Closed Braids and Heegaard Splittings 1

Closed Braids and Heegaard Splittings William W. Menasco University at Bualo Bualo, New York 14214 Dedicated to Joan Birman on her 70th birthday.

Abstract In this note we will be investigating a strategy for constructing 3-manifolds that have multiple strongly irreducible Heegaard splittings that are not equivalent. This strategy combines techiques involving framed links and a calculus on closed braids developed by Joan Birman and the author.

1 Introduction Let M be a closed oriented 3-manifold. A Heegaard splitting M = H1 [P H2 consists of an orientable surface P in M , together with two handlebodies H1 and H2 into which P divides M . P is the splitting surface and the genus of the Heegaard splitting is de ned as the genus of P . Two Heegaard splittings H1 [P H2 and H10 [Q H20 are equivalent if P is isotopic to Q in M . A meridian disc of one of the handlebodies is a properly embedded disc (Di ; @Di) (Hi; P ) such that @Di is homotopically non-trivial on P . (Other discussions on such discs have used the term essential in place of meridian.) A Heegaard splitting is reducible if there are meridian discs D1 and D2 such that @D1 = @D2 . A Heegaard splitting is weakly reducible if there are meridian discs D1 and D2 such that @D1 \ @D2 = ;. It is a straight forward matter to show that reducible implies weakly

partially supported by NSF grant #DMS 9626884

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reducible. For purposes of completeness of terminology we call a Heegaard splitting irreducible if it is not reducible and strongly irreducible if it is not weakly reducible. Let L = L1 [ : : : [ Ll S 3 be a link of l components and let r = (r1 ; : : : ; rl ) be an l-tuple of the rational numbers. We will denote by L(r) the closed 3-manifold obtained by performing ri surgery on the Li component of L. In this note we will be investigating a strategy for constructing 3-manifolds that have multiple strongly irreducible Heegaard splittings. This strategy combines techniques involving framed links and a calculus on closed braids [BM2]. Speci cally, we will show that when L is represented as a closed n-braid there is a canonical way of associating a Heegaard splitting of genus n with the framed link L(r) . Moreover, for a suciently complex closed braid L of one component all but nitely many framings r on L will yield a canonical Heegaard splitting that is strongly irreducible. When L can be represented by several non-conjugate closed n-braid there arises the possibility that we will have non-equivalent Heegaard splittings coming from this canonical association. ACKNOWLEDGEMENTS|The author wishes to thank Joan Birman and Xingru Zhang for numerous conversations. The author also grateful to the referee for a number of probing comments which helped clarify the exposition.

2 Construction of Heegaard splitting Let L(r) be a framed link as previously described and suppose L can be represented as a closed n-braid with a braid axis A. More speci cally, we think of the pair (S 3; A) as being equivalent to the pair (R3 [ 1; fz ? axisg [ 1). Then there is a choice of a bration of the open cylinder R3 ? fz ? axisg by half-planes H = fH j 2 [0; 2]g, such that L transversely intersects each H . In our discussion it will be more convenient to think of bration of S 3 ? A by 2-discs than a bration of R3 ?fz ? axisg by half-planes, so through the abuse of notation we will refer to H = fH j 2 [0; 2]g as such a 2-disc bration of S 3 ? A. Next, let H1 ; H2 H be two distinct disc bers and consider the 2-sphere S0 = H1 [A H2 that is geometrically punctured 2n times and algebraically zero times by L. Notice that S0 splits S 3 into two 3-balls, B1 and B2 , and Bj [ L Bj (j = 1; 2) is a trivial tangle of n-strands. Let P S 3 ? L be the closed surfaces

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of genus n contained in B1 that is obtained from S0 by performing the n peripheral tubings along each of the n trivial strands in B1. This tubing eliminates the 2n puncture points.

Theorem 1 P is a splitting surface for a Heegaard splitting of L(r). Moreover, the

construction of P is canonical in that it is independent of choice of disc bres H1 ; H2 .

