ON THE FIBRATION OF AUGMENTED LINK COMPLEMENTS DARLAN GIRA˜O 1 1 0 2 Abstract. We study the fibration of flat augmented link com- p e plements: simple combinatorial conditions on the diagram imply S that these links fiber. We further show that certain surgeries on 4 these links produce fibered manifolds. This is then used to prove 1 that within a very large class of links, called locally alternating augmented links, every link is fibered. ] T G 1. Introduction . h Let K be an oriented link in S3. By a Seifert surface S we mean an t a orientable spanning surface for K, i.e., ∂S = K and the orientation of m S agrees with that of K. It is known that every oriented link has a [ Seifert surface (see for instance [Ro]). We say the link K is fibered if 1 S3−K has the structure of a surface bundle over the circle, i.e., if there v exists a Seifert surface S such that S3−K ∼= (S×[0,1])/φ, where φ is 4 8 a homeomorphism of S. In this case we abuse terminology and say S 0 is a fiber for K. 3 Deciding whether or not a link K is fibered, or even virtually fibered, . 9 is in general a very hard problem. In the early 60’s Murasugi ([Mu]) 0 1 proved that an alternating link is fibered if and only if its reduced 1 Alexander polynomial is monic. Stallings [St] proved that a link K is : v fibered if and only if π (S3 −K) contains a finitely generated normal 1 i X subgroup whose quotient is Z. Murasugi’s work is constructive, but re- r stricts to alternating links. Stalling’s result is very general, but usually a very hard to verify. In [Ga] Gabai proved that if a Seifert surface S can be decomposed as the Murasugi sum of surfaces S ,...,S , then S is a 1 n fiber if and only if each of the surfaces S is a fiber. Goodman-Tavares i ([GT]) showed that under simple conditions imposed on certain span- ning surfaces, it is possible to decide whether or not these surfaces are fibers for pretzel links. Leininger ([Le]) provided the first examples of virtually fibered but not fibered knots. Walsh ([Wa]) showed that all two-bridge knots and links are virtually fibered. She also showed that spherical Montesinos knots and links are virtually fibered. Very recently Futer-Kalfagianni-Purcell ([FKP1], theorem 5.11) introduced 1 2 DARLAN GIRA˜O a new method for deciding whether or not a given spanning surface is fiber for a link K. From a diagram of the link they construct an associated surface (called A-state surface) and a certain graph. They show that this surface is a fiber if and only if the corresponding graph is a tree. On these notes we will be mainly concerned with the fibration of three classes of links: augmented links, locally alternating augmented links, and links obtained from augmented links by performing certain surgeries. Augmented links have played a central role in several recent devel- opments in 3-manifold topology. Lackenby and Agol-Thurston ([La1]) used them to estimate volumes of alternating link complements. Futer- Kalfagianni-Purcell ([FKP]) used them to obtain diagramatic volume estimates of many knots and links. Futer-Purcell ([FP]) also used them to prove that if K is a link with a twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling K is hyperbolic. Their combinatorial argu- ment further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and give a lower bound for the genus of K. Cheesebro-DeBlois-Wilton ([CDW]) proved that hyperbolic augmented links satisfy the virtual fibering conjecture. However they did not provide any information on the virtual fibers. Here we provide combinatorial conditions on the diagram, which imply that very often augmented links fiber and explicitly exhibit their fibers. We also show that when this is the case, then certain Dehn surgeries on these links produce fibered manifolds. This last result is thenusedtoshowthatwithintheclassoflocallyalternatingaugmented links every link is fibered, and explicitly exhibit their fibers. Wenextdefinetheseclasseswe’llbeworkingwithandstatethemain results. Acknowledgements I am very grateful to Alan Reid, for his extraordinary guidance and unwavering support. I am also thankful to Cameron Gordon for helpful conversationsandJoa˜oNogueiraandJessicaPurcellfortheircomments on an early draft of this work. The author was partially supported by CAPES/Fulbright Grant BEX 2411/05-9. 2. Augmented links, locally alternating augmented links and main results Thenotionofaugmented links wasfirstintroducedbyAdams([Ad1]) andfurtherexploredbyFuter-Kalfagianni-Purcell([FKP]),Futer-Purcell FIBRATION OF AUGMENTED LINKS 3 ([FP]) and Purcell ([Pu1, Pu2]). We recall it here. For more details see the very nice survey paper on augmented links by Purcell ([Pu]). Let K be a link in S3 with diagram D(K). Regard D(K) as a 4- valent graph in the plane. A bigon region is a complementary region of the graph having two vertices in its boundary. A string of bigon regions of the complement of this graph arranged end to end is called a twist region. A vertex adjacent to no bigons will also be a twist region. Encircle each twist region with a single unknotted component, called a crossing circle, obtaining a link J. S3 − J is homeomorphic to the complement of the link L obtained from J by removing all full twists from each twist region. The link L is called the augmented link associated to D(K). The original link complement can be obtained from