ar X iv : m at h / 06 02 47 6 v 1 [ m at h . G T ] 2 1 Fe b 20 06 JUGGLING BRAIDS AND LINKS

Using a simplistic model of juggling based on physics, a natural map is constructed from the set of periodic juggling patterns (or site swaps) to links. We then show that all topological links can be juggled. 1. Juggling sequences The art of juggling has been around for thousands of years. Over the past quarter of a century, the interplay between juggling and mathematics has been well studied. There has even been a book [6] devoted to this relationship, dealing with several combinatorial ideas. Numerous juggling software is also available; in particular, Lipson and Wright’s elegant and wonderful JuggleKrazy [4] program helped motivate much of this paper. Most of the information useful to juggling can be accessed via the Juggling Information Service webpage [3]. The goal of this paper is to construct and study a map from juggling sequences to topological braids. An early form of this idea providing motivation can be found in the work of Tawney [7], where he looks at some classic juggling patterns. In our discussion, we remove everything that is not mathematically relevant. Thus, assume the juggler in question is throwing identical objects, referred to as balls. By convention, there are some basic rules we adhere to in juggling. J1. The balls are thrown to a constant beat, occurring at certain equally-spaced discrete moments in time. J2. At a given beat, at most one ball gets caught and then thrown instantly. J3. The hands do not move while juggling. J4. The pattern in which the balls are thrown is periodic, with no start and no end to this pattern. J5. Throws are made with one hand on odd-numbered beats and the other hand on even-numbered beats. A throw of a ball which takes k beats from being thrown to being caught is called a k-throw. Condition J5 implies that when k is even (or odd), a k-throw is caught with the same (or opposite) hand from which it was thrown. Thus, a 5-throw starting in the left hand would end in the right hand 5 beats later, while a 4-throw starting in the left hand would end back in the left hand after 4 beats. In this notation, a 0-throw is a placeholder so that an empty 2000 Mathematics Subject Classification. Primary 57M25, Secondary 52C20. 1 2 SATYAN L. DEVADOSS AND JOHN MUGNO hand can take an action, where no ball gets caught or thrown on that beat. The following definition from [2] is used to embody some of the rules above. Definition 1. A juggling pattern is a bijection f : Z → Z : t → t+ df(t) where n ∈ N and df(t+n) = df(t) ≥ 0. A juggling sequence (or a site swap) is the sequence 〈 df(0), df(1), · · · , df(n− 1) 〉 arising from a juggling pattern. The number df(t) is the throw value at time t, and the number n associated to a juggling pattern is its period. Thus, a juggling sequence is simply used to keep track of successive throw values. Under this terminology, the sequence 〈 n 〉 yields the n-ball cascade for n odd, and the n-ball fountain for n even; the n-ball shower is given by the 〈 1, 2n− 1 〉 pattern. A juggling sequence can be represented graphically in several ways. A common way is the juggling diagram, with the height of the balls in the vertical direction drawn with respect to time. Figure 1 shows the juggling diagram of the 〈 5, 1 〉 sequence, the lines in this diagram corresponding to the path of the juggled balls traced out over time. We denote odd and Figure 1. Juggling diagram of the 〈 5, 1 〉 sequence. even numbered beats by solid and open circles, distinguishing the two hands. The juggling pattern associated to the 〈 5, 1 〉 sequence is