Proof. First, to see that P is canonical we need only show that if we construct a P 0

using disc bres H1 ; H2 then P 0 is isotopic to P in the complement of L. But this is easily seen since the pair (H1 ; H2 ) is isotopic to the pair (H1 ; H2 ) by a rotation about A, and a rescaling of the angle between 10 and 20 . Second, it is obvious that the construction of P leaves it bounding a handlebody H1 B1 . Thus, we need only establish the claim that H2 = L(r) ? int(H1 ) is a handlebody. For reasons having to do with encouraging a familiarity with the geometry, we will give two proofs of this claim. The rst proof will be in the special situation where r is a tuple of integers and we will show that H2 retracts onto a graph. The second proof we be in the general situation where r is a tuple of rationals. Consider a component Li L. Let fp1 ; : : : ; pk g = S0 \ Li be a cyclic ordering on Li of its puncture points with the 2-sphere S0 . Moreover, let (p1; p2) [ (p2; p3) [ : : : [ (pk ?1; p1 ) [ (pk ; p1) = Li be the arc decomposition of Li that has (pj ; pj+1) being the arc of Li \ B1 having endpoints pj and pj+1, and (pj?1; pj ) being the arc of Li \ B2 having endpoints pj?1 and pj . The genus n surface P is constructed by tubing along all of the -arcs for each component of L. This tubing construction can be speci cally described as follows. Let () be a regular neighborhood of an -arc in B1. Then () has the structure of D2 and, for each such -arc, P is obtained from S0 by removing @ D2 @B1 (producing a 2-sphere minus discs) and gluing in @D2 . So returning to the xed component Li, the above described tubing construction results in replacing disc neighborhoods of fp1 ; : : : ; pk g with tubes [f(p1; p2) @D2 ; : : : ; (pk ?1 ; pk ) @D2 g]. Next, we notice the following isotopy of P in L(r). Since the framing on Li is an integer, there is an isotopy between the tube (p1; p2) @D2 and the tube ( (p2; p3) @D2 ) [ ((p3; p4) @D2 ) [ : : : [ ((pk ?1; pk ) @D2) [ ( (pk ; p1) @D2 ) where the common boundary circles are held xed throughout the isotopy. This isotopy is achieved by pushing the tube (p1; p2) @D2 through the solid torus that is used in the Dehn surgery at Li. Thus, in L(r) the surface P |the 0

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surface obtained from S0 by tubing along arcs|is isotopic to the surface obtained from S0 by, rst, tubing along arcs f(p3; p4); : : : ; (pk ?1; pk )g|the short tubings| and, second, by tubing along the combined-arc [ (p2; p3) [ (p3 ; p4) [ : : : [ (pk ?1; pk )] |the long tubing. Notice that this long tubing will go \through" each short tubing exactly once. (In the case where Li intersect S0 twice then there is a single - and -arc and by above they are isotopic to each other.) Over all components of L we can choose n ? l short tubings and l long tubings. To extend this tubing construction to all components of L and at the same time see that the resulting surface bounds a handlebody in the complement of int(H1 ) we start with S0 = @B2 . The performance of all of the chosen n ? l short tubings on the components of L can be extended to the attachment of n ? l 1-handles on the 3-ball B2 where the -arcs associated with each short tubing are the cores of the attached 1-handles. Denote this handlebody in S 3 by H. Notice that, due to the fact that the -arcs form a trivial tangle in B1, H retracts to a wedge of n ? l circles, each circle being unknotted in S 3. Moreover, notice that each of the chosen combined-arcs that is associated with a long tubing is totally contained in H and, due to the facts that 1) the -arcs formed a trivial tangle in B2 , and 2) each 1-handle of H has only one combined-arc passing through it, these combined-arcs are trivial in H. (Speci cally, this means that there exists disjoint 2-discs f1 ; : : : ; l g H such that @ j = 1j [ 2j where 1j is the combined-arc, 2j @ H, 1 j l.) Finally, the attaching the long tubings to @ H can be extended to the deletion from H of a regular tubular neighborhood of each combined-arc. This deletion of tubular neighborhoods from H produces a \handlebody with wormholes" (see [MT] for further study) that is a retract of H2 which by abuse of notation we still call H2 . Since these combined-arcs in H were trivial, the deletion of their tubular neighborhoods produces a handlebody of genus n. In particular, we can add to the previous wedge of n ? l circles l meridian curves, one for each of the l combined-arcs, such that the resulting graph is a retract of H2 . This graph will link L l-times (one time for each meridian curve contained in the wedge) and each of the remaining n ? l circles of the wedge can be seen in S 3 as spanning 2-discs that are punctured by L. So the claim that H2 is a handlebody in the special situation where r is a tuple of integers is established. Establishing the handlebody claim in the general framing setting is very straight forward. First, notice that there are n ? 1 disjoint compressing discs in B2 that split it into n 3-balls such that each one of these balls contains a single component of B2 \ L. By the construction of P , it is easily seen that these compressing discs also exist in i

