Multi-switches and virtual knot invariants
MMULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS
VALERIY BARDAKOV, TIMUR NASYBULLOV
Abstract.
In the paper we introduce a general approach how for a given virtualbiquandle multi-switch (
S, V ) on an algebraic system X (from some category) and agiven virtual link L construct an algebraic system X S,V ( L ) (from the same category)which is an invariant of L . As a corollary we introduce a new quandle invariantfor virtual links which generalizes previously known quandle invariants for virtuallinks. Keywords: multi-switch, virtual knot, knot invariant, quandle, Yang-Baxter equa-tion.Mathematics Subject Classification: 57M27, 57M25, 20F36, 20N02, 16T25. Introduction
A set-theoretical solution of the Yang-Baxter equation is a pair (
X, S ), where X isa set and S : X × X → X × X is a bijective map such that( S × id )( id × S )( S × id ) = ( id × S )( S × id )( id × S ) . The problem of studying set-theoretical solutions of the Yang-Baxter equation wasformulated by Drinfel’d in [21]. If (
X, S ) is a set-theoretical solution of the Yang-Baxter equation, then the map S is called a switch on X (see [23]). A pair of switches( S, V ) on X is called a virtual switch on X if V = id and the equality( V × id )( id × S )( V × id ) = ( id × V )( S × id )( id × V )holds.Switches and virtual switches are connected with virtual braid groups and virtuallinks. Using a (virtual) switch one can construct an integer-valued invariant for(virtual) links, so called coloring-invariant [23, Section 6]. Also using a (virtual)switch on a set X it is possible to construct a representation of the (virtual) braidgroup on n strands by permutations of X n [23, Section 2]. Moreover, if X is analgebraic system, then under additional conditions using a (virtual) switch on X it ispossible to construct a representation of the (virtual) braid group by automorphismsof X . The Artin representation ϕ A : B n → Aut( F n ) (see [13, Section 1.4]), the Buraurepresentation ϕ B : B n → GL n ( Z [ t, t − ]) (see [13, Section 3]) and their extensionsto the virtual braid groups (cid:101) ϕ A : V B n → Aut( F n ), (cid:101) ϕ B : V B n → GL n ( Z [ t, t − ])(see [30, 42]) can be obtained on this way. a r X i v : . [ m a t h . A T ] J a n VALERIY BARDAKOV, TIMUR NASYBULLOV
Despite the fact that virtual switches can be used for constructing representationsof virtual braid groups, there are representations
V B n → Aut( G ), where G is somegroup, which cannot be obtained using any virtual switch on G . For example, theSilver-Williams representation ϕ SW : V B n → Aut( F n ∗ Z n +1 ) (see [39]), the Boden-Dies-Gaudreau-Gerlings-Harper-Nicas representation ϕ BD : V B n → Aut( F n ∗ Z )(see [14]), the Kamada representation ϕ K : V B n → Aut( F n ∗ Z n ) (see [8]) andthe representations ϕ M : V B n → Aut( F n ∗ Z n +1 ), ˜ ϕ M : V B n → Aut( F n ∗ Z n )of Bardakov-Mikhalchishina-Neshchadim (see [2, 3]) cannot be obtained using anyvirtual switch. In order to overcome this obstacle, in the paper [5] we introduce thenotion of a (virtual) multi-switch which generalizes the notion of a (virtual) switch,and introduce a general approach how a virtual multi-switch ( S, V ) on an algebraicsystem X can be used for constructing a representation ϕ S,V : V B n → Aut( X ). Allthe representations above can be obtained using this approach for appropriate virtualmulti-switches.In the present paper we continue studying applications of virtual multi-switches.Namely, we introduce a general approach how virtual multi-switches can be used forconstructing invariants of virtual links. For a given virtual multi-switch ( S, V ) onan algebraic system X (from some category) and a given virtual link L we introducethe algebraic system X S,V ( L ) (from the same category as X ) which is an invariant of L . As a corollary, we construct a new quandle invariant (cid:101) Q ( L ) for virtual links whichgeneralizes the quandle of Manturov [32] and the quandle of Kauffman [30].So, every multi-switch on an algebraic system leads to a virtual link invariant whichis an algebraic system. The invariants which are algebraic systems are of specialimportance in the theory since usually they are very strong (for example, the knotquandle is a complete invariant for classical links under weak equivalence [28, 33]),and they lead to a lot of different invariants (see, for example, [19, 20, 23]).The paper is organized as follows. In Section 2, we give necessary preliminariesabout virtual links. In Section 3, we give preliminaries about multi-switches. InSection 4, for a given multi-switch ( S, V ) on an algebraic system X and a givenvirtual link L we introduce the algebraic system X S,V ( L ) and prove that X S,V ( L ) is avirtual link invariant (Theorem 1). In Section 5, we find a way how to construct thealgebraic system X S,V ( L ) using the representation ϕ S,V : V B n → Aut( X ) introducedin [5] (Theorem 3). Finally, in Section 6, we introduce a new quandle invariant forvirtual links (Theorem 4 and Theorem 5). Acknowledgement.
The results are supported by the grant of the Russian ScienceFoundation (project 19-41-02005).2.
Virtual knots and links
In this section we recall definitions of virtual knots and links. Virtual links wereintroduced by Kauffman [30] as a generalization of classical links. Topologically,virtual links can be interpreted as isotopy classes of embeddings of classical linksin thickened orientable surfaces.
A virtual n -component link diagram is a regular ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 3 immersion of n oriented circles (i. e. an immersion, such that there are no tangentcomponents, and all intersection points are double points) such that every crossing iseither classical (see Figure 1 ( a , b )) or virtual (see Figure 1 ( c )). The crossing depicted ( a ) ( b ) ( c ) Figure 1.
Crossings in the virtual link diagram.on Figure 1 ( a ) is called a positive crossing , the crossing depicted on Figure 1 ( b ) iscalled a negative crossing , and the crossing depicted on Figure 1 ( c ) is called a virtualcrossing . A virtual 1-component link diagram is called a virtual knot diagram .Two virtual link diagrams D , D are said to be equivalent if the diagram D can be transformed to the diagram D by planar isotopies and three types of localmoves (generalized Reidemeister moves): classical Reidemeister moves R , R , R (seeFigure 2), virtual Reidemeister moves V R , V R , V R (see Figure 3), and mixedReidemeister moves V R (see Figure 4). The equivalence class of a given virtual n -component link diagram is called a virtual n -component link . A virtual 1-componentlink is called a virtual knot . R R R Figure 2.
Classical Reidemeister moves.
V R V R V R Figure 3.
Virtual Reidemeister moves.Let A be an arbitrary non-empty set. The map f from the set of all virtual linkdiagrams to A is called an ( A -valued) invariant for virtual links if it maps equivalentlink diagrams to the same element of A . Due to the definition of equivalent virtuallink diagrams, the map f is an invariant for virtual links if and only if f ( D ) = f ( D )whenever D is obtained from D using only one generalized Reidemeister move. If f is an invariant for virtual links, and L is a virtual link (i. e. an equivalence class ofsome virtual link diagram D ), then we write f ( L ) = f ( D ). VALERIY BARDAKOV, TIMUR NASYBULLOV
V R Figure 4.
Mixed Reidemeister moves.3.
Switches and multi-switches
Switches and the Yang-Baxter equation.
A set-theoretical solution of theYang-Baxter equation is a pair (
X, S ), where X is a set, and S : X → X is a mapsuch that(1) ( S × id )( id × S )( S × id ) = ( id × S )( S × id )( id × S ) . If (
X, S ) is a set theoretical solution of the Yang-Baxter equation such that themap S : X → X is bijective, then the map S is called a switch on X (see [23,Section 2]). Let S be a switch on X such that S ( x, y ) = ( S l ( x, y ) , S r ( x, y )) for somemaps S l , S r : X → X and x, y ∈ X . A switch S is said to be non-degenerate if forall elements a, b ∈ X the maps S la , S rb : X → X given by the formulas S la ( x ) = S l ( a, x ) , S rb ( x ) = S r ( x, b )for x ∈ X are invertible. A switch S is called involutive if S = id .If X is not just a set but an algebraic system: a group, a module etc, then everyswitch S on X is called, respectively, a group switch, a module switch etc. Example 1.
Let X be an arbitrary set, and T ( a, b ) = ( b, a ) for all a, b ∈ X . Themap T is a switch which is called the twist . It is clear that T is involutive andnon-degenerate. Example 2.