the link J by performing 1/n-Dehn filling on the crossing circles, for appropriate choices of n. Figure 1. (a) Initial link K; (b) link J obtained by adding crossing circles; (c) augmented link L; (d) corre- sponding flat augmented link. WhenallthetwistregionsinthediagramD(K)haveanevennumber of crossings, then all non-crossing circle components of the augmented link L will be embedded in the projection plane. We denote these types of links by flat augmented links. Given the diagram of a flat augmented link L we construct a Seifert surface S , called standard Seifert surface, and a graph G (L) (this L B will be done in section 4). We now state our main results. Theorem A. Let L be a flat augmented link. Then the standard Seifert surface S is a fiber if and only if the graph G (L) is a tree. L B Performing ±1 Dehn surgery on crossing circle components of links as above yield new ones which are again fibered. 4 DARLAN GIRA˜O Theorem B. Let L be an flat augmented link such that the graph G (L) is a tree. Then performing ±1 Dehn surgery on crossing circle B components yields a fibered link K. Given a flat augmented link L, one can construct a corresponding locally alternating augmented link L as follows: in each crossing circle a change two of the crossings so that the crossings in the crossing circles are alternating. This is described in Figure 2. Note that the resulting link need not to be alternating. Figure 2. (a) Pairs of crossings yielding locally alter- nating link; (b) Canonical Seifert surface for resulting link. We later show that every locally alternating augmented link L can a be obtained from ±1 surgery on the crossing circles of a flat augmented ˜ ˜ link L such that G (L) is a tree. This implies B Theorem C. Let L be an locally alternating augmented link obtained a from an flat augemented link L. Then L fibers. a Remark 1. We remark that our methods and the surfaces and graphs we construct are very different from those in [FKP1]. However, it is very interesting that we are obtaining the same type of results: a manifold fibers given that a certain associated graph is a tree. We also note that very often fibration of the links considered here cannot be detected from their construction but is detected by ours (examples for the converse can also be exhibited). The remainder of the paper is organized as follows: in section 3 we we recall the notion of Murasugi sum. This will be used in the con- struction of the standard surface. In section 4 we set up the terminolgy FIBRATION OF AUGMENTED LINKS 5 of standard surface and of the G (L) graph needed for the remaider B of the paper. In section 5 we prove theorem A. In section 6 we prove theorem B . Finally, in section 7 we use theorem B to prove theorem C. 3. Murasugi Sum In this section we recall the notion of Murasugi sum ([Ga1], [Le], [Mu]). Definition 1. We say that the oriented surface T in S3 is with bound- ary L is the Murasugi sum of the two oriented surfaces T and T with 1 2 boundaries L and L if there exists a 2-sphere S in S3 such bounding 1 2 the balls B and B with T ⊂ B for i = 1,2, such that T = T ∪T 1 2 i i 1 2 and T ∩T = D where D is a 2n-sided disk contained in S (see Figure 1 2 3). Figure 3. The result concerning Murasugi sum we need is the following, due to Gabai ([Ga]). Theorem 3.1. Let T ⊂ S3, with ∂T = L, be a Murasugi sum of oriented surfaces T ⊂ S3, with ∂T = L , for i = 1,2. Then S3 − L i i i is fibered with fiber T if and only if S3 −L is fibered with fiber T for i i i = 1,2. Remark 2. We abuse notation by saying that L is the Murasugi sum of L and L . 1 2 4. Set up The augmented links we consider are flat, i.e., there are no twists ad- jacent to crossing circles. We may isotope the link slightly so that in its diagram all crossing circles encircle locally vertical strings and so that 6 DARLAN GIRA˜O the overcrossings of the crossing circles are below the undercrossings (see Figure 4 (b)). Our goal is to find a condition under which such links fiber. Recall that Stallings ([St]) provided a method for checking whether or not a given Seifert surface is a fiber for the complement of an oriented link. Theorem 4.1 (Stallings). Let T ⊂ S3 be a compact, connected, ori- ented surface with nonempty boundary ∂T. Let T ×[−1,1] be a regular neigborhood of T and let T+ = T ×{1} ⊂ S3 −T. Let f = ϕ| , where T ϕ : T ×[−1,1] −→ T+ is the projection map. Then T is a fiber for the link ∂T if and only if the induced map f : π (T) −→ π (S3−T) is an ∗ 1 1 isomorphism. Let L be a flat augmented link with the description as above. Ori- ent planar components in the clockwise direction and crossing circle components in the counterclockwise direction, as seen from above the projecting plane (see Figure 4). Consider the Seifert surface given by the Seifert algorithm. This Seifert surface will be called the canonical Seifert surface and the above orientation for L the canonical orienta- tion. Consider the diagram of L as a plannar graph. This graph divides the plane into regions which are checkerboard colored. The unbounded region is colored white and the other regions are colored accordingly. There are three types of white regions: Type A regions, bounded by those crossing circles that bound two white regions; Type B regions, bounded by those crossing circles that bound a single white region; Type C regions, not bounded by crossing circles. Note that type A regions come in pairs. We will the denote the above regions by A ,A ,A ,A ,...,A ,A ,B ,...,B ,C ,...,C re- 11 12 21 22 p1 p2 1 q 1 r spectively. We denote the unbounded white region by C . 