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H2 . Second, observe that these compressing discs split P into the disjoint union of

all the peripheral tori of L. Finally, notice that the peripheral tori of L bound solid tori in L(r). } Based upon Theorem 1 we say that P is the canonical splitting surface of L(r) that is associated with the n-braid representative of L. This Heegaard splitting of L(r) has a number of interesting geometric features that may prove useful in establishing particular properties of L(r). Speci cally, notice that there is an alternate description of the construction of P . Let T be the peripheral tori of L embedded in S 3 such that T \ H is a union of n disjoint circles, for any H 2 H. Fixing on a particular H , let fc1 ; : : : ; cng (T \ H ) be this set of circles and let f1; : : : ; n?1g H be any collection of disjoint arcs such that: 1) i \ (S1jn cj ) = @i for 1 i n ? 1; and 2) the graph in H of (S1jn cj ) [ (S1in?1 i) is connected. Then the surface obtained from tubing T along the -arcs is equivalent to P . Since L can be pushed into T away from the endpoints of the -arcs, under this alternate construction of the splitting surface it is easily seen that L can be embedded in P . Furthermore, let L0 L(r) be the canonical link having the property that S 3 ? L = L(r) ? L0 . This alternate construction also allows us to easily see the following duality of properties with respect to P between L S 3 and L0 L(r). Speci cally, there exists disjoint discs 1; : : : ; n S 3 that are each once punctured by L and, when viewing P as being in S 3 , can be used to pairwise compress P to produce S0. Similarily, there exists disjoint discs 10 ; : : : ; n0 L(r) that are each once punctured by L0 and, when viewing P as being in L(r), can be used to pairwise compress P to produce a 2-sphere in L(r) that is punctured 2n-times by L0 . Moreover, L0 can be embedded in P when it is viewed in L(r).

Example 2 Poincare Homology Sphere| In [R] an extended example is worked out in section 9.D illustrating how the Poincare Homology Sphere as represented by the

right-hand trefoil with +1 framing is equivalently represented by a genus 2 Heegaard splitting. (The key fact used is that the trefoil is a tunnel number one knot.) Starting in Figure 2(a) we illustrate this same equivalence between representation modalities for the Poincare Homology Sphere by using the canonical splitting surface P associated to the 2-braid representation of the right-hand trefoil. Figure 2(b) shows an isotopic copy of P that is constructed by a short and long tubing. Figure 2(c) has furthered the isotopy of P so that it is easily observe that \below" P we have it retracting onto a graph that is the wedge of two circles. }

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isotopic tubes

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Figure 1: The isotopy of the two tubes between (a) and (b) can be accomplished by pushing the indicated tube in (a) through the Dehn surgery torus. We can next un-braid the resulting tube in (b) by sliding base of the tube around in the planar surface P n ftubesg. To obtain (c) we then slide this tube's base through the other tube that has remained static to this point in the isotopy.