Let X be a group. Then the map S A defined by S A ( a, b ) = ( aba − , a )for a, b ∈ X is a switch which is called the Artin switch on a group X . This switchis non-degenerate.In order to introduce the next example of a switch, let us recall that a quandle Q isan algebraic system with one binary algebraic operation ( a, b ) (cid:55)→ a ∗ b which satisfiesthe following three axioms.(1) For all a ∈ Q the equality a ∗ a = a holds.(2) The map I a : b (cid:55)→ b ∗ a is a bijection of Q for all a ∈ Q . For a, b ∈ Q wedenote by a ∗ − b = I − b ( a ).(3) For all a, b, c ∈ Q the equality ( a ∗ b ) ∗ c = ( a ∗ c ) ∗ ( b ∗ c ) holds.A quandle Q is called trivial if a ∗ b = a for all a, b ∈ Q , the trivial quandle with n elements is denoted by T n .Quandles were introduced in [28, 33] as an invariant for classical links. More pre-cisely, to each oriented diagram D of an oriented knot K in R one can associate the ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 5 quandle Q ( K ) which does not change if we apply the Reidemeister moves to the dia-gram D . Kauffman [30] extended the idea of the quandle invariant Q ( K ) from classi-cal links to virtual links by ignoring virtual crossings (ignoring virtual crossings makesthe quandle invariant for virtual links weaker than for classical links). Manturov [32]constructed a quandle invariant for virtual links which generalizes the quandle ofKauffman [30]. Over the years, quandles have been investigated by various authors forconstructing new invariants for knots and links (see, for example, [15,17,25,29,34,36]).Algebraic properties of quandles including their automorphisms and residual proper-ties have been investigated, for example, in [1, 6, 9, 12, 18, 37]. For more details aboutquandles see [15, 22, 38].The following example gives a quandle switch. Example 3.
Let ( X, ∗ ) be a quandle. Then the map S Q ( a, b ) = ( b, a ∗ b ) for a, b ∈ X is a quandle switch. This switch is non-degenerate.A lot of examples of switches on different algebraic systems can be found, forexample, in [5, Section 2], [10, Section 3] and [23, Section 2].3.2. Switches and invariants of virtual links.
Switches can be used for con-structing invariants for virtual links. Let S be a switch on X such that S ( a, b ) = ( S l ( a, b ) , S r ( a, b ))for maps S l , S r : X → X and a, b ∈ X . For a, b ∈ X denote by S l ( a, b ) = b a , S r ( a, b ) = a b , so, on X we have two binary operations ( a, b ) (cid:55)→ a b , ( a, b ) (cid:55)→ a b , whichare called the up operation and the down operation defined by S , respectively. TheYang-Baxter equation for S implies the following equalities a bc = a c b b c , a bc = a c b b c , ( a b ) c ba = ( a c ) b ca (2)for all a, b, c ∈ X . A switch S is called a biquandle switch on X if the following twoconditions hold.(1) The switch S is non-degenerate, i. e. the maps S la , S ra : X → X given by S la ( x ) = x a , S ra ( x ) = x a are bijective. We denote by b a − = ( S ra ) − ( b ) , b a − = ( S la ) − ( b ) . (2) a a − = a a a − and a a − = a a a − for all a ∈ X .The switches from Examples 1, 2, 3 are biquandle switches. If S is a biquandle switchon X , then the set X with the up and the down operations defined by S is called a biquandle and is denoted by ( X, S ). If condition (2) of the biquandle switch doesnot hold, then the algebraic system (
X, S ) is called a birack .Biquandles were introduced in [23] as a tool for constructing invariants for virtualknots and links. Let S be a biquandle switch on X with S ( a, b ) = ( S l ( a, b ) , S r ( a, b )) VALERIY BARDAKOV, TIMUR NASYBULLOV for maps S l , S r : X → X , and a, b ∈ X . Since S is a biquandle switch on X , thealgebraic system ( X, S ) is a biquandle. Let the inverse to S map S − : X → X hasthe form S − ( a, b ) = ( Q l ( a, b ) , Q r ( a, b ))for maps Q l , Q r : X → X , and a, b ∈ X .Let D be a virtual link diagram. A strand of D going from one crossing (classicalor virtual) to another crossing (classical or virtual) is called an arc of D . A labelling of the diagram D by elements of the biquandle ( X, S ) is a marking of arcs of D by elements of X such that near each crossing of D the marks of arcs are as onFigure 5. A diagram D can have zero, one or several (even infinitely many) labellings a b a b a bS l ( a, b ) S r ( a, b ) Q r ( a, b ) Q l ( a, b ) b a Figure 5.
Marks of arcs in D .by elements of the biquandle ( X, S ). The set of all labellings of the diagram D byelements of the biquandle ( X, S ) is denoted by lab ( X,S ) ( D ). If X is a finite set, thenthe set lab ( X,S ) ( D ) is finite.Fenn, Jordan-Santana and Kauffman in [23, Theorem 6.12] proved that if D , D are equivalent virtual link diagrams, then each labelling of the diagram D pro-vides a unique labelling of the diagram D . In particular, if X is a finite set, then (cid:12)(cid:12) lab ( X,S ) ( D ) (cid:12)(cid:12) = (cid:12)(cid:12) lab ( X,S ) ( D ) (cid:12)(cid:12) , i. e. the integer (cid:12)(cid:12) lab ( X,S ) ( D ) (cid:12)(cid:12) is an invariant for vir-tual links. This invariant is called the biquandle-coloring invariant defined by ( X, S )and is denoted by C ( X,S ) ( D ). So, every biquandle switch provides an invariant forvirtual links. Papers [7,11,16,24,27,31] give several application of biquandles in knottheory.3.3. Multi-switches and virtual multi-switches.
Let X be a non empty set, and X , X , . . . , X m be non-empty subsets of X . One can identify the sets( X × X × X × · · · × X m ) and X × X × X × · · · × X m and for elements A = ( a , a , . . . , a m ) , B = ( b , b , . . . , b m ) from X × X × X ×· · ·× X m denote the ordered pair ( A, B ) by(
A, B ) = ( a , b ; a , b ; a , b ; . . . ; a m , b m ) . We say that a map S : X × X × · · · × X m → X × X × · · · × X m is an ( m + 1) -switch , or a multi-switch on X (if m is not specified) if S is a switch on X × X × X × · · · × X m , such that S ( c , c , . . . , c m ) = ( S ( c , c , . . . , c m ) , S ( c ) , . . . , S m ( c m )) ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 7 for c ∈ X , c i ∈ X i for i = 1 , , . . . , m , and S , S , . . . , S m are the maps S : X × X × · · · × X m → X ,S i : X i → X i , for i = 1 , , . . . , m. If S is an ( m + 1)-switch on X defined by the maps S , S , . . . , S m , then we write S = ( S , S , . . . , S m ). Note that for i = 1 , , . . . , m the map S i is a switch on X i . If X is not just a set but an algebraic system: group, module etc, then every multi-switchon X is called, respectively, a group multi-switch, a module multi-switch etc. We donot require here that X , X , . . . , X m are subsystems of X . Example 4.
Every switch on X is a 1-switch on X . Example 5. If S is a switch on X , and S i is a switch on X i ⊂ X for i = 1 , , . . . , m ,then the map S × S × · · · × S m is an ( m + 1)-switch on X .The multi-switches from Example 4 and Example 5 are in some sense trivial. Thefollowing example introduced in [5, Proposition 1] gives a non-trivial example of amodule 2-switch. Example 6.
