Virtual Homotopy. February 26, 2009

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1 arxiv:math/ v2 [math.gt] 9 Nov 2006 Virtual Homotopy H. A. Dye MADN-MATH United States Military Academy 646 Swift Road West Point, NY hdye@ttocs.org Louis H. Kauffman Department of Mathematics, Statistics, & Computer Science University of Illinois at Chicago 851 South Morgan St Chicago, IL kauffman@uic.edu February 26, 2009 Abstract Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor s µ and µ invariants to welded and virtual links. We conclude this paper with several examples, and compute the µ invariants using the Magnus expansion and Polyak s skein relation for the µ invariants. 1 Introduction Virtual link diagrams are a generalization of classical link diagrams introduced by Louis H. Kauffman in 1996 [9]. A virtual link diagram is a decorated immersion of n copies of S 1 with two types of crossings: classical and 1

2 virtual. Classical crossings are indicated by over/under markings and virtual crossings are indicated by a solid encircled X. An example of a virtual link diagram is shown in figure 1. Figure 1: Kishino s knot Two virtual link diagrams are said to be virtually isotopic if one may be transformed into another by a sequence of classical Reidemeister moves (shown in 2) and the virtual Reidemeister moves shown in figure 3. Two welded links diagrams are said to be welded equivalent if one may be transformed in the other by a sequence of Reidemeister moves (classical and virtual) and the upper forbidden move (shown in figure 4). The forbidden move is not allowed during virtual isotopy. I. II. III. Figure 2: Classical Reidemeister moves I. II. III. IV. Figure 3: The virtual Reidemeister moves In this paper, we apply a variety of results from the area of virtual knot theory. We will extend the definition of link homotopy to virtual and welded link diagrams, and Milnor s µ invariants for virtual and welded links. 2

3 Figure 4: The upper forbidden move We define two virtual link diagrams to be virtually homotopic if one diagram may be transformed into the other by virtual isotopy and self crossing change. (By self crossing change, we mean changing the over/under markings at a crossing between two segments of the same link component.) Two welded diagrams are welded homotopic if one diagram may be transformed into the other by a sequence of classical and virtual Reidemeister moves, the forbidden move, and self crossing change. Homotopic classical link diagrams (and braids) have been studied extensively, starting with Milnor [13] [14], Goldsmith [6], and more recently Habeggar and Lin [7]. There are significant differences between the study of homotopy classes in the classical case and the virtual case. We introduce these differences by examining the homotopy class of several virtual and welded knot diagrams. We first consider virtual knot and link diagrams. To distinguish homotopy classes of virtual knot diagrams, we recall the category of flat virtual knot diagrams. A flat virtual knot diagram is a decorated immersion of S 1 into the plane with two types of crossings: flat crossings and virtual crossings. (Flat crossings are indicated by an unmarked X.) Two flat diagrams are equivalent if one may be transformed into another by a sequence of flat Reidemeister moves and virtual Reidemeister moves. The flat classical Reidemeister moves are shown in figure 5. The flat versions of virtual move IV. and the forbidden move are shown in figure 6. I. II. III. Figure 5: Flat Reidemeister moves We map the category of virtual knot diagrams into the category of flat virtual knot diagrams by flattening all the classical crossings in the diagram as indicated in figure 7. 3

4 IV. F. Figure 6: The flat virtual Reidemeister move and the flat forbidden move Figure 7: Mapping into the flat knot diagrams We observe that equivalence classes of homotopic virtual knot diagrams are in 1-1 correspondence with equivalence classes of flat virtual knot diagrams. The existence of non-trivial flat virtual diagrams demonstrates that not all virtual knot diagrams are homotopy equivalent to the unknot. (Recall that all classical knot diagrams are homotopic to the unknot.) In fact, we prove the following theorem: Theorem 1.1. The category of flat virtual link diagrams is an invariant image of the category of homotopic virtual link diagrams. In the case of knot diagrams, the category of flat virtual knot diagrams is equivalent to the category of homotopic virtual link diagrams. Proof: Let L and L be homotopic virtual link diagrams. Then there is a finite sequence of Reidemeister moves and self crossing changes transforming L into the L. The Flat version of this sequence may be applied to Flat(L). This demonstrates that the category of flat virtual links is an invariant image of the category of homotopic virtual link diagrams. We restrict our attention to knot diagrams. Let K and K be virtual knot diagrams. Suppose that Flat(K) is equivalent to Flat(K ). Then there exists a sequence of flat and virtual Reidemeister moves transforming F lat(k) into Flat(K ). Each step of the sequence may be realized as a virtual or classical Reidemeister move by switching the classical crossings appropriately. Hence K is homotopic to the K. We immediately obtain the following corollary. Corollary 1.2. A virtual knot diagram K is homotopic to the unknot if and only if Flat(K) is equivalent to the unknot. 4

