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PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2016 Mathematical Sciences Publishers PACIFIC JOURNAL OF MATHEMATICS Vol. 284, No. 1, 2016 dx.doi.org/10.2140/pjm.2016.284.1 BITWIST MANIFOLDS AND TWO-BRIDGE KNOTS JAMES W. CANNON, WILLIAM J. FLOYD, LEERLAMBERT, WALTER R. PARRY AND JESSICA S. PURCELL Though LeeR Lambert spent his life as an actuary and a musician and was a loving father of nine girls and one boy, he had always wanted to earn an advanced degree as a math- ematician. With the encouragement of his wife, he earned his Ph.D. in mathematics at the age of 68. Many of the results of this paper appeared in his Ph.D. dissertation at Brigham Young University. At the age of 71, LeeR died of bone cancer. We miss you, LeeR. We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bitwisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bitwist construction is of- ten efficient and natural. Finally, we give applications to computations of fundamental groups and homology of these branched cyclic covers. 1. Introduction Branched cyclic covers of S3 have played a major role in topology, and continue to appear in a wide variety of contexts. For example, branched cyclic covers of S3 branched over two-bridge knots have recently appeared in combinatorial work bound- ing the Matveev complexity of a 3-manifold[Petronio and Vesnin 2009], in algebraic and topological work determining relations between L-spaces, left-orderability, and taut foliations[Gordon and Lidman 2014; Boyer et al. 2013; Hu 2015], and in geometric work giving information on maps of character varieties[Nagasato and Ya- maguchi 2012]. They provide a wealth of examples, and a useful collection of mani- folds on which to study conjectures. Given their wide applicability, and their con- tinued relevance, it is useful to have many explicit descriptions of these manifolds. We give a new and elegant construction of the branched cyclic covers of two- bridge knots, using the bitwist construction of[Cannon et al. 2009]. While other presentations of these manifolds are known (see, for example,[Minkus 1982; Mulazzani and Vesnin 2001]), we feel our descriptions have several advantages. First, they follow from a recipe involving exactly the parameters necessary to describe a two-bridge knot, namely, continued fraction parameters. Our descriptions MSC2010: 57M12, 57M25. Keywords: bitwist manifolds, two-bridge knots, branched cyclic covers. 1 2 J. W. CANNON, W. J. FLOYD, L. LAMBERT, W. R. PARRY AND J. S. PURCELL apply uniformly to all two-bridge knots, and all branched cyclic covers of S3 branched over two-bridge knots. Second, they are obtained from a description of a two-bridge knot using a very straightforward bitwisted face-pairing construction, as in [Cannon et al. 2000; 2002; 2003; 2009]. Bitwisted face-pairings (read “bi-twisted”, as in twisted two ways) are known to produce all closed orientable 3-manifolds. The examples of this paper show, in addition, that bitwist constructions are often efficient and natural. While a generic face-pairing will yield a pseudomanifold, which, with probability 1, will not be an actual manifold[Dunfield and Thurston 2006], bitwisted face-pairings avoid this problem. (We will review necessary information on bitwisted face-pairings, so no prior specialized knowledge is required to understand our constructions.) Third, our description leads to immediate consequences in geometric group theory. We obtain a simple proof of the fact that the fundamental group of the n-fold branched cyclic covering of S3, branched over a two-bridge knot, has a cyclic presentation. Our description also gives immediate presentations of two well-known families of groups, the Fibonacci and Sieradski groups. These are known to arise as fundamental groups of branched cyclic covers of S3 branched over the figure-eight and trefoil knots, respectively. These groups have received considerable attention from geometric group theorists; see, for example,[Cavicchioli et al. 1998] for further references, and Section 6 for more history. Our methods recover the fact that the first homology groups of Sieradski manifolds are periodic. We also give a proof that their fundamental groups are distinct using Milnor’s characterization of these spaces. We consider orders of abelianizations of Fibonacci groups as well. These orders form an interesting sequence related to the Fibonacci sequence, which we shall see. 1A. Bitwisted face-pairing description. We will see that the bitwist description of any two-bridge knot is encoded as the image of the north–south axis in a ball labeled as in Figure 1, along with an associated vector of integer multipliers. For the branched cyclic cover, the description is encoded by adding additional longitudinal arcs to the sphere. We now describe the construction briefly, in order to state the main results of the paper. A more detailed description of the construction, with examples, is given in Section 2. Begin with a finite graph 0 in the 2-sphere S2 D @B3 that is the union of the equator e, one longitude NS from the north pole N to the south pole S, and 2k ≥ 0 latitudinal circles, such that 0 is invariant under reflection V S2 ! S2 in the equator. Then 0 divides S2 into 2.k C 1/ faces that are paired by . This face-pairing is shown in Figure 1. BITWISTMANIFOLDSANDTWO-BRIDGEKNOTS 3 N e v S Figure 1. The model face-pairing: a faceted 3-ball with dotted central axis and reflection face-pairing V @B3 ! @B3. As with any face-pairing, the edges fall into edge cycles. The equator e forms one edge cycle c0 since the reflection leaves e invariant. Each other edge of the graph is matched with its reflection to form another edge cycle ci . We number these edge cycles from 0 through 2k C 1, with even numbers associated with latitudinal edges, as indicated in Figure 2. Now choose nonzero integer multipliers, denoted m0; m1;:::; m2k; m2kC1, for the edge cycles ci . In the case at hand, restrict the choice of multipliers mi as follows. Each latitudinal edge cycle c2i is assigned either C1 or −1 as multiplier. Each longitudinal edge cycle c2iC1 may be assigned any integer multiplier m2iC1 whatsoever, including 0. The multiplier m2iC1 D 0 is usually forbidden, but in this case indicates that the two edges of edge-class c2iC1 must be collapsed to a point before the bitwist construction is engaged. N 2kC1 2k 2k−1 2k−2 4 3 e 2 1 0 v Figure 2.