Let R be an integral domain, X be a free module over R , and X bea subset of the multiplicative group of R (one can think about X as about a subsetof X identifying X ⊂ R with X x ⊂ X for a fixed non-zero element x from X ).Then the map S : X × X → X × X given by S ( a, b ; x, y ) = ((1 − y ) a + xb, a ; y, x ) a, b ∈ X, x, y ∈ X is a 2-switch on X .Since an ( m + 1)-switch on X is a switch on X × X × X × · · · × X m , the notionof the involutive ( m + 1)-switch follows from the same notion for switches. Let S, V : X × X × X × · · · × X m → X × X × X × · · · × X m be an ( m + 1)-switchand an involutive ( m + 1)-switch on X , respectively. We say that the pair ( S, V ) is a virtual ( m + 1) -switch on X (or a virtual multi-switch on X , if m is not specified)if the following equality holds(3) ( id × V )( S × id )( id × V ) = ( V × id )( id × S )( V × id ) , where id denotes the identity on X × X × X × · · · × X m . Some examples of virtualmulti-switches can be found in [5].4. Multi-switches and knot invariants
In Section 3.2 we noticed that biquandle switches can be used for constructinginvariants for virtual links. Namely, if S is a biquandle switch on a finite set X ,then the number C ( X,S ) ( D ) of labellings of a virtual link diagram D by elements ofthe biquandle X is an integer-valued invariant for virtual links. In this section weintroduce a general construction how a given multi-switch on an algebraic system X can be used for constructing an algebraic system which is an invariant for virtuallinks. Using this approach it is possible to construct a group of a link, a quandle of a VALERIY BARDAKOV, TIMUR NASYBULLOV link, a biquandle of a link, a module of a link, a skew brace of a link (see [26, 35, 40]for details about skew braces) etc.Let X be an algebraic system, and X , X , . . . , X m be subsystems of X such that(1) for i = 0 , , . . . , m the subsystem X i is generated by elements x i , x i , . . . (whichare not necessarily all different),(2) { x i , x i , . . . } ∩ { x j , x j , . . . } = ∅ for i (cid:54) = j ,(3) the set of elements { x ij | i = 0 , , . . . , m, j = 0 , , . . . } generates X ,(4) for every permutation α of N with a finite support the map x ij (cid:55)→ x iα ( j ) for i = 0 , , . . . , m , j = 1 , , . . . induces an automorphism of X . Remark . In this section we require that X , X , . . . , X m are subsystems of X , whilein the definition of the virtual multi-switch we do not require this condition. Remark . Condition (4) implies that for a fixed i the elements x i , x i , . . . are eitherall different, or all coincide.For n ≥ i = 0 , , . . . , m denote by X ( n ) i the subsystem of X i generated by x i , x i , . . . , x in , and by X ( n ) the subsystem of X generated by X ( n )0 , X ( n )1 , . . . , X ( n ) m . Remark . From condition (4) follows that for n > X ( n ) isisomorphic to the quotient of X ( n +1) by ( m + 1) relations x n +1 = x n , x n +1 = x n , . . . x mn +1 = x mn (4)(we add to X ( n ) generators x n +1 , x n +1 , . . . , x mn +1 and say that they are equal to x n , x n , . . . , x mn ). We will write ( m + 1) equalities from (4) as one equality of ( m + 1)-tuples ( x n +1 , x n +1 , . . . , x mn +1 ) = ( x n , x n , . . . , x mn ) . Let S = ( S , S , . . . , S m ), V = ( V , V , . . . , V m ) be a virtual ( m + 1)-switch on X such that S = ( S l , S r ) , V = ( V l , V r ) : X × X × X × · · · × X m → X ,S i = ( S li , S ri ) , V i = ( V li , V ri ) : X i → X i , for i = 1 , , . . . , m, and for i = 0 , , . . . , m the images of maps S li , S ri , V li , V ri are words over its argumentsin terms of the operations of X . From this fact follows, in particular, that ( S, V )induces a virtual ( m + 1)-switch on X ( n ) for all n . Since the map S : X × X × X · · · × X m → X × X × X · · · × X m is a switch, it is invertible, and we can speak about S − .Let D be a virtual link diagram with n arcs. Mark the arcs of D by ( m + 1)-tuples (cid:101) x j = ( x j , x j , . . . , x mj ) for j = 1 , , . . . , n , where x j , x j , . . . , x mj are generators of X described at the beginning of this section. Let the labels of arcs near some crossingbe as on Figure 6 (the big circle in the middle of the crossing means that the crossingcan be positive, negative, or virtual). ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 9 (cid:101) x i (cid:101) x p (cid:101) x j (cid:101) x q Figure 6.
Labels of arcs of D near some crossing.By the definition, a subsystem of X generated by all x ij from labels on arcs of D is X ( n ) . Denote by X S,V ( D ) the quotient of X ( n ) by the relations which can be writtenfrom the crossings of D in the following way: S ( (cid:101) x i , (cid:101) x j ) = ( (cid:101) x p , (cid:101) x q ) , if the crossing is positive ,S − ( (cid:101) x i , (cid:101) x j ) = ( (cid:101) x p , (cid:101) x q ) , if the crossing is negative ,V ( (cid:101) x i , (cid:101) x j ) = ( (cid:101) x p , (cid:101) x q ) , if the crossing is virtual , where the labels are as on Figure 6. Note that each crossing of D gives 2( m + 1)relations, so, the number of relations is equal to the number of crossings multipliedby 2( m + 1). Theorem 1.
Let X be an algebraic system, and X , X , . . . , X m be subsystems of X which satisfy conditions (1)-(4) from the beginning of this section. If ( S, V ) isa virtual ( m + 1) -switch on X such that the maps S, V are biquandle switches on X × X × X × · · · × X m , then X S,V ( D ) is an invariant for virtual links.Proof. Due to condition (4) from the beginning of this section, for every permutation α of N with a finite support the map x ij (cid:55)→ x iα ( j ) for i = 0 , , . . . , m , j = 1 , , . . . induces an automorphism of X . From this condition follows that the algebraic system X S,V ( D ) is well defined, i. e. it is not important how (in which sequence) we labelthe arcs in D .In order to prove that X S,V is an invariant for virtual links, it is enough to provethat X S,V ( D ) is isomorphic to X S,V ( D ) in the case when D is obtained from D using only one generalized Reidemeister move (see Figures 2, 3, 4). Depending onthis generalized Reidemeister move we consider several cases. Case 1: D is obtained from D by R -move or V R -move. We consider in detailsthe case of the move R . The diagrams D and D differ only in the small neighbor-hood, where D is as on the left part of Figure 7, and D is as on the right part ofFigure 7.Suppose that outside of the neighborhood where the R -move is applied diagrams D and D have n arcs. Label the arcs of D and D (which are not labeled yet)inside of the neighborhood where the R -move is applied by additional labels. Thenin the neighbourhood where we apply the R -move the arcs of the diagram D have R Figure 7.
The neighborhood, where D and D are different: D ison the left, D is on the right.the labels depicted on Figure 8. Therefore X S,V ( D ) is the quotient of X ( n ) by the (cid:102) x i (cid:102) x j Figure 8.
Labels of the arcs in D .relations which can be written from the part of the diagram outside of Figure 8 and m + 1 relations (cid:101) x i = (cid:101) x j from the part of the diagram on Figure 8.Let us find X S,V ( D ). In the neighbourhood where we apply the R -move thearcs of the diagram D have the labels depicted on Figure 9. By the definition the (cid:102) x i (cid:102) x j (cid:102) x n +1 Figure 9.
Labels of the arcs in D .algebraic system X S,V ( D ) is the quotient of X ( n +1) ( n arcs outside of Figure 9 andone extra arc inside of Figure 9) by the relations which can be written from the partof the diagram outside of Figure 9 and 2( m + 1) relations(5) S ( (cid:101) x n +1 , (cid:101) x j ) = ( (cid:101) x n +1 , (cid:101) x i )which are written from the crossing on Figure 9. For the elements A, B from the set X × X × X ×· · ·× X m denote by S ( A, B ) = ( B A , A B ), where B A , A B are the elementsfrom X × X × · · · × X m . Since S is a biquandle switch on X × X × X × · · · × X m ,the maps B (cid:55)→ B A , B (cid:55)→ B A are invertible and we can speak about B A − , B A − for ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 11
A, B ∈ X × X × · · · × X m . In these denotations equalities (5) can be written in theform (˜ x j ) ˜ x n +1 = ˜ x n +1 , (6) (˜ x n +1 ) ˜ x j = ˜ x i . (7)From equality (7) follows that ˜ x n +1 = (˜ x i ) ˜ x − j . Since the images of the maps S lk , S rk are words over its arguments in terms of operations of X for k = 0 , , . . . , m , theequality ˜ x n +1 = (˜ x i ) ˜ x − j implies that the generators x n +1 , x n +1 , . . . , x mn +1 can be ex-pressed as words over generators x i , x i , . . . , x mi , x j , x j , . . . , x mj . So, we can deletethe elements x n +1 , x n +1 , . . . , x mn +1 from the generating set of X S,V ( D ). Therefore X S,V ( D ) is the quotient of X ( n ) (we change n + 1 by n since we deleted all theelements x n +1 , x n +1 , . . . , x mn +1 from the generating set) by the relations which can bewritten from the part of the diagram outside of Figure 9 and m + 1 relations whichare obtained from equalities (6), (7) excluding ˜ x n +1 . From (6) we have the equality (cid:101) x j = ( (cid:101) x n +1 ) (cid:101) x − n +1 , (8)from (7) we have the equality (cid:101) x i = ( (cid:101) x n +1 ) (cid:101) x j = ( (cid:101) x n +1 ) ( (cid:101) x n +1 ) (cid:101) x − n +1 , (9)and using the second axiom of biquandles we can exlude (cid:101) x n +1 from equalities (8), (9)and obtain the equality ˜ x i = ˜ x j . Therefore X S,V ( D ) is the quotient of X ( n ) by therelations which can be written from the part of the diagram outside of Figure 9 and m + 1 relations(10) ˜ x i = ˜ x j . Comparing this description of X S,V ( D ) with the description of X S,V ( D ), we seethat X S,V ( D ) = X S,V ( D ).The case of the R -move with the negative crossing, and the case of the V R -moveare almost the same with small mutations. Case 2: D is obtained from D by R -move or V R -move. We consider in detailsonly the case of the move R , the case of the move V R is similar. The diagrams D and D differ only in the small neighborhood, where D is as on the left part ofFigure 10, and D is as on the right part of Figure 10. R Figure 10.
