Reidemeister moves for knots and links in lens spaces Текст научной статьи по специальности «Математика»

Вестник КемГУ № 3/1 2011 Геометрия трехмерных многообразий

УДК 515.162.8

REIDEMEISTER MOVES FOR KNOTS AND LINKS IN LENS SPACES Enrico Manfredi, Michele Mulazzani

ПРЕОБРАЗОВАНИЯ РЕЙДЕМЕЙСТЕРА ДЛЯ УЗЛОВ И ЗАЦЕПЛЕНИЙ В

ЛИНЗОВЫХ ПРОСТРАНСТВАХ

E. Манфреди, M. Мулаццани

We extend the concept of diagrams and associated Reidemeister moves for links in S3 to links in lens spaces, using a differential approach. As a particular case, we obtain diagrams and Reidemeister type moves for links in RP3 introduced by Y.V. Drobothukina.

В данной работе понятия диаграммы и преобразований Рейдемейстера, известные для зацеплений в S3, распространяются для зацеплений в линзовых пространствах. В частности, получены диаграммы и преобразования типа Рейдемейстера для зацеплений в RP3, введенные ранее Ю.В. Дро-ботухиной.

Ключевые слова: преобразования Рейдемейстера, линзовые пространства, трехмерные многообразия.

Keywords: Reidemeister moves, lens spaces, 3-manifolds.

1. Preliminaries

In this paper we work in the Diff category (of smooth manifolds and smooth maps). Every result also holds in the PL category, and in the Top category if we consider only tame links.

Definition 1. Let X and Y be two smooth manifolds.

A smooth map f : X ^ Y is an embedding if the differential dxf is injective for all x e X and if X and f (X) are homeomorphic. As a consequence, X and f (X) are diffeomorphic and f (X) is a submanifold of Y.

An ambient isotopy between two embeddings l0 and li from X to Y is a smooth map H : Y x [0,1] ^

Y such that, if we write at each t e [0,1], H(y,t) = ht(y), then ht : Y ^ Y is a diffeomorphism, h0 = Idy and l1 = h1 o l0.

Definition 2. (Links) A link in a closed 3-manifold M3 is an embedding of v copies of S1 into M3, namely it is l : S1 LI...US1 ^ M3. A link can also be denoted by L, where L = l(S1 U ... U S1) c M3. A knot is a link with v = 1.

Two links L0 and L1 are equivalent if there exists an ambient isotopy between the two embeddings l0 and l1 .

Definition 3. (Lens spaces) Let p and q be two integer numbers such that gcd(p, q) = 1 and 0 < q < p. Consider B3 := {(x1 ,x2,x3) e R3 | x2 + x| + x2 < 1} and let E+ and E_ be respectively the upper and the lower closed hemisphere of dB3. Call B2 the equatorial disk, defined by the intersection of the plane x3 = 0 with B3. Label with N and S respectively the Tnorth poley (0,0,1) and the Tsouth poley (0,0, -1) of B3.

Let gp,q : E+ ^ E+ be the rotation of 2nq/p around the x3 axis as in Figure 1, and let f3 : E+ ^ E_ be the reflection with respect to the plane x3 =0. The lens space L(p, q) is the quotient of B3 by the

equivalence relation on dB3 which identify x e E+ with f3 o gp,q(x) e E_. We denote by F : B3 ^ B3/ ~ the quotient map. Notice that on the equator dB0 = E+ n E_ there are p points in each class of equivalence.

Fig. 1. Representation of L(p, q)

It is easy to see that L(1,0) = S3 since g1j0 = Id_E+. Furthermore, L(2,1) is RP3, since we obtain the usual model of the projective space where opposite points of the boundary of the ball are identified.

Proposition 4. [1] The lens spaces L(p, q) and L(p', q') are diffeomorphic (as well as homeomorphic) if and only if p' = p and q' = ±q±1mod p.

2. Links in S3 2.1. Diagrams

One of the first tools used to study links in S3 are diagrams, that is to say, a suitable projection of the links on a plane.

