, that is, for any quandle we may define . Analogously, we will denote the element stipulated by property (2) by , i.e. . By convention, we interpret . In Section 6 we review Joyce’s geometric definition of the fundamental quandle of the complement of any -link.

While a quandle can be thought as a set with two binary operations, and a (strong) biquandle, [22, 11, 5], is a set together with four binary operations, often denoted using exponent notation as , , , . As for quandles, these operations are assumed to be associated from left to right unless otherwise specified with parentheses, for example, . These operations are required to satisfy the following properties:

  1. For any fixed , the functions sending to , , , are bijective functions on with an argument (this implies all four operations have right inverses).

  2. For any , iff , and iff .

  3. The following equalities hold for all : , , , , , , , , , .

Every quandle can be made into a biquandle by defining the biquandle operations to be , , . Another family of examples may be constructed by taking to be a module over . Then we define biquandle operations on to be , , , and . Then is a biquandle called an Alexander biquandle, [24].

Although the axioms for a biquandle appear somewhat abstract, they are all motivated by considering labelings of semi-arcs in diagrams for virtual links or labelings of sheets of broken surface diagrams for surfaces embedded in , and then requiring that the resulting algebraic structure must be invariant under the Reidemeister or -dimensional Roseman moves.

We may also define weak biquandles by dropping the first axiom (i.e. the axiom that each biquandle operation has a right inverse). However, for our purposes, we will always use the term “biquandle” to indicate a strong biquandle, unless otherwise noted.

There are also other conventions for denoting the biquandle operations; we have followed Carrell’s notation, [5], since that source discusses applications of biquandles to knotted surfaces. Biquandles can also be defined as sets with two right-invertible operations which satisfy the following identities (see [38] for a discussion of these): , , , .

Biquandles have been found to be useful for distinguishing virtual -knots that are indistinguishable by quandles. For example, the Kishino knot, Fig. 7, is a virtual knot whose quandle is isomorphic to the quandle of the unknot. However, it can be distinguished from the unknot using the biquandle, [21, 3].

Figure 7. A virtual knot diagram of the Kishino knot.

6. The Link Quandle and Group in Arbitrary Dimensions

Quandles as knot invariants were introduced in [17, 28]. In this section we discuss the geometric definition of the fundamental quandle of an -knot, originally given by Joyce in [17]. In that paper, it was shown that the quandle of a classical knot is determined by its knot group together with the specification of a meridian and the peripheral subgroup. We will show that this is true for arbitrary (orientable) knots in spheres, as well as for virtual -knots.

Let be a link in , with and assumed to be orientable, and let be an open tubular neighborhood of . Then is a manifold with boundary containing . The fundamental group of this space is called the link group. We will follow the convention that the product of two homotopy classes of paths goes from right to left, i.e. is the homotopy class of a path which follows and then . We call the image of the homomorphism induced by the injection the peripheral subgroup of Note that is defined only up to conjugation; any of its conjugates are also peripheral subgroups.

A meridian of is the boundary of the disk fibre of the tubular neighborhood of considered as a disk bundle over . Note that the orientations of and of determine an orientation of . A meridian is defined uniquely up to conjugation.

Figure 8. The quandle operation for the fundamental quandle of a link in any space . The open circle indicates the basepoint of , while the black circles represent points on the boundary of a tubular neighborhood of .

The definition of the quandle, and of the fundamental group of a link complement, requires concatenation of paths. We will use the following convention for path concatenation: given two paths , we will write to indicate the path that follows first, and then follows . This convention makes it easier to discuss the action of the link group on the quandle.

We define the quandle associated to the pair and a chosen base point as follows. The elements of are homotopy classes of paths in that start at and end in , where homotopies are required to preserve these two conditions. The quandle operation, , between two elements is defined as the homotopy class of , where the overbar indicates that the path is followed in reverse and is the meridian of passing through the endpoint of , cf. 8. It is a straightforward exercise to show that this binary operation is well-defined and it satisfies the definition of a quandle operation. We call the link quandle of (or the knot quandle if is a knot).