The neighborhood, where D and D are different: D ison the left, D is on the right. Suppose that outside of the neighborhood where the R -move is applied diagrams D and D have n arcs. Label the arcs of D and D (which are not labeled yet)inside of the neighborhood where the R -move is applied by additional labels. Thenin the neighbourhood where we apply the R -move the arcs of the diagram D havethe labels depicted on Figure 11. Therefore X S,V ( D ) is the quotient of X ( n ) by the (cid:102) x i (cid:102) x j (cid:102) x p (cid:102) x q Figure 11.
Labels of the arcs in D .relations which can be written from the part of the diagram outside of Figure 11 and2( m + 1) relations (cid:101) x i = (cid:101) x p , (cid:101) x j = (cid:101) x q from the part of the diagram on Figure 11.Let us find X S,V ( D ). In the neighborhood where we apply the R -move the arcsof the diagram D have the labels depicted on Figure 12. There are four possibilities (cid:102) x i (cid:102) x j (cid:102) x p (cid:102) x q (cid:102) x n +1 (cid:102) x n +2 Figure 12.
Labels of the arcs in D .for the orientation of the arcs on Figure 12: i) both arcs are oriented from up todown, ii) both arcs are oriented from down to up, iii) the left arc is oriented fromup to down, and the right arc is oriented from down to up, and iv) the left arc isoriented from down to up, and the right arc is oriented from up to down. Since casei) is very similar to case ii), and case iii) is very similar to case iv), we will considerin details only case i) and case iii).Let both arcs on Figure 12 are oriented from the top to the bottom. In this case X S,V ( D ) is the quotient of X ( n +2) ( n arcs outside of Figure 12 and 2 new arcs insideof Figure 12) by the relations which can be written from the part of the diagramoutside of Figure 12 and 4( m + 1) relations S − ( (cid:101) x i , (cid:101) x j ) = ( (cid:101) x n +1 , (cid:101) x n +2 ) , (11) S ( (cid:101) x n +1 , (cid:101) x n +2 ) = ( (cid:101) x p , (cid:101) x q )(12) ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 13 which are written from two crossings on Figure 12.Since for k = 0 , , . . . , m the images of the maps S lk , S rk are words over its argu-ments in terms of operations of X , from equality (11) follows that the generators x n +1 , x n +1 , . . . , x mn +1 , x n +2 , x n +2 , . . . , x mn +2 can be expressed as words over generators x i , x i , . . . , x mi , x j , x j , . . . , x mj . Hence, we can delete the elements x n +1 , x n +1 , . . . , x mn +1 , x n +2 , x n +2 , . . . , x mn +2 from the generating set of X S,V ( D ). Therefore X S,V ( D ) isthe quotient of X ( n ) (here we change n + 2 by n since we deleted all the elements x n +1 , x n +1 , . . . , x mn +1 , x n +2 , x n +2 , . . . , x mn +2 from the generating set) by the relationswhich can be written from the part of the diagram outside of Figure 12 and 2( m + 1)relations obtained from equality (12) changing ( (cid:101) x n +1 , (cid:101) x n +2 ) by S − ( (cid:101) x i , (cid:101) x j ) (that iswhat we have from equality (11)). This equality clearly has the form(13) ( (cid:101) x i , (cid:101) x j ) = ( (cid:101) x p , (cid:101) x q ) . Therefore, we proved that if both arcs on Figure 12 are oriented from the top tothe bottom, then X S,V ( D ) is the quotient of X ( n ) by the relations which can bewritten from the part of the diagram outside of Figure 12 and 2( m + 1) relations (13).Comparing this description of X S,V ( D ) with the description of X S,V ( D ), we see that X S,V ( D ) = X S,V ( D ).Let the left arc on Figure 12 is oriented from up to down, and the right arc onFigure 12 is oriented from down to up. In this case X S,V ( D ) is the quotient of X ( n +2) by the relations which can be written from the part of the diagram outside ofFigure 12 and 4( m + 1) relations S ( (cid:101) x n +1 , (cid:101) x i ) = ( (cid:101) x n +2 , (cid:101) x j ) , (14) S − ( (cid:101) x n +2 , (cid:101) x q ) = ( (cid:101) x n +1 , (cid:101) x p )(15)which are written from two crossings depicted on Figure 12. For elements A, B from X × X × X ×· · ·× X m denote by S ( A, B ) = ( B A , A B ), where B A , A B are the elementsfrom X × X × · · · × X m . Since S is a biquandle switch on X × X × X × · · · × X m ,the maps B (cid:55)→ B A , B (cid:55)→ B A are invertible and we can speak about B A − , B A − for A, B ∈ X × X × · · · × X m . In these denotations equalities (14), (15) can be writtenin the form (˜ x i ) ˜ x n +1 = ˜ x n +2 , (16) (˜ x n +1 ) ˜ x i = ˜ x j , (17) (˜ x p ) ˜ x n +1 = ˜ x n +2 , (18) (˜ x n +1 ) ˜ x p = ˜ x q . (19)Since for k = 0 , , . . . , m the images of the maps S lk , S rk are words over its argumentsin terms of operations of X , from equalities (16), (17) follows that˜ x n +1 = (˜ x j ) ˜ x − i , ˜ x n +2 = (˜ x i ) ˜ x n +1 = (˜ x i ) (˜ x j ) ˜ x − i , i. e. the generators x n +1 , x n +1 , . . . , x mn +1 , x n +2 , x n +2 , . . . , x mn +2 can be expressed aswords over generators x i , x i , . . . , x mi , x j , x j , . . . , x mj . So, we can delete the elements x n +1 , x n +1 , . . . , x mn +1 , x n +2 , x n +2 , . . . , x mn +2 from the generating set of X S,V ( D ). There-fore X S,V ( D ) is the quotient of X ( n ) (we changed n + 2 by n since we deleted theelements x n +1 , x n +1 , . . . , x mn +1 , x n +2 , x n +2 , . . . , x mn +2 from the generating set) by therelations which can be written from the part of the diagram outside of Figure 12 andrelations obtained from equalities (18), (19) changing ˜ x n +1 by (˜ x j ) ˜ x − i , and changing˜ x n +2 by (˜ x i ) (˜ x j ) ˜ x − i (that is what we have from equalities (16), (17)). It is easy to seethat these equalities will have the form˜ x i = ˜ x p , ˜ x j = ˜ x q . (20)Comparing this description of X S,V ( D ) with the description of X S,V ( D ), we see that X S,V ( D ) = X S,V ( D ) in the case when the left arc on Figure 12 is oriented from upto down, and the right arc on Figure 12 is oriented from down to up. Case 3: D is obtained from D by R -move, V R -move, or V R -move. We con-sider in details only the case of the move
V R , the case of the moves R and V R are similar. The diagrams D and D differ only in the small neighborhood, where D is as on the left part of Figure 13, and D is as on the right part of Figure 13. V R Figure 13.
The neighborhood, where D and D are different: D ison the left, D is on the right.There are eight possibilities for the orientation of the arcs on Figure 13. Turaev(see page 544 in [41]) proved that the third Reidemeister move for one orientation ofarcs on Figure 13 can be realized as a sequence of several Reidemeister moves R , V R and the third Reidemeister move for another fixed orientation of arcs on Figure 13.For example, if the top and the bottom arcs on Figure 13 are oriented from theleft to the right, and the middle arcs is oriented from the right to the left, then theReidemeister move V R can be realized as a sequence of moves: 1) two moves V R ,2) one move V R , where all arcs are oriented from the left to the right, 3) two moves V R (see Figure 14). Therefore it is enough to consider only one orientation of thearcs on Figure 13. Let us consider the case when all the arcs on Figure 13 are orientedfrom the left to the right.Suppose that outside of the neighborhood where the V R -move is applied diagrams D and D have n arcs. Label the arcs of D and D (which are not labeled yet)inside of the neighborhood where the V R -move is applied by additional labels. Thenin the neighbourhood where we apply the V R -move the arcs of the diagram D havethe labels depicted on Figure 15. Therefore X S,V ( D ) is the quotient of X ( n +3) ( n ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 15
Figure 14.