5 Note that as a result, Kishino s knot is not equivalent to the unknot. It was shown in [11] that there is no sequence of flat moves that transforms a flat Kishino s knot into the unknot. Theorem 1.3. If K is a welded knot then K is homotopic to the unknot. Proof:Two welded knots are equivalent if one can be transformed into the other by a sequence of virtual and classical moves, including the forbidden move. Note that in the forbidden move, two classical crossings pass over a virtual crossing. We can switch the crossings on this move, so that two classical crossing pass under a virtual crossing. Using both the under and over version of the forbidden move, any virtual knot can be unknotted [5]. The category of flat virtual links (more than one component) is not equivalent to homotopy classes of virtual links. We can not flatten crossings between the components. Similarly, the approach using both forbidden moves can not be applied to welded link diagrams. To unknot and unlink a link diagram, we may need to switch crossings between two different components of the link diagram. In the next section, we generalize Milnor s link group to virtual and welded link diagrams. 2 Link Groups In [13], Milnor defined the link group of a classical link and we extend this definition to virtual and welded link diagrams. Remark 2.1. The link group is also referred to as the reduced group in [7] and [13]. Let L be a virtual link diagram with n components: K 1, K 2,...K n. Recall that the fundamental group of the virtual link digram, π 1 (L), is a free group modulo relations determined by the classical crossings. We compute the fundamental group by assigning generators, a i1, a i2,...a ik, to the segments (of the i th component) that initiate and terminate at a classical crossing. We then record relations, r 1, r 2,...r m, that are determined by each crossing, as shown in figure 8. The fundamental group of a link L is: π 1 (L) = {a 11, a 12,...a 1k, a 21...a nˆk r 1, r 2,...r m } 5

6 a b a b a b 1 b c b = a c 1 b c b = a c no relation Figure 8: Fundamental group relations The commutators of the generators for the i th component have the form: a ij a ik a 1 ij a 1 ik and are denoted [a ij, a ik ]. We use A to denote the smallest normal subgroup generated by the commutators formed by the meridians of a fixed component: [a ij, b ik ] from all components (i {1, 2,...n}) of the diagram. The link group of L, denoted as G(L), is the fundamental group, π 1 (L), modulo A (G(L) = π 1 (L)/A). Milnor proved the following theorem. Theorem 2.1. Let L be a classical link diagram and let ˆL be the trivial link. If G(L) is isomorphic to G(ˆL), via an isomorphism preserving the conjugate classes corresponding to meridians, then L is homotopic to the trivial link. Proof: See [13]. For virtual link diagrams, the requirement that G(L) = G(ˆL) is necessary but not sufficient. (Kishino s knot demonstrates that two non-homotopic virtual links may have the same link group.) However, we can prove this theorem using the Wirtinger presentation and extend the result to virtual link diagrams. Theorem 2.2. If two virtual(welded) link diagrams, L and ˆL, are homotopic then G(L) = G(ˆL). Proof: The fundamental group is invariant under the Reidemeister moves. We need only prove that the link group is invariant under crossing change. We examine the case when two diagrams differ by exactly one crossing, as illustrated in figure 9. In this figure, the dotted line indicates that both segments of the crossing are in the same component. (Alternate choices for this connection are possible, but the arguments are similar to the one given below.) From the crossing, L, in figure 9 we obtain the following relationships: a = c bc = ad. 6

7 d a d a c b c b L L s Figure 9: Invariance under crossing change Since the segments b and c are part of the same component, we observe that b = gcg 1 where g π 1 (L). Substituting this equation into bc = ad, we compute that b = cdc 1. Since d and c are meridians of the same component, we observe that b d in G(L). In the diagram, L s, from figure 9: b = d da = bc Rewriting the last equation, we observe that bcb 1 = a. Note that b and c are meridians of the same component, so that c a in G(L s ). The two diagrams only differ at a single crossing and we conclude that G(L) and G(L s ) are equivalent. Many non-trivial virtual link diagrams have a fundamental group equivalent to the fundamental group of the unknot; indicating that link group does not determine the virtual homotopy class. For example, Kishino s knot has a trivial link group (since π 1 (K) = Z) but Kishino s knot is not virtually homotopic to the unknot. However, Kishino s knot is welded homotopic to the unknot. For welded link diagrams we make the following conjectures: Conjecture 2.3. If L is welded link diagram and π 1 (L) Z... Z then L is welded equivalent to the unlink. Conjecture 2.4. If L is a welded link diagram and G(L) Z... Z then L is homotopic to the unlink. We now extend Milnor s µ invariants by utilizing Milnor s original definition of the invariant. 7