0 A crossing circle bounding a type B region B will be called a B- i circle, also denoted by B . Note that given the diagram for L, as i described above, there will be a type C region to the right and another onetotheleftofeveryB-circle. WesaytheB-circleisadjacent tothese regions. A crossing circle bounding a pair of type A regions A ,A i1 i2 will be called a A-circle and denoted by A . Note that there will be i one type C region below and one above of every A-circle. We say the A-circle is adjacent to them. Let G (L) be the graph obtained from B the diagram of L as follows Vertices are type C regions; FIBRATION OF AUGMENTED LINKS 7 - - + + + + - - - - Figure 4. Canonical Seifert surface for augmented link and corresponding white regions determined by the dia- gram An edge joins C and C if there is a B-circle adjacent to both i j of them simultaneously. From the canonical Seifert surface with canonical orientation we con- struct the standard Seifert surface, denoted S , as follows: L 1. Remove the portion of the link and of the canonical surface contained in the interior of a sphere around A-circles. 2. For each A-circle removed, add an untwisted band as described in Figure 5. 3. Perform Murasugi sum of two Hopf bands along the untwisted band added on 2 (see Figure 6). Figure 7 illustrates two examples of the construction of the graph G (L) from the diagram of the link L. B 5. Proof of theorem A Theorem A. Let L be a flat augmented link. Then the standard Seifert surface S is a fiber if and only if the graph G (L) is a tree. L B Note that the the standard surface S is obtained as the Murasugi L sumofacollectionofHopfbandstogetherwithaleftoversurface. Each of these Hopf bands is a fiber for the complement of their boundary Hopf links. Denote by S the resulting leftover surface and by L(cid:48) its L(cid:48) boundary link. L(cid:48) is itself a flat augmented link in which all crossing circles are B-circles. Observe also that, by construction, G (L) = B 8 DARLAN GIRA˜O Figure 5. Replacing A-circle by untwisted band (steps 1 and 2 above) = + Figure 6. Obtaining A-circle from Murasugi sum of two bands. G (L(cid:48)). In view of Theorem 3.1 we need to find conditions under B which the surface S is a fiber for the link L(cid:48). This is the content of L(cid:48) Lemma 5.1. The oriented link L(cid:48) fibers with fiber S if and only if its L(cid:48) associated graph G (L(cid:48)) is a tree. B This lemma concludes the proof of Theorem A. Remark 3. Note that, for L(cid:48), its canonical and standard Seifer surface coincide. We now proceed to prove the lemma. Proof of lemma 5.1. Observe first that the link L(cid:48) is itself a flat aug- mented link, obtained from L by removing its A-circles and adding FIBRATION OF AUGMENTED LINKS 9 Figure 7. Original link; replace A-circles; associated graph. untwisted bands. The diagram of L(cid:48) divides the plane into type B and type C regions. Observe also that the fundamental group of the surface S is free L(cid:48) with a generating set given by the loops on the surface around white regions. This can be easily seen since a regular neighborhood of this surface is a handlebody. Let B ,...,B and C ,...,C be the type B 1 q 1 r and C regions given by the diagram of L(cid:48). Denote the loops on S L(cid:48) around these regions by u ,...,u , u ,...,u respectively. They will b1 bq c1 cr be oriented counterclockwise. The fundamental group of the complement S3 − S of the above L(cid:48) Seifert surface is also free with generating set given by the loops going through white regions perpendicular to the projecting plane in S3−S L(cid:48) (thisistruesincearegularneigborhoodofthissurfaceisahandlebody). 10 DARLAN GIRA˜O These loops are denoted by x ,...,x and x ,...,x , according to the b1 bq c1 cr type of region. All these loops are to be oriented from above to below the projecting plane. The surface S is two sided and our convention throughout is that L(cid:48) from above the diagram of L(cid:48) we see the “−”side of the pieces bounded by a crossing circle (see Figure 4). Let S+ be the copy of S in S3−S L(cid:48) L(cid:48) L(cid:48) parallel to S obtained from S by pushing it in the “+”direction. L(cid:48) L(cid:48) This is formally defined by the map f : S −→ S3 −S described in L(cid:48) L(cid:48) Theorem 4.1. With this in mind it is not hard to describe the induced map f (see ∗ Figure 8). We have:  u (cid:55)→ x x−1  bi cm cn u (cid:55)→ x x−1w ,where w is a word without the letter x ; cm bi cm m m bi u (cid:55)→ x x−1w ,where w is a word without the letter x . cn cn bi n n bi Remark 4. If a B-circle is adjacent to the unbounded region C and 0 another region C , then we have u (cid:55)→ x±1. m bi cm Figure 8. (a) Loops u ,u and u ; (b) Their images bi cm cn under f . ∗ The strategy now is as follows: by Theorem 4.1 S is a fiber iff the L(cid:48) map f : π (S ) −→ π (S3 − S ) is an isomorphism. We show that ∗ 1 L(cid:48) 1 L(cid:48) if G (L(cid:48)) is not a tree then f is not injective. When G (L(cid:48)) is a tree B ∗ B we show that f is surjective and therefore an isomorphism (recall that ∗ free groups are Hopfian).