Definition 5. Let L be a link in S3 = R3 U {<»}. Since L is compact, up to an affine transformation of R3, we can suppose that L belongs to intB3.

Вестник КемГУ № 3/1 2011 Геометрия трехмерных многообразий

Let p : B3 \ {N,S} ^ Bg be the projection defined in the following way: take x e B3 \ {N, S}, construct the circle (or the line) c(x) through N, x and S and set p(x) := c(x) n B2.

Take L c intB3 and project it using p|L : L ^ B2. For P e p(L), p_]1(P) may contain more than one point; in this case, we say that P is a multiple point. In particular, if it contains exactly two points, we say that P is a double point. We can assume, by moving L via a small isotopy, that the projection p|L : L ^ B0 of L is regular, namely:

1. the arcs of the projection contain no cusps;

2. the arcs of the projection are not tangent to each other;

3. the set of multiple points is finite, and all of them are actually double points.

These requests correspond to violations represented in Figure 2.

Fig. 2. Violations V1 , V2 and V3

Now let Q be a double point and consider p—1 (Q) = {P1 ,P2}. We suppose that P2 is nearer to S than P1. Take U as an open neighborhood of P2 in L such that p(U) does not contain other double points. We call U underpass. Take the complementary set in L of all the underpasses. Every connected component of this set (as well as its projection in B2) is called overpass. The underpasses are visualized in the projection by removing U from L' before projecting the link (see Figure 3). Observe that we may have components of the link which are single overpasses.

Fig. з. A link in S3 and corresponding diagram

Definition 6. A diagram of a link L in S3 is a regular projection of L on the equatorial disk B2, with specified overpasses and underpasses.

2.2. Reidemeister moves

There are three (local) moves that allow us to determine when two links in S3 are equivalent from

their diagrams. Reidemeister proved this theorem for PL links. For the Diff case a good reference is [Т], where the proof involves links in arbitrary dimensions, so it is rather complicated.

Definition 7. The Reidemeister moves on a diagram of a link L с S3 are the moves R1,R2, R3 of Figure

4.

Fig. 4. Reidemeister moves

Вестник КемГУ № З/1 2011 Геометрия трехмерных многообразий

Theorem S. [Т] Two links L0 and L1 in S3 are equivalent if and only if their diagrams can be joined by a finite sequence of Reidemeister moves R1, R2, R3 and diagram isotopies.

Proof. It is easy to see that each Reidemeister move produces equivalent links, hence a finite sequence of Reidemeister moves and isotopies on a diagram does not change the equivalence class of the link.

On the other hand, if we have two equivalent links L0 and L1 , then we have an ambient isotopy, namely: H : S3 x [G, 1] ^ S3, such that l1 = h1 о l0. At each t є [G, 1] we have a link Lt, defined by ht(l0). Thanks

to general position theory (see [Т] for details), we can assume that the projection of Lt is not regular only a finite number of times, and that at each of these times it violates only one condition.

From each type of violation a transformation of the diagram appears, that is to say, a Reidemeister move, as it follows (see Figure 5):

- from violation V1 we obtain move R1;

- from violation V2 we obtain move R2;

- from violation V3 we obtain move R3.

So diagrams of two equivalent links can be joined by a finite sequence of Reidemeister moves R1,R2, R3 and diagram isotopies. □

3. Links in RP3

3.1. Diagrams

The definition given by Drobothukina [3] of diagram for links in the projective space makes use of the model of the projective space RP3 explained in Section 1, as a particular case of L(p, q) with p = 2 and q = 1. Namely, consider B3 and identify diametrically opposed points on its boundary (let ~ be the equivalence relation), so RP3 = B3/ ~ and the quotient map is denoted by F.

Definition 9. Let L be a link in RP3. Consider L' := F-1 (L); by moving L via a small isotopy in RP3, we can suppose that:

i) L' does not meet the poles S and N of B3;

ii) L' n dB3 consists of a finite set of points.

So L' is the disjoint union of closed curves in intB3 and arcs properly embedded in B3 (i.e. only the boundary points belong to dB3).