Other orientation of arcs. ˜ x i ˜ x j ˜ x k ˜ x r ˜ x q ˜ x p ˜ x n +2 ˜ x n +3 ˜ x n +1 Figure 15.
Labels of the arcs in D .arcs outside of Figure 15 and 3 new arcs depicted on Figure 15) by the relationswhich can be written from the part of the diagram outside of Figure 15 and 6( m + 1)relations V (˜ x j , ˜ x i ) = (˜ x n +2 , ˜ x n +1 ) , (21) S − (˜ x k , ˜ x n +2 ) = (˜ x r , ˜ x n +3 ) , (22) V (˜ x n +3 , ˜ x n +1 ) = (˜ x q , ˜ x p )(23)from the part of the diagram on Figure 15. Equalities (21), (22), (23) can be rewrittenin the following form ( id × V )(˜ x k , ˜ x j , ˜ x i ) = (˜ x k , ˜ x n +2 , ˜ x n +1 ) , (24) ( S − × id )(˜ x k , ˜ x n +2 , ˜ x n +1 ) = (˜ x r , ˜ x n +3 , ˜ x n +1 ) , (25) ( id × V )(˜ x r , ˜ x n +3 , ˜ x n +1 ) = (˜ x r , ˜ x q , ˜ x p ) . (26)Since for t = 0 , , . . . , m the images of the maps S lt , S rt are words over its argumentsin terms of operations of X , from equalities (24), (25) one can express the generatorsfrom tuples ˜ x n +1 , ˜ x n +2 , ˜ x n +3 as words over generators from tuples ˜ x i , ˜ x j , ˜ x k . So, we candelete the elements from tupels ˜ x n +1 , ˜ x n +2 , ˜ x n +3 from the generating set of X S,V ( D ).Therefore X S,V ( D ) is the quotient of X ( n ) (we changed n + 3 by n since we deletedthe elements from tupels ˜ x n +1 , ˜ x n +2 , ˜ x n +3 from the generating set) by the relationswhich can be written from the part of the diagram outside of Figure 15 and relationsobtained from equalities (24), (25), (26) excluding ˜ x n +1 , ˜ x n +2 , ˜ x n +3 . It is easy to see that these equalities will have the form( id × V )( S − × id )( id × V )(˜ x k , ˜ x j , ˜ x i ) = (˜ x r , ˜ x q , ˜ x p ) . (27)The algebraic system X S,V ( D ) can be found similarly. Let in the neighbourhoodwhere we apply the V R -move the arcs of the diagram D have the labels depicted onFigure 16. Then similarly to X S,V ( D ), the algebraic system X S,V ( D ) is the quotient ˜ x i ˜ x j ˜ x k ˜ x r ˜ x q ˜ x p ˜ x n +2 ˜ x n +3 ˜ x n +1 Figure 16.
Labels of the arcs in D .of X ( n ) by the relations( V × id )( id × S − )( V × id )(˜ x k , ˜ x j , ˜ x i ) = (˜ x r , ˜ x q , ˜ x p ) . (28)Comparing this equality with equality (27) due to the fact that ( S, V ) is a virtualmulti-switch on X we conclude that X S,V ( D ) = X S,V ( D ). (cid:3) Theorem 1 gives a powerful tool for constructing invariants for classical and virtuallinks. In particular, a lot of known invariants can be constructed using Theorem 1.For example, if X = X = F ∞ is a free group with the free generators x , x , . . . , and S, V : X → X are the maps given by S ( x, y ) = ( y, yxy − ) , V ( x, y ) = ( y, x ) , then ( S, V ) is a biquandle 1-switch on X . If L is a virtual link presented by a diagram D , then the algebraic system X S,V ( D ) defined in Theorem 1 is the group of a virtuallink L introduced by Kauffman in [30]. If D represents a classical link, then thisgroup is the classical knot group.If X = X = F Q ∞ is the free quandle with the free generators x , x , . . . , and S, V : X → X are the maps given by S ( x, y ) = ( y, x ∗ y ) , V ( x, y ) = ( y, x ) , then ( S, V ) is a biquandle 1-switch on X , and the algebraic system X S,V ( D ) intro-duced in Theorem 1 is the quandle of a virtual link L (represented by the diagram D )introduced by Kauffman in [30]. If D represents a classical link L , then this quandleis the fundamental quandle of L introduced by Joyce [28] and Matveev [33]. ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 17
Let X = F ∞ ∗ Z ∞ be the free product of the infinitely generated free group F ∞ with the free generators x , x , . . . and the free abelian group Z ∞ with the canoni-cal generators which are grouped into three groups x , x , . . . , x , x , . . . , x , x , . . . ,where x = x = . . . Let X = (cid:104) x , x , . . . (cid:105) = F ∞ , X = (cid:104) x , x , . . . (cid:105) = Z ∞ , X = (cid:104) x , x , . . . (cid:105) = Z ∞ , X = (cid:104) x , x , . . . (cid:105) = Z and the maps S, V : X × X × X × X → X × X × X × X are given by the formulas S ( a, b ; x, y ; p, q ; r, s ) = ( ab x a − ry , a s ; y, x ; q, p ; s, r ) ,V ( a, b ; x, y ; p, q ; r, s ) = ( b p − , a q ; y, x ; q, p ; s, r ) , for a, b ∈ X , x, y ∈ X , p, q ∈ X , r, s ∈ X . Then ( S, V ) is a virtual 4-switch on X , and the algebraic system X S,V ( D ) introduced in Theorem 1 is the group G M ( D )introduced in [3].Let R = Z [ t ± , s ± ] be the ring of Laurent polynomials in two variables t , s . If X = X is the free R -module with the basis x , x , . . . , and S, V : X → X are themaps given by S ( x, y ) = ( sy, tx + (1 − st ) y ) , V ( x, y ) = ( y, x )for x, y ∈ X , then ( S, V ) is a biquandle 1-switch on X , and the algebraic system X S,V ( D ) introduced in Theorem 1 is the Alexander module of a virtual link repre-sented by D . The elementary ideals of this module determine both the generalizedAlexander polynomial (also known as the Sawollek polynomial) for virtual knots andthe classical Alexander polynomial for classical knots.The examples above show that a lot of known invariants for classical and virtuallinks can be constructed using Theorem 1. In Section 6 using Theorem 1 we willconstruct a new quandle invariant for virtual links.In the examples above we see that a lot of known invariants which are algebraicsystems can be constructed as X S,T , where T is the twist. This fact reflects thegeometric interpretation of the virtual crossing in the virtual link diagram: virtualcrossing in the virtual link diagram is a defect of the representation of the virtualknot on the plane. Thus, the following question is natural. Problem . Let (
S, V ) be a virtual biquandle 1-switch on an algebraic system X .Is there a biquandle switch S (cid:48) on X , such that the invariants X S,V and X S (cid:48) ,T areequivalent, where T is the twist on X ?At the moment we do not knot the answer to this question.5. Another representation of X S,V ( D )Multi-switches were introduced in [5] as a tool for constructing representationsof virtual braid groups. Namely, if ( S, V ) is a virtual multi-switch on an algebraicsystem X , then under additional conditions one can construct a representation ϕ S,V : V B n → Aut( X ) of the virtual braid group V B n by automorphisms of X . In this section we recallthe construction of the representation ϕ S,V and show how this representation can beused for finding the set of generators and the set of defining relations of the algebraicsystem X S,V ( D ) introduced in Theorem 1.Let us recall that the virtual braid group V B n on n strands is the group with2( n −
1) generators σ , σ , . . . , σ n − , ρ , ρ , . . . , ρ n − (see Figure 17) and the following i i + 1 n i i + 1 n i i + 1 n Figure 17.