8 3 The µ Invariants We extend Milnor s µ and µ invariants to k component virtual (and welded) link diagrams by computing the representatives of the longitude of each component in the Chen group: π 1 (L)/π 1 (L) n. (The classical analog of this proof is given in [14].) We then apply the Magnus expansion to to the longitude of each component. For each component i, we obtain a polynomial, P i, in k variables. We then compute the modulus of each polynomial, producing the µ invariants. Finally, we will show that this invariant unchanged by link homotopy. Let ˆL be an k component virtual (or welded) braid diagram with all strands oriented downwards such that the closure of ˆL is the link diagram L. Label the uppermost meridian of these components as: a 10, a 20, a 30,...a k0. Using the Wirtinger presentation, we can compute π 1 (L). Let a i0, a i1,...a iji be the generators of the meridians of the i th component. Then L has a presentation of the form: π 1 (L) = {a 10, a 11...a 1j1, a 20, a 21...a k,mk r i1 a i0 r 1 i1 = a i1,...r ij a ij r 1 ij = a ij+1, a i0 = a iki for i 1, 2,...k} where r ij π 1 (L). Let l ij = r i0 r i1...r ij then a ij = l ij a i0 l 1 ij. Now, let l i denote l iki and a i denote a i0. Then a i = l i a i l 1 i. Each l i is word consisting of conjugates of the terms a i and represents the longitude of the longitude of the i th component. Let A denote the free group generated by {a 1, a 2,...a n }. We may rewrite the presentation of the fundamental group as: {a 1, a 2,...a n [a i, l i ] = 1}. Let A 1 = A and A n = [A, A n 1 ] (the n th commutator subgroup). In this context, we will denote the Chen group, π 1 (L)/π 1 (L) (n), as G(n). A presentation of G(n) for the link L is given by: {a 1, a 2...a n [a i, l i ] = 1, A n }. We will use the following lemma to simplify the presentation of the Chen group. Lemma 3.1. Let g be an element of A. Let g ĝ modulo A n 1 then in A/A n. gbg 1 b 1 ĝbĝ 1 b 1 8

9 Proof: Let gbg 1 b 1 ĝbĝ 1 b 1 modulo A n 1. If ĝ 1 gbg 1 ĝb 1 is not equivalent to 1 modulo A n then ĝ 1 g is not an element of A n 1. This contradicts our assumption. Let l (n 1) i denote a representative of l i modulo A n 1 then G(n) = {a 1, a 2,...a n [a i, l (n 1) i ] = 1, A n }. Note that G(1) = 1, the trivial group. The group G(2) is the abelianization of the fundamental group, since any conjugate of a i a i modulo A 2. We have now demonstrated that the Chen group of any n component link diagram has a standard presentation: Proposition 3.2. Let L be an k component link diagram. Then we may write the presentation of n th Chen group as: G(n) = {a 1, a 2...a k [a i, l (n 1) i ] = 1, A n }. (1) where w i is a reduced word in A/A n 1. We now apply the Magnus expansion to the group G(n). The Magnus expansion is a homomorphism ψ such that: ψ : G(n) Z[x 1, x 2,...x k ] where Z[x 1, x 2,...x k ] is a non-commutative ring. We define the Magnus expansion on G(n) by specifying the generators: ψ(a i ) = 1 + x i ψ(a 1 i ) = 1 x i + x 2 i x3 i... Notice that ψ(a i a 1 i ) = ψ(1) = 1 but ψ(a i a j ) ψ(a j a i ) since the x i do not commute. For convenience, we will denote l (n 1) i as w i for the remainder of this paper. The image of the longitude of the i th component, w i, can be written as a polynomial in n variables. ψ(w i ) = 1 + Σµ(i 1,...i s, w i )x i1 x i2...x is. (2) Let J is a proper subset of {i 1, i 2,...i s }. We define (i 1, i 2,...i s, w i ) as follows: { the g.c.d. of µ{j, w i } if i / {i 1, i 2,...i s }, (i 1, i 2,...i s, w i ) = 0 if i {i 1, i 2,...i s }. 9