Note that there is a right action of on : is the equivalence class including the homotopy class of the path .

6.1. Quandles From Groups

Given a group , a subgroup , and an element in the center, of , Joyce defines a quandle operation on the set of right cosets of , , as follows: . This operation is well-defined, because if , then for some and

(1)

since . We will denote a quandle constructed in this way by , or simply when there is a canonical choice for , for example, when is a meridian and a peripheral subgroup of a knot.

6.2. Group Actions on Quandles

The results in this section are due to Joyce, [17].

For any link in , acts on by setting to be the homotopy class of the path .

Consider now a knot and a path connecting a fixed base point with (Clearly, ) Note that defines a peripheral subgroup of Its elements are of the form where is any loop in based at the endpoint of In particular, determines a meridian

Lemma 10.

For any knot ,
(1) the action on is transitive, i.e. every element of is equal to for some .
(2) is the stabilizer of the action on i.e. iff

Proof.

(1) Choose representatives of with an endpoint coinciding endpoints on . Then for given by the path .
(2) Since every is of the form , which can be homotoped along to . Conversely, suppose that The homotopy transforming into moves the endpoint of along a closed loop in That means that and are homotopic in with their endpoints fixed. Consequently, i.e.

Note that Lemma 10(2) implies that the stabilizer of is .

Consider the peripheral subgroup and the meridian determined by a path from the base point in to , as above. Then the map sending to is well defined by Lemma 10(2).

Theorem 11.

is an isomorphism of quandles. Consequently, is determined by with its peripheral structure.

Proof.

is - by Lemma 10(2), as well as onto, by Lemma 10(1). To show that is a quandle homomorphism, observe that

(2)

It follows that

(3)

Therefore, is a bijective quandle morphism. ∎

6.3. Remarks

Joyce has also proved a theorem for classical knots stating that the triple can be reconstructed from . We generalize his result here.

Let be an -knot in in general position with respect to Denote its diagram by Let be a group with the following presentation: its generators are the faces of . Let be the pure double point set. Then for each connected component of consider the relation where are the three faces meeting at with being the overcrossing face, and with the normal vector to pointing toward , as illustrated in Fig. 9.

Figure 9. A double point curve as shown yields relations on the group or quandle.

Similarly we associate with a quandle . Its generators are faces of The relations between faces are where are as above. Note that, in general, this quandle is not isomorphic to the group quandle of the knot group of . For example, the knot group and its group quandle do not distinguish the square knot from the granny knot, but the knot quandles for these two knots are distinct.

Theorem 12.

([19, Theorem 2]) For any knot simply connected, and in general position with respect to , with the knot diagram , and are isomorphic to the knot group and knot quandle of .

This theorem is proved by an iterative application of the Van Kampen theorem. Note that in [19] only groups are considered, but the proof for quandles is analogous. The Van Kampen theorem holds for quandles, by [17, Theorem 13.1].

Theorem 13.

For any -knot in , the triple is uniquely determined (up to an isomorphism) by the knot quandle .

Proof.

For any quandle , we may define the group to be the group generated by the elements of , whose relations are exactly the quandle relations, with the quandle operation interpreted as conjugation in the group. By Theorem 12, has a presentation whose generators correspond to the faces of a knot diagram of , and whose relations are the Wirtinger relations corresponding to each meeting of faces in a double point set. It is then easy to check that has a presentation as a group whose generators correspond to the faces of a knot diagram of , with relations being exactly the Wirtinger relations from each double point component. It follows that is canonically isomorphic to (up to a choice of a basepoint). There is also a canonical map from the elements of to the elements of , sending a word in the generators of to the corresponding word in the generators of , with the quandle operation again interpreted as group conjugation. For any denote the corresponding element in by . Then can be assumed to be any of the generators corresponding to a face in the knot diagram. In general, acts on by . To see that this action is well-defined, note that if an element of has two presentations, and , this implies that those products of Wirtinger generators are homotopic rel basepoints, and this homotopy passes to a homotopy of to . It is then straightforward to check that this action of is just the geometrically defined action of on . Then will be the stabilizer of under this action, and so we have constructed . Note that is unique (up to conjugation) and is determined by . Hence, this triple is unique up to an isomorphism. ∎

Quandle invariants are particularly simple for knotted -spheres, , since their peripheral groups are cyclic and, hence , is unique. (The choice between and is determined by the orientations of and of ). It follows that for classical knots, the quandle does not contain much more information than the fundamental group. On the other hand, when is a more complicated subgroup of , the knot quandle may be able to capture more information than .