Geometric interpretation of σ i (on the left), σ − i (in themiddle), and ρ i (on the right).defining relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 i = 1 , , . . . , n − ,σ i σ j = σ j σ i | i − j | ≥ ,ρ i ρ i +1 ρ i = ρ i +1 ρ i ρ i +1 i = 1 , , . . . , n − ,ρ i ρ j = ρ j ρ i | i − j | ≥ ,ρ i = 1 i = 1 , , . . . , n − ,ρ i +1 σ i ρ i +1 = ρ i σ i +1 ρ i i = 1 , , . . . , n − ,σ i ρ j = ρ j σ i | i − j | ≥ . Let X be an algebraic system, and X , X , . . . , X m be subsystems of X such that(1) for i = 0 , , . . . , m the subsystem X i is generated by elements x i , x i , . . . , x in ,(2) { x i , x i , . . . , x in } ∩ { x j , x j , . . . , x jn } = ∅ for i (cid:54) = j ,(3) the set of elements { x ij | i = 0 , , . . . , m, j = 1 , , . . . , n } generates X .Let S = ( S , S , . . . , S m ), V = ( V , V , . . . , V m ) be a virtual ( m + 1)-switch on X suchthat S = ( S l , S r ) , V = ( V l , V r ) : X × X × X × · · · × X m → X ,S i = ( S li , S ri ) , V i = ( V li , V ri ) : X i → X i , for i = 1 , , . . . , m, and for i = 0 , , . . . , m the images of the maps S li , S ri , V li , V ri are words over itsarguments in terms of the operations of X . For j = 1 , , . . . , n − R j , G j ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 19 the following maps from { x ij | i = 0 , , . . . , m, j = 1 , , . . . , n } to XR j : x j (cid:55)→ S l ( x j , x j +1 , x j , x j +1 , . . . , x mj , x mj +1 ) ,x j +1 (cid:55)→ S r ( x j , x j +1 , x j , x j +1 , . . . , x mj , x mj +1 ) ,x j (cid:55)→ S l ( x j , x j +1 ) ,x j +1 (cid:55)→ S r ( x j , x j +1 ) , ... x mj (cid:55)→ S lm ( x mj , x mj +1 ) ,x mj +1 (cid:55)→ S rm ( x mj , x mj +1 ) , (29) G j : x j (cid:55)→ V l ( x j , x j +1 , x j , x j +1 , . . . , x mj , x mj +1 ) ,x j +1 (cid:55)→ V r ( x j , x j +1 , x j , x j +1 , . . . , x mj , x mj +1 ) ,x j (cid:55)→ V l ( x j , x j +1 ) ,x j +1 (cid:55)→ V r ( x j , x j +1 ) , ... x mj (cid:55)→ V lm ( x mj , x mj +1 ) ,x mj +1 (cid:55)→ V rm ( x mj , x mj +1 ) , (30)where all generators which are not explicitly mentioned in R j , G j are fixed, i. e. R j ( x ik ) = G j ( x ik ) = x ik for k (cid:54) = j, j + 1, i = 0 , , . . . , m , and assume that R j , G j arewell defined: since the elements x i , x i , . . . , x in are not necessary all different, some ofthese elements can coincide. The fact that R j , G j are well defined means that theimages of equal elements are equal. For example, if x ij = x ij +1 , then we assume that S li ( x ij , x ij +1 ) = R j ( x ij ) = R j ( x ij +1 ) = S ri ( x ij , x ij +1 ) . If for j = 1 , , . . . , n − R j , G j induce automorphisms of X , then wesay that ( S, V ) is an automorphic virtual multi-switch (shortly, AVMS) on X withrespect to the set of generators { x ij | i = 0 , , . . . , m, j = 1 , , . . . , n } . The followingtheorem is proved in [5, Theorem 1]. Theorem 2.
Let ( S, V ) be an AVMS on X with respect to the set of generators { x ij | i = 0 , , . . . , m, j = 1 , , . . . , n } . Then the map ϕ S,V : V B n → Aut( X ) which is defined on the generators of V B n as ϕ S,V ( σ j ) = R j , ϕ S,V ( ρ j ) = G j , f or j = 1 , , . . . , n − , where R j , G j are defined by equalities (29), (30), is a representation of V B n . The map ϕ S,V from Theorem 2 is given explicitly on the generators σ , σ , . . . , σ n − , ρ , ρ , . . . , ρ n − of the virtual braid group V B n by formulas (29), (30). Let β be anarbitrary braid from V B n , and x be an arbitrary element from X . In order to find the value ϕ S,V ( β )( x ) express the braid β in terms of the generators σ , σ , . . . , σ n − , ρ , ρ , . . . , ρ n − , i. e. write β in the form β = β β . . . β k , where β i ∈ { σ , σ . . . , σ n − , σ − , σ − , . . . , σ − n − , ρ , ρ , . . . , ρ n − } for i = 1 , , . . . , k , and then reading β from the left to the right calculte the images x ϕ S,V ( β ) −−−−−→ x ϕ S,V ( β ) −−−−−→ x ϕ S,V ( β ) −−−−−→ . . . ϕ S,V ( β k ) −−−−−→ x k . The last calculated value x k is the image of ϕ S,V ( β )( x ). This agreement means that ϕ S,V ( β β . . . β k )( x ) = ϕ S,V ( β k ) ϕ S,V ( β k − ) . . . ϕ S,V ( β )( x ) . Let X be an algebraic system, and X , X , . . . , X m be subsystems of X whichsatisfy conditions (1)-(4) from the beginning of Section 4. Let ( S, V ) be a virtual( m + 1)-switch on X such that S = ( S l , S r ) , V = ( V l , V r ) : X × X × X × · · · × X m → X ,S i = ( S li , S ri ) , V i = ( V li , V ri ) : X i → X i , for i = 1 , , . . . , m, and for i = 0 , , . . . , m the images of maps S li , S ri , V li , V ri are words over its argumentsin terms of operations of X . From this fact follows, in particular that ( S, V ) induces avirtual ( m + 1)-switch ( S ( n ) , V ( n ) ) on X ( n ) for all n . Suppose that for all n = 2 , , . . . the virtual ( m + 1)-switch ( S ( n ) , V ( n ) ) is AVMS on X ( n ) with respect to the setof generators { x ij | i = 0 , , . . . , m, j = 1 , , . . . , n } . According to Theorem 2 for n = 2 , , . . . we have representations ϕ S ( n ) ,V ( n ) : V B n → Aut (cid:0) X ( n ) (cid:1) . Since ϕ S ( n ) ,V ( n ) and ϕ S ( n +1) ,V ( n +1) are obtained from the same virtual multi-switch( S, V ) restricted to different sets, we have ϕ S ( n +1) ,V ( n +1) | V B n = ϕ S ( n ) ,V ( n ) , so, automorphisms ϕ S ( n ) ,V ( n ) “agree” with each other for different n . Denote by V B ∞ = (cid:83) n V B n , and by ϕ S,V : V B ∞ → Aut( X ) the homomorphism which is equalto ϕ S ( n ) ,V ( n ) on V B n (this homomorphism is well defined since ϕ S ( n ) ,V ( n ) agree witheach other). Now we can write ϕ S,V : V B n → Aut( X ( n ) ) meaning the restriction of ϕ S,V to V B n .If ( S, V ) is a virtual ( m + 1)-switch on X such that, S, V are biquandle switcheson X × X × X × · · · × X m , then by Theorem 1 one can construct the invariant X S,V ( D ) for virtual links. The following theorem gives a method how to calculate X S,V ( D ) using the representation ϕ S,V : V B ∞ → Aut( X ). Theorem 3.
Let β ∈ V B n be a virtual braid, and D = ˆ β be the closure of β . Then X S,V ( D ) is equal to the quotient of X ( n ) by the relations ϕ S,V ( β )( x ij ) = x ij for i = 0 , , . . . , m , j = 1 , , . . . , n . ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 21
Proof.
The braid β can be written in the form β = β β . . . β k where, β i ∈ { σ , σ . . . , σ n − , σ − , σ − , . . . , σ − n − , ρ , ρ , . . . , ρ n − } for i = 1 , , . . . , k . Label by˜ x = ( x , x , . . . , x m ) , ˜ x = ( x , x , . . . , x m ) , . . . ˜ x n = ( x n , x n , . . . , x mn )the arcs of the diagram ˆ β which contain the upper points of β (see Figure 18 for thecase β = σ j ). ˜ x ˜ x ˜ x j ˜ x j +1 ˜ x n Figure 18.
Labels of arcs in β = σ j .In a similar way for i = 2 , , . . . , k label by˜ x ( i − n +1 = (cid:0) x i − i +1 , x i − i +1 , . . . , x m ( i − n +1 (cid:1) ˜ x ( i − n +2 = (cid:0) x i − i +2 , x i − i +2 , . . . , x m ( i − n +2 (cid:1) . . . ˜ x in = (cid:0) x in , x in , . . . , x min (cid:1) the arcs of the diagram ˆ β which contain the upper points of β i . Since the top pointsof β i +1 coincide with the bottom points of β i , some arcs are labeled several times (seeFigure 19 for the case β i = σ j , where ˜ x ( i − n +1 = ˜ x in +1 ). ˜ x ( i − n +1 ˜ x ( i − n +2 ˜ x ( i − n + j ˜ x ( i − n + j +1 ˜ x in ˜ x in +1 ˜ x in +2 ˜ x in + j ˜ x in + j +1 ˜ x ( i +1) n Figure 19.