10 Then µ(i 1, i 2,...i s, w i ) Z/ (i 1, i 2,...i s, w i )Z. We use to define an ideal D i,n. For any w i G(n), the Magnus expansion of w i ( ψ(w i )) modulo D i,n determines the µ invariants. That is, µ(j, w i ) Z[x 1, x 2,...x n ]/D i,n. Let v = cx i1 x i2...x is. Then v D i,n if {i 1, i 2,...i s } n c 0 modulo (i 1, i 2,...i s, w i ) i {i 1, i 2,...i s } We observe that if µ(j, w i ) 0 then the cardinality of J is less than n ( J < n). Remark 3.1. The definition of (i 1, i 2,...i s, w i ) in the virtual (welded) case differs from the classical case. In the classical case, if i {i 1, i 2,...i s } then µ(i 1, i 2,...i s, w i ) = 0. We will discuss the implications of this fact in sections 5 and 6. We need several facts about this ideal in order to prove that Z [x 1, x 2,...x k ] modulo D i,n is a homotopy invariant. We first prove the following technical lemmas about D i,n. Lemma 3.3. If v = cx i1 x i2...x is is an element of D i,n, that is, v is equivalent to 0 in Z[x 1, x 2,...x k ]/D i,n then vx j and x j v are elements of D i,n. Proof: Let v = cx i1 x i2... x is. If j = i then vx j and x j v are elements of D i,n. If s n 1 then x j v and vx j are elements of D i,n. If s < k 1, we observe that c 0 modulo (i 1, i 2,...i s, w i ). Note that {i 1, i 2,...i s } is a subset of {i 1, i 2,...i s, j}. As a result, (i 1, i 2,...i s, j, w i ) divides (i 1, i 2,...i s ). Then (i 1, i 2,...i s, j, w i ) divides c and vx j D i. Similarly, we may argue that x j v D i We have demonstrated that D i,n is a two sided ideal. Lemma 3.4. If m A n then ψ(m) 1 modulo D i,n. Proof: To demonstrate that ψ(m) D i,n, we will show that each nonconstant term has degrees greater than or equal to n. We consider m A 2. Without loss of generality, we assume that m = [a i, a j ]. Now, ψ(m) = (1 + x i )(1 + x j )(1 x i + x 2 i x 3 i...)(1 x j + x 2 j x 3 j...) (3) 10

11 Let v i = (1 x i + x 2 i x 3 i...) v j = (1 x j + x 2 j x3 j...) Rewriting equation 3: ψ(m) = (1 + x i + x j + x i x j )v i v j = (1 + x j v i + x i x j v i )v j = (1 + x j + ( x j x i + x j x 2 i x j x 3 i...) + x i x j v i )v j = 1 + ( x j x i + x j x 2 i x jx 3 i...)v j + x i x j v i v j. All non-constant terms have degrees greater than or equal to two. Let m A n 1, ψ(m) = 1 + M and ψ(m 1 ) = 1 + ˆM. By assumption, all terms in M and ˆM have degree greater than or equal to n 1. Note that (1+M)(1+ ˆM) = (1 + ˆM)(1 + M) = 1. Now, without loss of generality, we may assume that all elements of A n have the form: We compute that [m, a i ] where m A n 1. ψ([m, a i ]) = (1 + M)(1 + x i )(1 + ˆM)(1 x i + x 2 i x3 i...) Letting v i denote 1 x i + x 2 i x 3 i..., we multiply: ψ([m, a i ]) = (1 + M + x i + x i M + ˆM + M ˆM + x i ˆM + xi M ˆM)v i = (v i + x i v i + Mx i v i + x i ˆMvi + Mx i ˆMvi ) = 1 + Mx i v i + x i ˆMvi + Mx i ˆMvi. Since each term in M and ˆM has degree greater than or equal to n 1, we observe that ψ([m, a i ]) 1 modulo D i,n. 4 Invariance of µ Under Homotopy We first prove that µ is a link invariant. The fundamental group is invariant under virtual isotopy and the upper forbidden move. As a result, G(n) is also invariant under virtual isotopy and the upper forbidden move. We need only show that the image of G(n) is invariant under change of base point and invariant under multiplication by elements of A n and [a i, w i ]. 11