3 Multiple canonical surfaces If we are to have the possibility of non-equivalent canonical splitting surfaces for L(r) we will need to look at links L that have multiple non-conjugant closed n-braid representatives. In [BM2] a calculus on closed braids is developed and investigated. In particular, for any xed n there are nitely many \structured isotopies" that allow one


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Figure 2: The strands entering and exiting the blocks can be weighted. In particular, for the

G- ype in (b) to be index preserving we must have the strands entering the top of block B weighted m and s (from left to right) and existing the bottom of block B weighted s and m (from left to right).

to jump between conjugacy classes of closed n-braid representations of any link. These structured isotopies fall into two categories, exchange moves and generalized ypes (or G- ypes). (One more isotopy is required for jumping between braid representation of diering index|destabilization as illustrated in Figure 3.) Figure 2(a) illustrates the standard \block and strand diagram" for the exchange move. There is no similar \standard block strand diagram" for G- ypes since the pathologies associated with \ yping" isotopies can be extensive and a complex book-keeping method is needed to enumenate them for each braid index. The simplest G- ype diagram is illustrated in Figure 2(b). (The reader should understand that the strands of the diagrams in Figure 2 can be weighted and inside the block any braiding is permitted.)

Proposition 3 Suppose L is represented by closed n-braids ^1 and ^2. Assume that ^1 and ^2 are related to each other by an exchange move. Then the canonical splitting surface for L(r) that is associated with ^1 is equivalent to the canonical splitting surface associated with ^2 .

Proof. Referring to Figure 2(a), consider the initial punctured surface S0 that is the

union of the two disc bres at angles 1 and 2 . The canonical splitting surface that starts with this S0 and adds the two short tubings that do not pass through braiding blocks. Although the individual disc bers are not invariant under an exchange move, the puncture 2-sphere S0 is, so it is easily seen that the resulting surface P is also

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Conjecture 4 Suppose L is a knot that is represented by closed n-braids ^1 and ^2.

Assume that ^1 and ^2 are related to each other by a G- ype (that cannot be achieved by exchange moves). Then the canonical splitting surface for L(r) that is associated with ^1 is not equivalent to the canonical splitting surface associated with ^2 .

Remark 5 At rst sight this conjecture might be seen as risky speculation, since

many inequivalent splitting surfaces in the knot complement could become equivalent after surgery on the knot. For example, the manifold resulting from surgery could be a lens space and, by the main result of [BO], it is know that Heegaard splittings of genus g is unique for lens spaces. However, there is some evidence that close braids that admit a G- ype cannot yield such simple manifolds under Dehn surgery. It is known that surgery on a torus knot can produce a lens space. (More generally, surgery on one-bridge knots contained in a solid torus can also produce a lens space [B]|torus knots are a sub-class of such one-bridge knots.) In [L] it was established that any closed braid representative of a torus knot can be reduced to minimal braid index through the uses of destabilizations and exchange moves, i.e. G- ypes are not needed. (As applied to torus knots, Proposition 3 would then be consistent with the main result in [BO].) The arguments in [L] may be adaptable to the general setting of one-bridge knots. A more speculative \conjecture" would be the case where L is a link of more than one component. } We say that a canonical splitting surface P in L(r) is doubly compressible if there exists 2-discs 1; 2 S 3 ? L such that i Hi (i = 1; 2;) are meridian discs and @ 1 \ @ 2 = ;. Examples of doubly compressible canonical splitting surfaces are easily constructed. Speci cally, any canonical splitting surface that is associated with a closed braid that admits a destabilization will be doubly compressible. Moreover, since by Proposition 3 we know that the canonical splitting surface is invariant under exchange moves, any canonical splitting surface that is associated with a closed braid which after some sequence of exchange moves admits a destabilization is also doubly compressible. Thus, by Theorem 1 of [BM1] every canonical surface associated with U (r) where U is a closed link representing an unlink is doubly compressible. Moveover, one can produce examples of closed braids of minimal braid index greater than two where the canonical splitting surface is doubly compressible and