Eisermann, [10], has in fact shown that the quandle cocycle invariants of classical knots are a specialization of certain colorings of their fundamental groups. His construction makes use of the full peripheral structure of the classical knot (that is, the longitude and the meridian), and thus does not generalize immediately to higher dimensions. However, in light of our result here, we pose the question of whether a similar construction might not be possible in higher dimensions.

7. Geometric Definition of Virtual Links in Any Dimension

Kauffman defined virtual links as a combinatorial generalizations of classical link diagrams or, alternatively, as the Gauss codes of classical links diagrams. However, there is a geometric definition of virtual links as well. We will follow essentially the treatment given in [18] and [6] for this approach. Note that while Carter et al. work with link diagrams on surfaces, we work with links in thickened surfaces; By Theorem 3 and the remarks below it, these approaches are equivalent. As we will see, the definition of [18] and [6] can be extended to higher dimensions in a purely geometric manner, without reference to any combinatorial constructions.

Let be the set of pairs where is a compact -manifold (with possibly non-empty boundary), is given the structure of a trivial -bundle over , and where is an -manifold embedded in the interior of such that meets every component of . The -bundle structure on will always be implied to be the canonical bundle coming from projection when we refer to elements of in this way. For convenience we will either simply write , with the bundle structure implied, or, if using the notation , we will assume is identified with some trivial bundle , with the bundle structure coming from projection onto the first factor of the Cartesian product. (Our theory will not be affected by the fact that may have corners.) Define a relation on by the condition iff there exists an embedding such that . We define an equivalence relation on , , to be the equivalence relation generated by together with smooth isotopy of the link. When , we say they are virtually equivalent.

Consider first the case . Note that any link diagram in a surface defines a Gauss code and, consequently, a virtual link. Since this construction is invariant under Reidemeister moves, it descends to a map {Kauffman’s virtual links}. Furthermore, since for it factors to

Definition 14.

An element of , with in general position with respect to , is an abstract -link iff is a deformation retract of .

We now wish to define an inverse map to . Given a virtual link diagram , we may construct an abstract link using the following construction. Let be a neighborhood of the graph in . Replace the virtual crossings by changing the thickened “+” shape in a neighborhood of each virtual crossing into two strips with an arc along each of them, as shown in Fig. 10. The result is a -manifold with a link diagram on it, which therefore defines a link in . Let us denote its equivalence class in by If two virtual link diagrams and are equivalent under the virtual Reidemeister moves, then it is easily checked that . We may therefore regard the map as a map from equivalence classes of virtual link diagrams to equivalence classes of virtual -links.

Figure 10. At each virtual crossing in , we modify the disk into two disks with an arc crossing each of them.

The relation between and Kauffman’s virtual link is shown by the following theorem, which is proved by a fairly easy verification.

Theorem 15.

([18, 6]) and are bijective inverses between the set of virtual link diagrams modulo the virtual Reidemeister moves and .

Therefore, gives a completely geometric definition for Kauffman’s virtual links. It has the particular advantage that it does not explicitly rely upon diagrams or Gauss codes for its definition, which makes it possible to generalize to higher dimensions without first needing to establish a higher-dimensional analog for diagrams or Gauss codes.

This equivalence of to Kauffman’s virtual links makes it geometrically natural to simply define virtual -links as elements of .

Definition 16.

A virtual -link is an element of . We denote the equivalence class of by

Elements such that has exactly one component will be termed virtual -knots. The set of virtual -knots will be denoted by . Furthermore, it may easily be seen that for , only if and are diffeomorphic.