12 Lemma 4.1. Let G(n) denote the Chen group, π 1 (L)/π 1 (L) n, of a link diagram L. Let w i = l (n 1) i and let a j represent the image of the generator of j t h component in G(n). 1. ψ(a j w i a 1 j ) = ψ(w i ) and ψ(a j a i a 1 j ) = ψ(a i ) (Invariance under conjugation by a j, representing a change of base point). 2. ψ([a j, w j ]) = 1 ( Verification of [a j, w j ] = 1). 3. ψ(m) = 1 for m A n (Verification of the identity m 1 for m A n ). Proof of Part 1: We compute ψ(a j a i a 1 j ). ψ(a j a i a 1 j ) = (1 + x j )(1 + x i )(1 x j + x 2 j x3 j...) = 1 + (x i x i x j + x i x 2 j...) + (x jx i x j x i x j + x i x 2 j...) Note that the term x i has coefficient 1. Since {i} is a proper subset of {j, i} and {j, i}, it follows that ψ(a j a i a 1 j ) ψ(a i ) mod D i,n. It immediately follows that ψ(w i ) ψ(a j w i a 1 j ). Proof of Part 2: We now demonstrate that ψ([a j, w j ]) = 1. From part 1, we note that Now, ψ(a j w j a 1 j ) = ψ(w j ). ψ([a j, w j ]) = ψ(w j (w j ) 1 ) = ψ(1). Proof of Part 3: See Lemma 3.4. Theorem 4.2. The term µ(x i1, x i2,...x is, w i ) is a link invariant. Proof: We note that the fundamental group of a virtual (welded) link diagram is invariant under the Reidemeister moves and the extended Reidemeister moves. To show that this is a link invariant, we need only apply Lemma 4.1. This demonstrates that µ is not altered by a change of base point or multiplication by elements equivalent to 1. We now consider the effect of crossing changes on the µ invariants. 12

13 Lemma 4.3. Let L be a k component link, and let w i represent the longitude of the i th component. 1. The Magnus expansion of w i is unchanged when multiplied by an element of the normal subgroup generated by the [a j ] 2 (a crossing change on the j th strand). 2. The Magnus expansion of w i is unchanged when multiplied by an element of the group generated by conjugates of a i, the generator of the meridian of the i th component. (a crossing change on the i th strand). To demonstrate that µ is a homotopy invariant, we must prove that the image of w i in Z[x 1, x 2,...x n ]/D i,n is invariant under crossing change. We consider the effect of a self crossing change on w i. Suppose that a crossing change occurs on the j th component. Locally, the meridians of the j th arc are replaced by a different conjugate of a j. Let w i = b 1 b 2...b m. Let b 1 = ca j c (where c is a conjugate of a j, indicating a self-crossing). The term b 1 represents the i th component under passing the j th strand. After a self-crossing change, w i = ˆb 1 b 2...b n and ˆb 1 = ĉa j ĉ 1 where ĉ is a conjugate of a j. To replace b 1 with ˆb 1, we multiply w i by ĉa j ĉ 1 a 1 j a j ca 1 j c 1 and element of the commutator subgroup generated by a j (elements of the form: [k, ˆk] where both k and ˆk are conjugates of a j ). We now consider a crossing change on the i th strand. Let w i = b 1 b 2... b m and suppose that b 1 represents the i th strand under passing itself, that is b 1 = ca ±1 i c 1. After the crossing change, w i = b 2...ˆb k...b m, where ˆb k is the new over crossing (generated by the crossing change). Note that ˆb k = ĉb 1 1 ĉ 1, and to replace b 1 with ˆb k,we multiply w i by (b 2... b k 1 ĉb 1 1 ĉ 1 b 1 k 1...b 1 2 )b 1 1, an element of the subgroup generated by conjugates of a i. Therefore, we need only show that w i is invariant under multiplication by an element of a i, the subgroup generated by conjugates of a i. Proof of Lemma 4.3, Part 1: Suppose X is an element of [a j ] 2. Then we may assume that X = [ca j c 1, a 1 j ] = [c, a j ][c, a 1 j ]. Note that [c, a j ] and [c, a 1 j ] are elements of A 2. Hence, each non-constant term in ψ([c, a j ]) and ψ([c, a 1 j ]) has degree 2 or greater and each term has contains at least two copies of x j since ψ(c) = 1 + x j.... Now, each non-constant term of ψ(x) has degree 2 or higher and contains at least two copies of x j. However, since µ(j, w i ) 0 if J contains duplicate entries, we see that ψ(xw i ) ψ(w i ). 13