Labels of arcs in β i = σ j .Due to Remark 3 from Section 4 the algebraic system X S,V ( ˆ β ) is isomorphic tothe quotient of X ( nk ) (for i = 1 , , . . . , k each β i gives n tuples of generators) by the k families of relations which can be written from the crossing in β , β , . . . , β k . Therelations which are obtained from β depending on β have the following form:if β = σ j , then the relations are˜ x n + r = ˜ x r for r (cid:54) = j, j + 1 , (˜ x n + j , ˜ x n + j +1 ) = S (˜ x j , ˜ x j +1 ) , (31) if β = σ − j , then the relations are˜ x n + r = ˜ x r for r (cid:54) = j, j + 1 , (˜ x n + j , ˜ x n + j +1 ) = S − (˜ x j , ˜ x j +1 ) , (32)if β = ρ j , then the relations are˜ x n + r = ˜ x r for r (cid:54) = j, j + 1 , (˜ x n + j , ˜ x n + j +1 ) = V (˜ x j , ˜ x j +1 ) , (33)(see Figure 19 for i = 1 and the definition of X S,V ( D )). From formulas (29), (30)and Theorem 2 (where ϕ S,V is defined) we see that relations (31), (32), (33) can berewritten in the unique way˜ x n + r = ϕ S,V ( β )(˜ x r ) r = 1 , , . . . , n, (34)where ϕ S,V ( β )(˜ x r ) = ( ϕ S,V ( β )( x r ) , ϕ S,V ( β )( x r ) , . . . , ϕ S,V ( β )( x mr )). In a similarway the relations which are obtained from β depending on β have the followingform:if β = σ j , then the relations are˜ x n + r = ˜ x n + r for r (cid:54) = j, j + 1 , (˜ x n + j , ˜ x n + j +1 ) = S (˜ x n + j , ˜ x n + j +1 ) , (35)if β = σ − j , then the relations are˜ x n + r = ˜ x n + r for r (cid:54) = j, j + 1 , (˜ x n + j , ˜ x n + j +1 ) = S − (˜ x n + j , ˜ x n + j +1 ) , (36)if β = ρ j , then the relations are˜ x n + r = ˜ x n + r for r (cid:54) = j, j + 1 , (˜ x n + j , ˜ x n + j +1 ) = V (˜ x n + j , ˜ x n + j +1 ) . (37)From formulas (29), (30) and Theorem 2 we see that relations (35), (36), (37) can berewritten in the unique way˜ x n + r = ϕ S,V ( β )(˜ x n + r ) r = 1 , , . . . , n. (38)From relations (34) and (38) it follows that˜ x n + r = ϕ S,V ( β β )(˜ x r ) r = 1 , , . . . , n. Using the same argumentation for i = 1 , , . . . , k − β i can be written in a unique way˜ x in + r = ϕ S,V ( β β . . . β i )(˜ x r ) r = 1 , , . . . , n. (39) ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 23
Since in ˆ β we identify the top points of β (which are the top points of β ) withthe bottom points of β (which are the bottom points of β k ), the relations which areobtained from β k can be written in a unique way˜ x r = ϕ S,V ( β β . . . β k )(˜ x r ) = ϕ S,V ( β )(˜ x r ) r = 1 , , . . . , n. (40)Therefore X S,V ( ˆ β ) is the quotient of X ( nk ) by the relations (39), (40). Using relations(39) we can delete the elements from tuples ˜ x in + r for i = 1 , , . . . , k − n ( k − X S,V ( D ). Therefore the algebraicsystem X S,V ( ˆ β ) is the quotient of X ( n ) ( nk − n ( k −
1) = n ) by relations (40). (cid:3) New quandle invariant for virtual links
In this section using Theorem 1 we construct a new quandle invariant for virtuallinks. Definition of a quandle is given in Section 3. The information about freeproducts of quandles can be found, for example, in [4]. In [5, Corollary 2] we foundthe following virtual 2-switch on quandles.
Proposition 1.
Let Q be a quandle, X be a trivial quandle, and X = Q ∗ X be thefree product of Q and X . Let S, V : X × X → X × X be the maps defined by S ( a, b ; x, y ) = ( b, a ∗ b ; y, x ) , V ( a, b ; x, y ) = ( b ∗ − x, a ∗ y ; y, x ) for a, b ∈ X , x, y ∈ X . Then ( S, V ) is a virtual -switch on X . Moreover, the maps S, V are biquandle switches on X × X . Using the virtual 2-switch (
S, V ) introduced in Proposition 1 we are going toconstruct a new quandle invariant (cid:101) Q ( L ) for virtual links which generalizes the quandleof Manturov [32] and the quandle of Kauffman [30].Let X = F Q ∞ be the free quandle on the generators x , x , . . . , X = T ∞ be thetrivial quandle on the elements y , y , . . . , and X = X ∗ X . If for j = 1 , , . . . wedenote by x j = x j , x j = y j , then it is clear that(1) for i = 0 , X i is generated by elements x i , x i , . . . ,(2) { x , x , . . . } ∩ { x , x , . . . } = ∅ ,(3) the set of elements { x ij | i = 0 , , j = 0 , , . . . } generates X ,(4) for every permutation α of N with a finite support the map x ij (cid:55)→ x iα ( j ) for i = 0 , j = 1 , , . . . induces an automorphism of X .Hence, conditions (1)-(4) from Section 4 hold for X, X , X and for a given virtuallink diagram D we can define the algebraic system X S,V ( D ). Let the virtual linkdiagram D has n arcs. In order to find X S,V ( D ) label the arcs of D by the tuples( x i , y i ) for i = 1 , , . . . , n . In the neighborhood of some crossings let the labels of arcsare as on Figure 20.Then X S,V ( D ) is the quotient of X ( n ) = F Q n ∗ T n by the relations which can be written ( x i , y i ) ( x j , y j )( x p , y p ) ( x q , y q ) Figure 20.
Labels of arcs in D near a crossing.from the crossings of D in the following way (described right before Theorem 1). x p = x j , x q = x i ∗ x j , y p = y j , y q = y i , positive crossing ,x p = x j ∗ − x i , x q = x i , y p = y j , y q = y i , negative crossing , (41) x p = x j ∗ − y i , x q = x i ∗ y j , y p = y j , y q = y i , virtual crossing . Since the maps
S, V introduced in Proposition 1 are biquandle switches on X × X ,then from Theorem 1 we conclude that X S,V ( D ) is the virtual link invariant. Let L bea virtual link represented by a virtual link diagram D . Denoting by (cid:101) Q ( L ) = X S,V ( D )we have the following result. Theorem 4.
The quandle (cid:101) Q ( L ) is a virtual link invariant. Since
F Q n ∗ T n is the quotient of the free quandle F Q n with the generators x , x , . . . , x n , y , y , . . . , y n by the relations y i ∗ y j = y i for all i, j = 1 , , . . . , n , thequandle (cid:101) Q ( L ) can be written as the quotient of F Q n (where n is the number of arcsin the diagram D which represents the virtual link L ) by relations (41) which can bewritten from the crossings of D and relations y i ∗ y j = y i for all i, j = 1 , , . . . , n .The quandle Q ( L ) introduced by Manturov in [32] can be found in the followingway. Let the diagram D which represents L has n arcs. Label the arcs of D by thesymbols x , x , . . . , x n . Then the quandle Q ( L ) is the quotient of the free quandle F Q n +1 on n + 1 generators x , x , . . . , x n , y by the relations which can be writtenfrom the crossings of D in the following way x p = x j , x q = x i ∗ x j , positive crossing ,x p = x j ∗ − x i , x q = x i , negative crossing ,x p = x j ∗ − y, x q = x i ∗ y, virtual crossing , where the labels of arcs are as on Figure 21. Comparing this description with thedescription of (cid:101) Q ( L ) we see, that the map which maps x i to x i for i = 1 , , . . . , n , andmaps y i to y for i = 1 , , . . . , n is a surjective homomorphism (cid:101) Q ( L ) → Q ( L ). It meansthat the quandle (cid:101) Q ( L ) generalizes the quandle Q ( L ) of Manturov (and therefore italso generalizes the quandle of Kauffman [30]). At the same time Q ( L ) and (cid:101) Q ( L ) are ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 25 x i x j x p x q Figure 21.