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Figure 3: Figure 4 illustrates that the 3-braid representation of the gure-eight knot, K , yields a canonical splitting surface that is doubly compressible. (The author thanks Xingru Zhang for the example in Figure 4.) Referring to Figure 4, the six-punctured 2sphere corresponds to S0 , the exterior of S0 corresponds to B0, and the interior of S0 corresponds to B1 . The shaded 2-disc 1 B1 has it boundary being @ 1 = [ where S0 and B1 \ K . Such a 2-disc results in a meridian of the canonical surface P when we perform the 3 peripheral tubings to S0 . B0 also contains a 2-disc which is labeled 0 and @ 0 intersects the @ 1 in exactly one point. The existence of such a pair of 2-discs is sucient to guaranty that P is doubly compressible|take two parallel copies of the disc resulting from 1 and band them together non-trivially using an arc in @ 0 . The example in Figure 4 suggests that if the canonical splitting surface of a closed braid is to be doubly incompressible then the closed braid must be suciently \complex". To this end, we introduce the follow notion of complexity. Recall that our construction of P has L H2 . Let f1 ; : : : ; ng H2 be the set of once punctured (by L) 2-discs that could be used to produce S0 from P by pairwise compression, i.e. surgerying P along the i0s. Let C be a symbol from the set fL; i ; j ; P g. A digon disc, or D-disc, is a 2-disc D H2 such that @ D = P [ , where C C , with @ D being homotopically non-trivial in P [ i . A rectangular disc, or R-disc, is a 2-disc R H2 such that @ R = L [ [ [ P , where C C . (We allow for the possibility of i = j .) We say that a braid is exhaustive if for every pair of 2-discs, (m ; X ) (H1 H2), we have @ m \ @ X 6= ;, where m is a meridian disc and X i

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∆1 ∆0

equator=axis

Figure 4: The curve on the 6-punctured 2-sphere labeled \equator" is the 3-braid axis for the

gure-eight knot. This 2-sphere is S0 and is the union of two disc bers in the briad bration.

is either a D-disc or R-disc.

Conjecture 6 Suppose L is represented by an exhaustive closed braid ^. Then the

canonical splitting surface for L(r) that is associated with ^ is not doubly compressible.

The reader should notice that the 3-braid representative of the gure-eight in Figure 4 is not exhaustive.

Theorem 7 Suppose L is an exhaustive closed n-braid such that the associated canon-

ical splitting surface P is not doubly compressible. Assume each component of L has individual braid index greater than one. Then either r is a boundary slope, or P is a strongly irreducible Heegaard splitting surface for the 3-manifold L(r)

Proof. Suppose for the framing r the canonical splitting surface P associated with

the braid L is a weakly reducible splitting surface. Then there exists two complete sets of meridian discs|two nonempty sets of non-parallel disjoint meridian discs


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f11 ; : : : ; 1s g H1 and f21; : : : ; 2s g H2 such that these two sets are disjoint 1

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and any other meridian is either not disjoint from these sets, or is parallel to a disc in one of these sets. Our construction of P has L H2. Since P is not doubly compressible each 2j H2 (1 j s2) can be represented by a properly embedded essential planar surface (j2 ; @j2 ) (H2 ? (L); P [@ (L)) such that @ 2j = j2 \P , @j2 \@ (L) 6= ;, and the slope of the boundary curves of j2 on the boundary torus @ (Li ) is ri (1 i l). We will use @iP as notion for the single boundary component of j2 on P . As shown in [CG], if we compress P in L(r) along our complete sets of meridian we will produce either a connected essential surface Pc L(r) or a 2-sphere. We rst deal with the case where Pc is an essential surface. Since each 2j has an associated j2 that has boundary curves on the peripheral tori @ (Li )), there is an associate essential surface F = Pc \ (S 3 ? (L)) (S 3 ? (L)). The slopes of the boundary curves @F correspond to the framing r. Next we deal with the case where Pc L(r) is a 2-sphere. By construction Pc \ P 6= ;. Thus, we can apply Haken's lemma [J] to produce a new 2-sphere, which we still call Pc, such that Pc \ P = c where c is a simple closed curve. (Speci cally, Pc illustrates that P is a reducible splitting of L(r).) Let Pc \ Hi = Pi; i = 1; 2. Since P is not doubly compressible and L H2 , we have that: P1 is a meridian disc of H1 ; and H2 ? L contains a planar surface } (which is not a disc) such that } \ P = c with the slope of the boundary curves of } on the peripheral tori of L corresponding to the framing r. Borrowing from the notation in alternate construction of P , we can assume that }0 s intersection with the set of once punctured discs f1 ; : : : ; n g is transverse. We concern the intersection set I = f} \ (S1jn j )g which is a disjoint union of circles and arcs that can be embedded in both } and S1jn j . We wish to consider how this intersection set can be simpli ed. First notice that, since each component of @} n c must intersect at least one of the j discs, each component of @} n c contains the endpoint of at least one arc of I . Let I be such an arc and, without loss of generality, assume 1 . Then either has an endpoint on c or both endpoints on @} n c. If it is the latter case then, in 1 , splits o a half-disc that illustrates that } is boundary compressible in H2 . Thus, we could have chosen a planar surface with fewer boundary components on the peripheral one of the tori of L. So we may assume that has one endpoint on c and one on @} n c. Moreover, since each component of L individually has braid