14 Proof of Lemma 4.3, Part 2: Let ψ(w i ) = 1+W. We may assume that y = ca i c 1. Applying Lemma Let ψ(y) = 1+Y, where each term in Y has at least one copy of x i, so that 1+Y 1 D i,n. Now, ψ(yw i ) = (1+Y )(1+W) so that ψ(yw i ) = 1 + Y + W + Y W 1 + W modulo D i,n We apply this lemma to prove that µ(x i1, x i2,...x is, w i ) is a link homotopy invariant. Theorem 4.4. The term µ(x i1, x i2,...x is, w i ) is a homotopy link invariant. Proof: We apply lemma 4.3 to demonstrate that µ(x i1, x i2,...x is, w i ), obtained from the Magnus expansion of the reduced representative of the i th longitude, is unchanged by a self crossing change. Combined with theorem 4.2, this demonstrates that µ is a link homotopy invariant. 5 Computational Examples In this section, we compute some simple examples using the Magnus expansion. We then apply Polyak s skein relation [15] for the µ invariant to the same examples. The proof of this skein relation appeals directly to the classical proof of µ invariance, which uses the Magnus expansion. The relationship between the Magnus expansion and Fox s free calculus (applied to the fundamental group), allows us to apply Polyak s skein relation to virtual and welded links. Remark 5.1. Although we can use Polyak s results to compute the µ invariant, we can not use his result to directly determine the µ invariants. To compute the µ invariants we need to reduce modulo D i,n. We should also note that the closure of two inequivalent braids may result in two welded (virtual) braids. We recall the skein relation given by Polyak in [15] in figure 10. We refer to a crossing between the strand l + and L k or a crossing between the 14

15 Case R: l + Lk l Lk l l0 Case L: L k l L l l l + k 0 (Dashed lines indicate deleted edges.) Figure 10: Polyak s skein relation strand l and L k. In case R, the µ invariant is determined by the following equations: µ k (l + ) µ k (l ) = 1 µ i1...i nk(l + ) µ i1...i nk(l ) = µ i1...i n (l ) µ ki1...i n (l + ) µ ki1...i n (l ) = µ i1...i n (l 0 ) µ i1...i k 1 ki k+1...i n (l + ) µ i1...i k 1 ki k+1...i n (l ) = µ i1...i k 1 (l )µ ik+1...i n (l 0 ) In case L, the µ invariant is determined by: µ k (l ) µ k (l + ) = 1 µ i1...i nk(l ) µ i1...i nk(l + ) = µ i1...i n (l ) µ ki1...i n (l ) µ ki1...i n (l + ) = µ i1...i n (l 0 ) µ i1...i k 1 ki k+1...i n (l ) µ i1...i k 1 ki k+1...i n (l + ) = µ i1...i k 1 (l )µ ik+1...i n (l 0 ) 5.1 Virtual Hopf Link The closure of the braid shown in figure 11 is the virtual Hopf link. We compute the fundamental group of the virtual Hopf link is generated by a and b modulo the following relation: a = bab 1 The longitude of the strands is given by: w a = b w b = 1 15

16 a b Figure 11: Virtual Hopf link We apply the Magnus expansion (sending b 1 + b and a 1 + a ) to the longitudes: ψ(b) = 1 + b and ψ(1) = 1. There are no proper subsets of {b} and {a}. As a result, µ(b, w a ) = 1 and µ(a, w b ) = 0. Note that link(b, a) = 1 and that link(b, a) = 0. We apply the skein relation to the virtual Hopf link and observe that it is correct. µ(l, L + ) µ(l, L ) = Virtual Link Example The closure of the of the braid shown in figure 12 is a two component virtual link. Observe that link(b, a) = 1 and link(a, b) = 2. The fundamental group of the link is generated by a and b. The relations in fundamental group are: a = bab 1 b = ba 2 ba 2 b 1. The longitudes of the strands are: w a = b and w b = ba 2. 16

17 a b Figure 12: Virtual link example Applying the Magnus expansion to the reduced longitudes: ψ(w a ) = 1 + b ψ(w b ) = (1 + b)(1 + a) 2 = 1 + b + 2a + 2ba + ba 2. Since there are two strands, all elements with degree greater than 1 are in D a,2 and D b,2. Now, µ(b, w a ) = µ(b, w a ) = 1 and µ(a, w b ) = µ(a, w b ) = 2. Applying the skein relation, we again verify that µ(l, L + ) µ(l, L 1 ) = Modification of the Borromean Links The closure of the braid shown in figure 13 is obtained by adding a classical and virtual twist to the Borromean links on 2 nd and 3 rd strands. If the 2 nd or 3 rd strand is removed from the closure of this braid, we obtain an unknotted 2 component unlink. If the first strand is removed, we obtain a 2 component link. The fundamental group of this link is generated by a, b, c. We obtain the following relations: a = a 1 b 1 ac 1 a 1 bacac 1 a 1 b 1 aca 1 ba b = cac 1 a 1 baca 1 c 1 c = cac 1 b 2 aca 1 c 1 a 1 b 2 aca 1 c 1. 17