Definition 17.

We will denote the set of virtual -links with a fixed by .

Our definition of virtual links may easily be generalized to virtual tangles. Let consist of pairs , for some -manifold , and is an -manifold properly embedded in . Note that this implies that is embedded in . Let iff there exists an embedding such that . We define to be the equivalence relation generated by and by smooth isotopy of , keeping the boundary of embedded in .

Definition 18.

A virtual -tangle is an element of .

Definition 19.

A realizable -link is a virtual -link whose equivalence class in includes a pair .

Equivalently, a realizable -link is a virtual -link which is virtually equivalent to some pair . We will use either characterization interchangeably.

It is known that not every element of is realizable. For higher , the in may have the diffeomorphism type of a manifold that does not embed into , in which case this pair cannot be realizable. Some simple examples of this may be found in odd-dimensional real projective spaces; for example, cannot be embedded in , [9].

Question 20.

For a fixed number , are there non-realizable elements of where is a manifold that embeds in ?

8. Homotopy Invariants for Virtual -Links

In this section we define some invariants of virtual -links. If these invariants are extensions of classical invariants, i.e. they agree with some classical invariant of -links whenever is realizable, then they also can be used to study the relationship between classical -links and virtual -links. We will begin with the following

Remark 21.

Any element is virtually equivalent to an abstract -link where is a tubular neighborhood of in , and is the canonical projection .

For a pair or , let denote the space , where is the relation for all Note that this is equivalent to gluing a cone over to . We can then use the pair of spaces to study virtual links, although this pair is not an element of .

Theorem 22.

The homotopy type of is invariant under virtual equivalence.

Proof.

It is immediate that isotopy of in will not change the homotopy type of . Let . We wish to show that has the same homotopy type as . We will define a deformation retraction defined as follows: is the identity on and it is given by a projection of onto as depicted in Fig. 11. Hence, maps to It is easy to see that this map is homotopic to the identity and, hence, as a deformation retraction, it is a homotopy equivalence between and

Figure 11. The map projects points that lie outside a neighborhood of straight up, while points in this neighborhood are pushed up and toward the .
Remark 23.

For , is homotopy equivalent to .

It follows that any homotopy invariants of links in extend to invariants of virtual links.

Remark 24.

We may also consider , where is the relation for all This space is homotopy equivalent to , for links in , but in general it is not clear whether this will be the case.

Definition 25.

(1) We define the link group of a virtual link to be .
(2) The image of in is called the peripheral subgroup of the virtual link group.
(3) For a knot , consider a base point which lies in the boundary of That base point defines the meridian given by the boundary of a disk in normal to , passing through . Note that a meridian is defined up to conjugation in if a base point is not specified).
(4) An oriented closed curve on passing though such that the algebraic intersection number between and vanishes in is called a longitude.

Note that for virtual -knots a longitude as above is unique and that commute. Furthermore, are uniquely defined up to mutual conjugation by an element of , if the base point is not specified. In other words, define a peripheral structure on .

Observe, however, that higher-dimensional knots may have many longitudes (even if is specified), which are not conjugate, depending on the topology of the knotted space.

Theorem 26.

Suppose as elements of . Then and are ambient isotopic in .

Proof.

The knot group, meridian, and longitude of are preserved under virtual equivalences. Since by a theorem of Waldhausen, [39], and a result of Gordon and Luecke, [16], these form a complete invariant for knots in , and since knots in are equivalent to knots in , the theorem follows. ∎

Joyce’s geometric definition of the fundamental quandle for a codimension- link, [17], recalled in Section 6, applies to links in spaces despite the fact that is not a manifold. In Joyce’s construction, we consider paths from a basepoint to , up to homotopy through such paths. Note that , a circle bundle over . Given two such paths, and , we define to be the path , where is the path circling the fiber of at the terminal point of , as illustrated in Fig. 8. The proof that this defines a quandle, and the theorems relating the quandle of to the fundamental group of and its peripheral structure when is an -knot, follow without change from those given in Section 6.

Definition 27.

We define the link quandle of a virtual link