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index greater than one then each component of @} n c has two such arcs of I attached to it. If ! I is a circle then, as above, let ! 1 . ! must bound a disc in 1, for otherwise we could construct a 2-sphere in S 3 that intersects L exactly once. Taking an innermost such circle in 1 we can compress } along the disc such an ! bounds and reduce the number of circles in I . Thus, we may assume that I contains only arcs which have either an endpoint on c and an endpoint on @} n c, or has both endpoints on c.

∆D

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Figure 5: The outer boundary circle is c and the inner circles are the curves of @} that are on the peripheral tori.

Now consider I } and refer to Figure 5 for an example of such an interesction set inside a planar surface. Notice that since each component of @} n c is assumed to have at least two arcs of I attached to it and, as argued above, each one of these arcs is also attached to c, a nesting feature of these arcs in } forces the existence of a rectangular region R } such that @ R is the union of four arcs c \ \ L \ where C C for C 2 fc; i ; L; j g. If int(R ) \ I = ; then R is an R-disc and, after a small isotopy, R \ P1 = ;. This contradicts our assumption that the braid representing L is exhaustive. Thus int(R ) \ I 6= ; and must contain an arc having both endpoints in c. But an outermost such arc in R splits o a subdisc in } that can be isotopied to produce a D-disc that is disjoint from P1. This, again, contradicts i

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our assumption that the braid is exhaustive.

}

Corollary 8 Suppose L is an exhaustive closed n-braid representing a non-trivial

knot such that the associated canonical splitting surface P is not doubly compressible. Then for all but nitely many nontrivial r, P is a strongly irreducible Heegaard splitting surface for the 3-manifold L(r).

Proof. By the main result in [H] we know that all but nitely many framing r will not correspond to the slope of the boundary curves of an essential surface in S 3 ? (L). Thus, statement follows as an application of Theorem 7. } Finally, if conjectures 4 and 6 are valid then understanding what words of the nbraid group correspond to exhaustive closed braids is the main obstacle for employing Corollary 8 to produce examples of closed 3-manifolds having numerous strongly irreducible Heegaard splittings of the same genus can be readily constructed. Any reasonable familiarity with closed braids would lead one to suspect that exhaustive braids are plentiful enough so that ones admitting G- ype would exist.

References [B] J. Berge, Some knots with surgeries yielding lens spaces, preprint. [BM1] J. Birman & W. Menasco, Studying Links via Closed Braids V: Closed braid representations of the unlink, Transactions of the AMS, Vol. 329, No. 2, 585{606, February 1992. [BM2] J. Birman & W. Menasco, Stabilization in the Braid Group, preprint in preparation. [BO] F. Bonahon & J. P. Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. E cole Norm. Sup., (4) 16, no. 3, 451{466, (1984). [CG] A. Casson & C. McA. Gordon, Reducing Heegaard splittings, Topology and its applications, 27 (1987), 275-283. [J] W. Jaco, Lectures on Three-Manifold Topology, AMS Regional conference series, no. 43.

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[L] J. Los, Knots, braid index and dynamical type, Topology, 33, no. 2, 257{270, (1994). [MT] W. Menasco & A. Thompson, Compressing Handlebodies with Holes, Topology Vol. 28, No. 4, (1989), 485{494. [H] A. E. Hatcher, On the Boundary Curves of Incompressible Surfaces, Pac. Joun. of Math., Vol. 99, No. 2, (1982), 373{377. [R] D. Rolfsen, Knots and Links, Mathematics Lecture Series 7, Publish or Perish, Inc., 1976.