18 Figure 13: Modification of the Borromean links The longitude of these components: w a = a 1 b 1 ac 1 a 1 bac w b = cac 1 a 1 w c = cac 1 a 1 b 2 aca 1 c 1 a 1 b 1 a. We now reduce the longitudes modulo A 2. w a (a 1 b 1 ac 1 a 1 ba)(c 0 ) b 1 c 1 bc modulo A 2 w b cac 1 a 1 w c (cac 1 a 1 b 2 aca 1 c 1 )(a 1 b 1 a) b 2 a 1 b 1 a We now apply the Magnus expansion to the reduced form of w a, that is: b 1 c 1 bc. ψ(b 1 c 1 bc) = (1 b + b 2...)(1 c + c 2...)(1 + b)(1 + c) = (1 b + b 2... c + bc b 2 c... + c 2 bc 2 + b 2 c 2...)(1 + b + c + bc) = 1 b + b 2... c + bc b 2 c... + c 2 bc 2 + b 2 bc b b 2 cb + c bc... + bc) All terms with degree greater than 2 are in D a,3. Collecting coefficients, we 18

19 observe that: µ(b, w a ) = 0 µ(c, w a ) = 0 µ(b, c, w a ) = 1 µ(c, b, w a ) = 1 The computation for ψ(w b ) is similar. Hence: µ(a, w b ) = 0 µ(c, w b ) = 0 µ(a, c, w b ) = 1 µ(c, a, w b ) = 1 We compute ψ(w c ) = ψ(b 2 a 1 b 1 a). Note that (1 + b) 2 (1 a + a 2...)(1 b + b 2...)(1 + a) = 1 + b + ab ba b 2... From this equation, we obtain: µ(a, w c ) = 0 µ(b, w c ) = 1 µ(a, b, w c ) = 1 µ(b, a, w c ) = 1 Note that since µ(i, w j ) = 0 for all i, j except for the case where i = b and j = c, that µ(i, k, w j ) = µ(i, k, w j ). We now apply Polyak s skein relation to this braid, as shown in figure 14. We apply case L of the skein relation to determine that: We also obtain: µ ab (l ) µ ab (l + ) = µ a (l )µ ba (l ) µ ba (l + ) = µ a (l 0 ) µ b (l ) µ b (l + ) = 1 which may be verified by computing link(b, l ) and link(b, l + ). Note that µ ab (l + ) = 0, µ ba (l + ) = 0, µ a (l ) = 1, and µ a (l 0 ) = 1. Hence, µ ab (l ) 0 = 1 µ ba (l ) 0 = 1 These results agree with those obtained from the Magnus expansion. 19

20 a b l a b l a a + l l 0 Figure 14: Application of Polyak s skein relation, case L 5.4 A Virtual Link with Linking Number Zero The closure of the link shown in figure 15 is obtained by changing two classical crossings in a link homotopic to the identity into virtual crossings. Remark 5.2. Note that link(a, b) = 0 and link(b, a) = 0 for this link. A classical, 2 component link with linking number zero is homotopically trivial. If this link is homotopic to a classical link then it is homotopic to the unknotted, 2 component unlink. b a Figure 15: Modification of a link homotopic to the identity 20

21 The fundamental group is generated by a and b with the relations: a = b 1 aba 1 baba 1 b 2 ab 2 ab 1 a 1 b 1 aba 1 b b = aba 1 baba 1. From these relations, the longitudes of the components are: w a = b 1 aba 1 baba 1 b 2 w b = aba 1. We apply the Magnus expansion to w a : ψ(w a ) 1 mod D a,2 Applying the Magnus expansion to w b : ψ(w b ) = 1 + (b ba + ba 2 ba 3...) + (ab aba + aba 2 aba 3...) We conclude that µ(b, w b ) = 1. Note that µ(b, w b ) 0 by definition. In the next section, we demonstrate that µ(i, w i ) = 0 for all components of a classical link and a link homotopic to a classical link, allowing us to conclude that this link is not homotopic to any classical link. 6 The µ invariant and virtual link diagrams In this section, we consider the invariant µ(a, w b ) for a b and a = b. We recall that the definition of linking number for virtual link diagrams. The sign of a crossing, c is determined by the local orientation as shown in figure Figure 16: Crossing sign 21