Let p : B3 \ {N, S} ^ B2 be the projection defined in the following way: take x e B3 \ {N, S},

construct the circle (or the line) c(x) through N, x and S and set p(x) := c(x) n B2.

Take L' and project it using p\L> : L' ^ B2. For P e p(L'), p—1 (P) may contain more than one point; in this case, we say that P is a multiple point. In particular, if it contains exactly two points, we say that P is a double point. We can assume, by moving L via small isotopies, that the projection p(L') is regular, namely:

1) the arcs of the projection contain no cusps;

2) the arcs of the projection are not tangent to each other;

3) the set of multiple points is finite, and all of them are actually double points;

4) the arcs of the projection are not tangent to 3B00;

5) no double point is on dB2.

These requests correspond to violations represented in Figures 2 and 6.

Fig. б. Violations Vi and V5

Now let Q be a double point and consider p-1 (Q) = {P1,P2}. We suppose that P2 is nearer to S than P1 . Take U as an open neighborhood of P2 in L' such that p(U) does not contain other double points and does not meet dB2. We call U underpass. Take the complementary set in L' of all the underpasses. Every connected component of this

set (as well as its projection in B2) is called overpass. The underpasses are visualized in the projection by removing U from L' before projecting the link (see Figure 7).

Definition 10. A diagram of a link L in RP3 is a regular projection of L' = F-1(L) on the equatorial disk B2, with specified overpasses and underpasses.

Fig. Т. A link in L(2, 1) and corresponding diagram

We label the boundary points of the link projection, in order to show the identifications. Assume that the equator is oriented counterclockwise if we look at it from N, and that the number of boundary points is 2t. Choose a point of p(L') on the equator and call it 1 as well as the antipodal point, then following the orientation of dB2, label the points of p(L') on the equatorial circle, as well as the antipodal ones, 2,... ,t (see Figure 7).

3.2. Generalized Reidemeister moves

We want to look for an analogue of the Reidemeister moves for links in S3, in order to understand when two diagrams of links in RP3 represent equivalent links.

Definition 11. The generalized Reidemeister moves on a diagram of a link L с RP3 are the moves R1 ,R2, R3 of Figure 4 and the moves R4, R5 of Figure

S.

1 " 2

Fig. S. Generalized Reidemeister moves for projective space

Вестник КемГУ № З/1 2011 Геометрия трехмерных многообразий

Theorem 12. [3] Two links L0 and L1 in the the projective space are equivalent if and only if their diagrams can be joined by a finite sequence of generalized Reidemeister moves R1,...,R5 and diagram isotopies.

Proof. It is easy to see that each Reidemeister move produces equivalent links, hence a finite sequence of Reidemeister moves and isotopies on a diagram does not change the equivalence class of the link.

On the other hand, if we have two equivalent links L0 and L1 , then we have an ambient isotopy, namely: H : RP3 x [0,1] ^ RP3, such that l1 = h1 o l0. At each t e [0,1] we have a link Lt, defined by ht(l0). As for links in S3, using general position theory we can

assume that the projection of Lt is not regular only a finite number of times, and that at each of these times it violates only one condition.

From each type of violation a transformation of the diagram appears, that is to say, a generalized Reidemeister move, as it follows (see Figures Б and 9).

So diagrams of two equivalent links can be joined by a finite sequence of generalized Reidemeister moves R1,...,R5 and diagram isotopies. □

- from violations V1 , V2 and V3 we obtain the classic Reidemeister moves R1, R2 and R3;

- from violation Vi we obtain move R4;

- from violation V5 we obtain move R5.

Fig. 9. Regularity violations produce generalized Reidemeister moves

4. Links in L(rp,q)

4.1. Diagrams

We improve the definition of diagram for links in lens spaces given by Gonzato [4]. We can assume p > 2, since we have already seen in the previous sections the particular cases L(1,0) = S3 and L(2,1) = RP3. Consider the construction of the lens space L(p, q) = B3/ ~ we give in the preliminaries, where F is the quotient map.

Definition 13. Let L be a link in L(p,q). Consider L' := F-1 (L); by moving L via a small isotopy in L(p, q), we can suppose that:

i) L' does not meet the poles S and N of B3;

ii) L' n dB3 consists of a finite set of points.

