1 Introduction To construct a branched covering of a 3-manifold M, we start with a tamely embedded knot or link L ρ M (the branch set) and a represent

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1 Kirby diagrams from branched-covering presentations Frank J. Swenton Department of Mathematics Middlebury College Middlebury, VT Abstract We present an algorithm that converts any description of a 3-manifold as a finite-sheeted cover of the 3-sphere, branched over some knot or knotted graph, into a Kirby diagram for that manifold, thus resolving the problem posed in [4]. AMS Classification numbers Primary: 57M12 Secondary: Keywords: 57R65 branched covering, Kirby diagram, manifold, algorithm 1

2 1 Introduction To construct a branched covering of a 3-manifold M, we start with a tamely embedded knot or link L ρ M (the branch set) and a representation ρ : ß 1 (M L)! ± n, where ± n denotes a finite symmetric group. Taking an open tubular neighborhood N of L, ρ determines an n-sheeted covering ß : M ~! M N ; we produce the branched cover from M ~ by gluing solid tori to its boundary components via homeomorphisms in such away that meridians project under ß to meridians ρ M. It is clear from the construction that this branched cover is a 3-manifold; under certain conditions on the representation ρ, this construction extends to produce a 3-manifold from a knotted graph ρ M as well. Branched coverings arise naturally in the field of algebraic geometry (indeed, they are modeled after the simple case z 7! z 2 for the Riemann sphere), and for example, such 3-manifolds arise as the links of singularities of algebraic varieties. From a topological viewpoint, the branched-covering construction is sufficiently general to produce any closed, connected, orientable 3-manifold as a branched cover of the 3-sphere. In fact, we need only consider 3-sheeted simple coverings of knots to do so [3],[1], and given a Kirby diagram [2] for a 3-manifold, such a branched covering description can be algorithmically constructed [6]. The converse problem of algorithmically converting a description of a 3-manifold as a branched cover of the 3-sphere into a Kirby diagram for that manifold is posed in [4]; in this paper, we provide such an algorithm. 2 Represented knots Before describing the details of the algorithm, we set notation and touch upon some preliminaries. By a knot, we mean a smooth (or PL) closed 1-submanifold of 3-space, considered up to ambient isotopy (note that we do not exclude knots of multiple components). The knots with which we'll be dealing will carry a representation of the knot group into some symmetric group ± n. In terms of knot diagrams, such a homomorphism is determined by its assignment of permutations to the Wirtinger generators for the arcs of the knot diagram. Such an assignment determines a valid homomorphism of the knot group just when at each crossing of the diagram, the cyclic product of the permutations undercrossing the arcs is trivial (i.e., the relations in the Wirtinger presentation must map to the trivial elementin± n ). Equivalently, the permutation attached 2

3 to an overcrossing strand acts by conjugation on the permutation for each strand that undercross it. When each Wirtinger generator for the knot diagram maps to a transposition (ij) 2 ± n, we call the group representation (and the corresponding branched covering) simple. In this case, we can view the permutations as coloring the arcs of the diagram themselves, for the direction of undercrossing is no longer a concern since (ij) = (ij) 1. We use the prefix fi n - to refer to an object that carries a coloring by transpositions in ± n, whereby a diagram for such a represented knot is concisely termed a fi n -knot diagram. 3 Knots with bands Given a knot k, aband ontheknotisanembedded I I intersecting the knot k precisely in the I. We'll refer to the first factor of this I I as the length (with its boundary, the ends) of the band and the second factor as the band's width (its boundary, thesides). The purpose a band on a knot is to prescribe a particular splicing of the knot specifically, the replacement of the band's ends by its sides. Figure 1: A band and the splicing that it prescribes A knot with bands is a knot k along with a finite collection of (pairwise) disjoint bands on k. Knots with bands will be considered equivalent just when they differ by ambient isotopy, as usual. To express a knot k as a band-sum of unlinked components is to find a knot with bands for which the knot is an unlink and the knot k results from splicing along all bands. Note that a band determines, and is determined by, a framed arc through its length and that bands may, of course, intertwine with each other and with the knot itself. We must be a bit careful when using bands with fi n -knot diagrams, for we insist that a fi n -knot with bands must give avalid coloring not only of the knot itself but also of the knot that results after splicing. The sides of each band will splice to parallel fi n -strands of the same color, so we consider each arc of a band in the diagram to carry a virtual coloring by some transposition. This virtual coloring 3