22 We define the link(b,a) to be: sign(c) (4) c where c is a crossing such that the strand a passes under strand b. Note that link(b, a) is not necessarily equivalent to link(a, b). Remark 6.1. In the classical case, link(a, b) = link(b, a) for a = b and a b. Remark 6.2. In [14], Milnor observes that µ(j, k, w i ) = µ(k, j, w i ) and develops the idea of shuffles, symmetry relations on the µ invariants. The symmetry relations need to be re-examined for virtual and welded diagrams. We will demonstrate that µ(b, w a ) = µ(b, w a ) = link(b, a) for a 0. Theorem 6.1. Let L be a link with at least two components, a and b. Then µ(b, w a ) = µ(b, w a ) = link(b, a). Proof: Consider the strand a. If strand strand passes under strand b with a positive orientation k times and with a negative orientation j times, then link(b, a) = k j. We compute the longitude of a, w a. When a i undercrosses b j, we note that a i+1 = (b c )a i (b c ) 1, where b c is a conjugate of b. Hence, for each positive undercrossing, w a contains a b and for each negative crossing a b 1. Now, applying the Magnus expansion to w a, we see that k factors have the form (1 + b) and j factors have the form (1 b + b 2 b 3...). All other terms are of the form (1 + c) or (1 c + c 2 c 3...). To compute the coefficient of b in the Magnus expansion, (1 + b) k (1 b + b 2 b 3...) j = (1 + kb +...)(1 jb +...) (5) = 1 + (k j)b +... (6) Hence µ(b, w a ) = k j. We observe that {b} does not contain a proper subset. Then µ(b, w a ) = k j. We now consider µ(a, w a ). In the classical case, µ(a, w a ) = 0 for all components. For the example given in section 5.4, µ(a, w a ) = 1. Theorem 6.2. Let L be a classical n component link diagram. Then µ(i, w i ) = 0 for any component i of the braid whose closure is L. 22

23 Proof: Let L be a classical n component link diagram. Then L is homotopic to the closure of a pure classical braid and does not have any self-crossings. Consider the generator of the meridian of component i, a i. The longitude of this component, w i has the form: w i = l 1 l 2...l n where each l k is conjugate of the generator of a component that underpasses component i. Hence, l k is not a conjugate of a i for all k {1, 2,...n}. The meridian a i occurs only as a part of a conjugate pair in the longitude. Hence, µ(i, w i ) = 0. Corollary 6.3. Let L be a link diagram. Let i be a component of L with longitude w i. If µ(i, w i ) 0 then L is not a classical link diagram. Proof: The proof of this corollary is immediate. Remark 6.3. We can amend Polyak s skein relation to include µ(i, w i ). Let L be an n component diagram and let w + i represent the i th component with a fixed positive self crossing. Let w i represent the diagram with the fixed self crossing changed to a negative crossing. Then µ(i, w + i ) µ(i, w i ) 0 modulo 2. Acknowledgments: The authors would like to thank Xiao Song Lin for the suggestion to investigate virtual link homotopy. It gives the second author great pleasure to acknowledge support from NSF Grant DMS This research was performed while the first author held a National Research Council Research Associateship Award jointly at the Army Research Laboratory and the United States Military Academy. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright annotations thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the United States Military Academy or the U.S. Government. (Copyright 2006.) References [1] Artin, E., Theory of braids, Ann. of Math. (2) 48, (1947), p [2] Bardakov, Valerij G., The virtual and universal braids. Fund. Math. 184 (2004), p

24 [3] Fenn, Roger, Techniques of geometric topology. London Mathematical Society Lecture Note Series, 57. Cambridge University Press, Cambridge, [4] Fenn, Roger; Kauffman, Louis H.; Manturov, Vassily O., Virtual knot theory unsolved problems. Fund. Math. 188 (2005),p [5] Finite Type Invariants Goussarov, Mikhail; Polyak, Michael; Viro, Oleg Finite-type invariants of classical and virtual knots. Topology 39 (2000), 5, p [6] Goldsmith, Deborah Louise, Homotopy of braids in answer to a question of E. Artin. Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973), p Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974 [7] Habegger, Nathan and Lin, Xiao-Song, The classification of links up to link-homotopy. J. Amer. Math. Soc. 3 (1990), 2, [8] Kamada, Seiichi, Invariants of virtual braids and a remark on left stabilizations and virtual exchange moves. Kobe J. Math. 21 (2004), 1-2, p [9] Kauffman, Louis H., Virtual Knot Theory, European Journal of Combinatorics, 20 (1999), 7, p , [10] Kauffman, Louis H. and Lambropoulou, Sofia, Virtual braids. Fund. Math. 184 (2004), p [11] Kadokami, Teruhisa, Detecting non-triviality of virtual links. J. Knot Theory Ramifications 12 (2003), 6, p [12] Kishino, Toshimasa and Satoh, Shin, A note on non-classical virtual knots. J. Knot Theory Ramifications 13 (2004), 13, p [13] Milnor, John, Link groups. Ann. of Math. 59 (1954), 2, p [14] Milnor, John, Isotopy of Links. Algebraic geometry and topology. A symposium in honor of S. Lefschetz, p Princeton University Press, Princeton, N. J., [15] Polyak, Michael, Skein relation for Milnor s µ invariants. Algebraic and Geometric Topology. 5 (2005), p