So L' is the disjoint union of closed curves in intB3 and arcs properly embedded in B3 (i.e. only the boundary points belong to dB3).

Let p : B3 x {N, S}

B02

be the projection

defined in the following way: take x e B3 \ {N, S}, construct the circle (or the line) c(x) through N, x and S and set p(x) := c(x) n B^

Take L' and project it using pi, : L'

Bl

For P e p(L'), pj], (P) may contain more than one point; in this case, we say that P is a multiple point. In particular, if it contains exactly two points, we say that P is a double point. We can assume, by moving L

via a small isotopy, that the projection p|L, : L' ^ B2 of L is regular, namely:

1) the arcs of the projection contain no cusps;

2) the arcs of the projection are not tangent to each other;

3) the set of multiple points is finite, and all of them are actually double points;

4) the arcs of the projection are not tangent to dB00;

5) no double point is on dB2;

6) L' n dB22 = 0.

Now let Q be a double point and consider p_1 (Q) = {P1,P2}. We suppose that P2 is nearer to S than P1. Take U as an open neighborhood of P2 in L' such that p(U) does not contain other double points and does not meet dB^ We call U underpass. Take the complementary set in L' of all the underpasses. Every connected component of this set (as well as its projection in B2) is called overpass. The underpasses are visualized in the projection by removing U from L' before projecting the link (see Figure 10).

Definition 14. A diagram of a link L in L(p, q) is a regular projection of L' = F-1(L) on the equatorial disk B2, with specified overpasses and underpasses.

Вестник КемГУ № 3/1 2011 Геометрия трехмерных многообразий

Fig. 10. A link in L(9,1) and corresponding diagram

We label the boundary points of the link projection, in order to show the identifications. Assume that the equator is oriented counterclockwise if we look at it from N. Consider the t endpoints of the overpasses that come from arcs of L' that are above the equator. Label them +1,..., +t according to the orientation of dE“^. Then label the other t points on the boundary, that come from arcs of L' under the equator, as -1,..., -t, where for each i = 1,..., t, we have +i ~ —i. An example is shown in Figure 10.

We want to explain which diagram violations

arise from condition 1)-6). For conditions 1), 2) and

3) we already know that the corresponding violations are V1 ,V2 and V3 of Figure 2.

Condition 4), as in the projective case, has a corresponding violation V4. On the contrary, condition 5) does not behave as in the projective case. Indeed two diagrammatic violations arise from it (V5 and V6), as Figure 11 shows. The difference between the two violations is that V5 involves two arcs of L' that end in the same hemisphere of dB3, on the contrary V6 involves arcs that end in different hemispheres.

Fig. 11. Violations V4, V5 and V6

Finally condition 6) produces a family of is that V7,1 has the arcs of the projection identified violations called V7,1,..., directly by gp,q, while V7,k has the arcs identified by

V7,p_1 (see Figure 12). The difference between them gk q, for k = 2,... ,p — 1.

Fig. 12.Violations V7,i,V7,2, ■ ■ ■, V7p-1

It is easy to see what kind of small isotopy on L is necessary, in order to make the projection of the link regular when we deal with violations V1,... ,V6. Now we explain how the link can avoid to meet dB2 up to isotopy, that is to say, avoid Vrt1,..., V7,p_1.

We start with a link with two arcs that ends on dB2. If we suppose that the endpoints of the arcs

are connected by gp,q, (a Vr,1 violation), then we can label the endpoints B and C, in a way such that C = gp,q(B). In this case the required isotopy is the one that lift up a bit the arc ending in B and lower down the other one.

Now if we suppose that the endpoints of the arcs are connected by a power of gp,q, (a V7,k violation

Вестник КемГУ № 3/1 2011 Геометрия трехмерных многообразий

with k > 1), then we can label the points B and C such that C = gp,q (B). In this case the required

isotopy is similar to the one of the example in L(9, 1) of Figure 13. In lens spaces with q =1, the new arcs end in the faces specified by the map f3 o gp,q.