4 is acted upon by arcs of the knot that overcross it, just as before thus, to ensure that the knot is assigned a valid coloring after the splicing, we insist that at each end, the virtual color of the band matches that of the arc to which it is attached. Note that although a band's virtual color is acted upon by that of knot arcs that overcross it, the width of the band represents two arcs of the same color (ij), and thus a band's color does not act on the color of anything that undercrosses it. 4 Branched covering preliminaries Our last task before outlining the algorithm is to set out a few basic facts about branched coverings of knots, each of which is easily verified geometrically. First, if we have a strand of knot whose Wirtinger generator is assigned some permutation ff 2 ± n, then a tubular neighborhood of this strand will produce a number of disjoint open balls in the branched cover, each ball corresponding to one orbit of f1; 2;:::;ng under ff. We note that, in terms of homeomorphism type in the branched cover, this is the same as if we'd replaced that single strand with a set of parallel fi n -strands colored in such away that the product of their colors gives a minimal decomposition of ff, as in Figure 2. We can braid these fi n -strands as we please and the branched cover will be preserved, because the branched cover of this neighborhood will remain a homeomorphic collection of open balls and the sheet-action of moving under this neighborhood will remain the same. A slightly less trivial case is encountered when we consider two parallel fi n - strands of the same color (note that (ij)(ij) is not a minimal transpositiondecomposition of any permutation, so we are not in the case discussed above). In this situation, we can easily construct the branched cover of a closed ball containing these two strands, and we find that it consists of a solid torus and (n 2) 3-balls. Noting that the branch set intersects the boundary of our neighborhood in just four points, we see that if we splice the two parallel fi n - strands, the effect on the branched cover is a surgery on the solid torus. Thus, the transformation from the branched cover of the knot contained in a given fi n -knot with bands and that of the knot after splicing is effected by a certain set of surgeries; a neighborhood of each band will branched-cover to one solid torus (the branching lift) and a collection of 3-balls (the nonbranching lifts). Our third example of a branched cover is that of a particularly simple type of fi n -knot in the 3-sphere. Let n 2 be fixed, and consider a fi n -knot consisting of (n 1) unlinked fi n -components, one colored (1j) for each 1 <j» n. We 4

5 can construct the associated branched cover geometrically by slicing open the 3-sphere along a spanning disk of each component, taking n disjoint copies of the resulting space (one for each sheet of the cover), and gluing them together as prescribed by thefi n -coloring. Specifically, onsheet1, we'll have a3-sphere with (n 1) spherical boundary components (one for each unlinked knot component). On each sheet j > 1, no branching occurs except at the component labeled (1j), so all other components' spanning disks can be re-glued as they started; what remains is a 3-sphere with one spherical boundary component, i.e., a 3-ball. Gluing the (n 1) 3-balls produced from each sheet j > 1 into the corresponding holes in sheet 1, we obtain the branched cover, which is a 3-sphere. Combining the three ingredients above, we obtain the core of the algorithm. Given a diagram for a represented knot or graph, we first convert it into a fi n - knot diagram having the same branched cover. This fi n -knot diagram is then algorithmically decomposed into a diagram for a fi n -unlink with bands that, when spliced, produces the given fi n -knot. We then further modify this fi n - unlink with bands so that it is of the third form outlined above and (pre-splicing) has a 3-sphere as its branched cover. Post-splicing, we obtain the branched cover of the given fi n -knot; by following the bands through the process, we find the surgeries necessary to obtain our manifold from the 3-sphere, with the relative twists of the bands determining the coefficients of these surgeries. Thus, we finish with a planar Kirby diagram for the associated branched covering manifold. 5 The algorithm Our algorithm takes as its input a description of a 3-manifold as a (possibly irregular) finite-sheeted branched cover of the 3-sphere, where the branch set its some knot or knotted graph. This is taken to be presented via a planar diagram in which a Wirtinger generator for each arc is assigned a permutation in ± n. Given such a planar diagram, we beginbyconverting it into a fi n -knot diagram describing the same manifold. We achieve this by first splitting each arc in the diagram into parallel fi n -strands, according to any minimal-length decomposition into transpositions of the permutation coloring the arc (Figure 2). Note that although the choice of fi n -strands is not uniquely determined by the original arc's permutation, the number of strands and the product of their colors are well-defined. Due to this, any two simplifications of that arc can be braided 5