Labels of arcs in D near a crossing.different. For example, if U is a trivial link with 2 components, then Q ( U ) = F Q ,while (cid:101) Q ( U ) = F Q ∗ T .Theorem 3 says how to use the representation ϕ S,V in order to find X S,V ( D ).If ( S, V ) is a virtual 2-switch described in Proposition 1, then the representation ϕ S,V : V B n → Aut(
F Q n ∗ T n ) is denoted by ϕ Q , it is described in [5, Theorem 3],and it has the following form. ϕ Q ( σ i ) : x i (cid:55)→ x i +1 ,x i +1 (cid:55)→ x i ∗ x i +1 ,y i (cid:55)→ y i +1 ,y i +1 (cid:55)→ y i , ϕ Q ( ρ i ) : x i (cid:55)→ x i +1 ∗ − y i ,x i +1 (cid:55)→ x i ∗ y i +1 ,y i (cid:55)→ y i +1 ,y i +1 (cid:55)→ y i , (42)From Theorem 3 and description (42) of the representation ϕ Q we have the followingdescription of (cid:101) Q ( L ). Theorem 5.
Let
F Q n be the free quandle on the set of generators { x , x , . . . , x n } , T n = { y , y , . . . , y n } be the trivial quandle, and β ∈ V B n be a virtual braid. Thenthe quandle (cid:101) Q ( ˆ β ) can be written as (cid:101) Q ( ˆ β ) = (cid:42) x , x , . . . , x n , y , y , . . . , y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ Q ( β )( x i ) = x i ,ϕ Q ( β )( y i ) = y i , i = 1 , , . . . , n,y r ∗ y s = y r , r, s = 1 , , . . . , n. (cid:43) , where ϕ Q is defined by (42). References [1] V. Bardakov, P. Dey, M. Singh, Automorphism groups of quandles arising from groups,Monatsh. Math., V. 184, N. 4, 2017, 519–530.[2] V. Bardakov, Yu. Mikhalchishina, M. Neshchadim, Representations of virtual braids by auto-morphisms and virtual knot groups, J. Knot Theory Ramifications, V. 26, N. 1, 2017, 1750003.[3] V. Bardakov, Yu. Mikhalchishina, M. Neshchadim, Virtual link groups, Sib. Math. J., V. 58,N. 5, 2017, 765–777.[4] V. Bardakov, T. Nasybullov, On embeddings of quandles into groups, J. Algebra and App.,2020, https://doi.org/10.1142/S0219498820501364.[5] V. Bardakov, T. Nasybullov, Multi-switches and representations of braid groups,ArXiv:Math/1907.09230. [6] V. Bardakov, T. Nasybullov, M. Singh, Automorphism groups of quandles and related groups,Monatsh. Math., V. 189, N. 1, 2019, 1–21.[7] V. Bardakov, T. Nasybullov, M. Singh, General constructions of biquandles and their sym-metries, ArXiv:Math/1908.08301.[8] V. Bardakov, M. Neshchadim, On a representation of virtual braids by automorphisms, Alge-bra Logic, V. 56, N. 5, 2017, 355–361.[9] V. Bardakov, M. Singh, M. Singh, Free quandles and knot quandles are residually finite, Proc.Amer. Math. Soc., V. 147, N. 8, 2019, 3621–3633.[10] A. Bartholomew, R. Fenn, Biquandles of small size and some invariants of virtual and weldedknots, J. Knot Theory Ramifications, V. 20, N. 7, 2011, 943–954.[11] A. Bartholomew, R. Fenn, Erratum: Biquandles of small size and some invariants of virtualand welded knots, J. Knot Theory Ramifications V. 26, N. 8, 2017, 1792002.[12] G. Bianco, M. Bonatto, On connected quandles of prime power order,ArXiv:Math/1904.12801.[13] J. Birman, Braids, links, and mapping class groups, Annals of Math. Studies 82, PrincetonUniversity Press, 1974.[14] H. Boden, E. Dies, A. Gaudreau, A. Gerlings, E. Harper, A. Nicas, Alexander invariants forvirtual knots, J. Knot Theory Ramifications, V. 24, N. 3, 2015, 1550009.[15] J. Carter, A survey of quandle ideas, Introductory lectures on knot theory, Ser. Knots Every-thing, World Sci. Publ., Hackensack, NJ, V. 46, 2012, 22–53.[16] J. Carter, D. Silver, S. Williams, M. Elhamdadi, M. Saito, Virtual knot invariants from groupbiquandles and their cocycles, J. Knot Theory Ramifications, V. 18 N. 7, 2009, 957–972.[17] A. Cattabriga, T. Nasybullov, Virtual quandle for links in lens spaces, Rev. R. Acad. Cienc.Exactas Fs. Nat. Ser. A Mat. RACSAM, V. 112, N. 3, 2018, 657–669.[18] W. Clark, M. Saito, Algebraic properties of quandle extensions and values of cocycle knotinvariants, J. Knot Theory Ramifications, V. 25, N. 14, 2016, 1650080.[19] E. Clark, M. Saito, L. Vendramin, Quandle coloring and cocycle invariants of composite knotsand abelian extensions, J. Knot Theory Ramifications, V. 25, N. 5, 2016, 1650024.[20] A. Crans, A. Henrich, S. Nelson, Polynomial knot and link invariants from the virtual biquan-dle, J. Knot Theory Ramifications, V. 22, N. 4, 2013, 134004.[21] V. Drinfel’d, On some unsolved problems in quantum group theory, Quantum groups(Leningrad, 1990), 1-8, Lecture Notes in Math., 1510, Springer, Berlin, 1992.[22] M. Elhamdadi, S. Nelson, Quandles–an introduction to the algebra of knots, Student Mathe-matical Library, V. 74, American Mathematical Society, Providence, RI, 2015.[23] R. Fenn, M. Jordan-Santana, L. Kauffman, Biquandles and virtual links, Topology Appl.,V. 145, N. 1–3, 2004, 157–175.[24] R. Fenn, Tackling the trefoils, J. Knot Theory Ramifications, V. 21, N. 13, 2012, 1240004.[25] R. Fenn, C. Rourke, Racks and links in codimension two, J. Knot Theory Ramifications, V. 1,N. 4, 1992, 343–406.[26] L. Guarnieri, L. Vendramin, Skew braces and the Yang-Baxter equation, Math. Comp., V. 86,N. 307, 2017, 2519–2534.[27] D. Hrencecin, L. Kauffman, Biquandles for virtual knots, J. Knot Theory Ramifications, V. 16,N. 10, 2007, 1361–1382.[28] D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, V. 23,1982, 37–65.
ULTI-SWITCHES AND VIRTUAL KNOT INVARIANTS 27 [29] S. Kamada, Knot invariants derived from quandles and racks, Invariants of knots and 3-manifolds (Kyoto, 2001), Geom. Topol. Monogr., Geom. Topol. Publ., Coventry, V. 4, 2002.103–117.[30] L. Kauffman, Virtual knot theory, Eur. J. Comb., V. 20, 1999, 663–690.[31] L. Kauffman, V. Manturov, Virtual biquandles, Fund. Math., V. 188, 2005, 103–146.[32] V. Manturov, On invariants of virtual links, Acta Appl. Math., V. 72, N. 3, 2002, 295–309.[33] S. Matveev, Distributive groupoids in knot theory, (in Russian), Mat. Sb. (N.S.), V. 119(161),N. 1(9), 1982, 78–88.[34] N. Nanda, M. Singh, M. Singh, Knot invariants from derivations of quandles,ArXiv:Math/1804.01113.[35] T. Nasybullov, Connections between properties of the additive and the multiplicative groupsof a two-sided skew brace, J. Algebra, V. 540, 2019, 156–167.[36] S. Nelson, The combinatorial revolution in knot theory, Notices Amer. Math. Soc., V. 58,2011, 1553–1561.[37] S. Nelson, C.-Y. Wong, On the orbit decomposition of finite quandles, J. Knot Theory Rami-fications, V. 15, N. 6, 2006, 761–772.[38] T. Nosaka, Quandles and topological pairs, Symmetry, knots, and cohomology, Springer Briefsin Mathematics, Springer, Singapore, 2017.[39] D. Silver, S. Williams, Alexander groups and virtual links, J. Knot Theory Ramifications,V. 10, N. 1, 2001, 151–160.[40] A. Smoktunowicz, L. Vendramin, On skew braces (with an appendix by N. Byott and L. Ven-dramin), J. Comb. Algebra, V. 2, N. 1, 2018, 47–86.[41] V. Turaev, The Yang-Baxter equation and invariants of links, Invent Math, V. 92, N. 3, 1988,527–553.[42] V. Vershinin, On homology of virtual braids and Burau representation, J. Knot Theory Ram-ifications, V. 10, N. 5, 2001, 795–812.
Valeriy Bardakov , , , ([email protected]),Timur Nasybullov , , ([email protected]) Tomsk State University, pr. Lenina 36, 634050 Tomsk, Russia, Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia, Novosibirsk State University, Pirogova 1, 630090 Novosibirsk, Russia,4