Fig. 13.Avoiding дB0 in L(9,1)

4.2. Generalized Reidemeister moves

Again, with the aim of discovering when two diagrams represent equivalent links in L(p,q), we generalize Reidemeister moves for diagrams of links.

Definition 15. The generalized Reidemeister moves on a diagram of a link L с L(p, q) are the moves R1,R2,R3 of Figure 4 and the moves R4,Rs, Re and R7 of Figure 14.

Fig. 14. Generalized Reidemeister moves

Theorem 16. Two links L0 and L1 in L(p,q) are equivalent if and only if their diagrams can be joined by a finite sequence ofgeneralized Reidemeister moves R1,... ,R7 and diagram isotopies.

Proof. It is easy to see that each Reidemeister move produces equivalent links, hence a finite sequence of Reidemeister moves and isotopies on a diagram does not change the equivalence class of the

link.

On the other hand, if we have two equivalent links L0 and L1, then we have an ambient isotopy between the two ambient spaces, namely: H : L(p, q) x [0,1] ^ L(p, q). At each t e [0,1] we have a link Lt, defined by ht(l0). Again, as for links in S3, using general position theory we can assume that the projection p(Lt) is not regular only a finite number of times, and that at each of these times it violates only one condition.

Вестник КемГУ № 3/1 2011 Геометрия трехмерных многообразий

Fig. 15. Regularity violations produce generalized Reidemeister moves

From each type of violation a transformation of the diagram appears, that is to say, a generalized Reidemeister move, as it follows (see Figures 5 and 15):

- from violations V1 , V2 and V3 we obtain the classic Reidemeister moves R1, R2 and R3;

- from violation V4 we obtain move R4;

- from violation Vs, we obtain two different moves, if the arcs L' with endpoints on the boundary are from the same side with respect to equator, then we obtain Rs, on the contrary we obtain Re;

- for condition б we have a family of violation

V? ... ,V?p-1, from which we obtain the

,R

?,p-l.

Indeed, if an arc cross the equator during the isotopy, then we have a class of moves, R7j1 = R7, R7,2,..., R7,p_1. All these moves can be seen as the composition of R7, R6, R4 and R1 moves. More precisely, the move R7,k with k = 2,... ,p — 1, can be obtained by the following sequence of moves: first we perform an R7 move on one overpass that end on the equator and the corresponding point in a small arc, then we repeat for k — 1 times the three moves R6-R4-R1 necessary to retract the small arc with same sign ending point (see an example in Figure 16).

So we can exclude R7,2,..., R7,p_1 from the move set and keep only R7 1 = R7. As a consequence, any

pair of diagrams of two equivalent links can be joined by a finite sequence of generalized Reidemeister moves R1,... ,R7 and diagram isotopies. □

References

[1] Brody, E. J. The topological classification of the lens spaces / E. J. Brody // Ann. of Math. -1960. - Vol. 71. - P. 163 - 184.

[2] Burde, G. Knots / G. Burde, H. Zieschang -Walter de Gruyter. - Berlin; New York, 2003.

[3] Drobotukhina, Y. V. An analogue of the Jones polynomial for links in RP3 and a generalization of the Kauffman-Murasugi theorem / Y. V. Drobotukhina // Leningrad Math. J. - 1991. -Vol. 2. - P. 613 - 630.

[4] Gonzato, M. Invarianti polinomiali per link in spazi lenticolari/ M. Gonzato // Degree thesis. -University of Bologna, 2007.

[5] Manfredi, E. Fundamental group of knots and links in lens spaces/ E. Manfredi // Degree thesis. -University of Trieste, 2010.

[6] Prasolov, V. V. Knots, links, braids and 3-manifolds. An introduction to the new invariants in low-dimensional topology / V. V. Prasolov, A. B. Sossinsky // Transl. of Math. Monographs. -Vol. 154. - Amer. Math. Soc.: Providence, RI, 1997.

[7] Roseman, D. Elementary moves for higher dimensional knots / D. Roseman // Fund. Math. -2004. - Vol. 184. - P. 291 - 310.

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Fig. 16. How to reduce a composite move