6 until their fi n -ends match; this braiding can be achieved algorithmically by braiding each strand in sequence around the subsequent strands. As remarked earlier, the original arc will generate the same branched cover as any (braided) simplification of that arc. Figure 2: Simplification of an arc Each vertex (if any) of the branch set can be resolved into a fi n -tangle by spherically braiding the incident fi n -ends produced so that adjacent ends in the diagram have the same coloring, then joining such adjacent ends (Figure 3). Figure 3: Resolution of a vertex Because the branched cover is a manifold, we are assured that link of this vertex branched-covers to a sphere, and thus, neighborhoods of our original vertex and its fi n -simplification will generate the same branched cover. This completes our simplification process, which transforms the given diagram into a fi n -knot diagram describing the same manifold. Our next step is to decompose this fi n -knot, modulo two local moves which preserve the branched cover, into a band-sum of fi n -unlinks; this can be effected algorithmically, as in [5]. Note that what we obtain is a trivial diagram for a fi n -unlink, with bands connecting and interleaved with the unlinked components in such a way that after splicing all bands, we obtain a fi n -knot representing the manifold in question. Before continuing, we further modify this fi n -unlink (with bands) so that it contains just one component each of color (12),(13),:::,(1n), as follows. (Note that this is only possible if the given representation into ± n is transitive; but if 6

7 not, the cover will be disconnected, so we split the knot into maximal transitive pieces and treat each piece separately.) Our first step will be to conjugate the coloring of each fi n -unlink by pulling it through other fi n -unlinks in such a way that its coloring becomes (1j) for some j, as illustrated in the left side of Figure 4. This achieved, we can further modify the fi n -unlink with bands by merging any identically-colored components via bands, as in the right side of Figure 4. Figure 4: Arranging a fin -unlink into standard form Our next step is to arrange the bands of the diagram so that they meet the disks bounded by the unlinked components of the fi n -knot diagram only at their ends and in right clasps, i.e., clasps between knot and band with the band twisted in such away that its width stays parallel to the fi n -arc it clasps, as illustrated at the far left in Figure 5. Figure 5: Construction of the cover, near the (1k) component (We now emulate our construction of the 3-sphere as teh branched cover of this unlink, carrying the bands through the process to find the surgeries required to produce our 3-manifold.) We begin by duplicating our diagram into n copies, 7

8 one for each sheet of the cover, then cut the clasps on each of the n sheets as follows: on the k th diagram, cut open the clasps at the spanning disk of each unlinked component whose color involves the sheet k. By our arrangement of the fi n -unlinks as above, on sheet 1 we'll cut along all spanning disks, and on each other sheet k > 1, we'll cut only along the spanning disk of the single component colored (1k). On each sheet k > 1, we flip the diagram into the [branching] unlinked component colored (1k); in terms of the diagram, this entails a planar inversion in the circle formed by that component, followed by a mirroring in the plane of the diagram, as illustrated at the bottom of Figure 5. This accomplished, we paste each diagram k > 1 into the corresponding hole in sheet 1's diagram, then attach the matching ends across the unlinked components, as at the the right in the figure. To finish, we erase the nonbranching bands and the fi n -unlinked components. The bands that remain have doubled into loops, each loop corresponding to a surgery locus in the manifold. These loops each give a component in the Kirby diagram, with the framing of that component computed as the algebraic twist of the band relative to the natural zero-framing of the component, as usual. 6 Remarks The above construction can be readily performed by hand for relatively simple knots and coverings, though it clearly becomes cumbersome as the complexity of the knot and the number of sheets increase. It can be fully implemented by computer, a project that the author intends to undertake in the context of a larger package for manipulation of diagrams for knots, represented knots, Kirby diagrams, and 2-knots. We note in closing that the decomposition of a fi n -knot into a band-sum of fi n -unlinks demonstrates the fi n -knot as the boundary of a smoothly-embedded fi n -surface in the 4-ball, by performing the bands' splices smoothly as we move inward from S 3 4 and then smoothly collapsing the resulting fi n -unlinked components. This determines a covering of B 4 branched over the surface, and demonstrating the branched-covering M 3! S 3 as the restriction of a branched covering N 4! B 4 of 4-manifolds. A moment's reflection reveals that this 4- manifold is represented by thesame Kirby diagram as generated by the above algorithm, with the components of the Kirby diagram considered, as usual, to represent 2-handles attached to B 4. 8

9 References [1] Hugh M. Hilden, Every closed orientable 3-manifold is a 3-fold branched covering space of S 3, Bull. Amer. Math. Soc. 80 (1974) [2] Robion Kirby, Acalculus for framed links in S 3,Invent. Math. 45 no. 1 (1978) [3] José Mar a Montesinos, A representation of closed orientable 3-manifolds as 3-fold branched coverings of S 3, Bull. Amer. Math. Soc. 80 (1974) [4] V. V. Prasolov, A. B. Sossinsky, Knots, links, braids and 3-manifolds. An introduction to the new invariants in low-dimensional topology, Translations of Mathematical Monographs 154, AMS (1997) 164 [5] F. Swenton, A structure theorem for simply-represented knots, to appear in the Journal of Knot Theory and Its Ramifications [6] F. Swenton, Banded knots and simple branched covers of the 3-sphere, Dissertation, Princeton University (1999)