ARC INDEX OF PRETZEL KNOTS OF TYPE (−p, q, r)

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2, . . . , αk with the following properties: (1) Each αi has no self-crossing. (2) If αi crosses over αj at a crossing, then i > j and it crosses over αj at any other crossings with αj . (3) For each i, there exists an embedded disk di such that ∂di = C and αi ⊂ di. (4) di ∩ dj = C, for distinct i and j. Then the pair (D,C) is called an arc presentation of L with k arcs, and C is called the axis of the arc presentation. Figure 1 shows an arc presentation of the trefoil knot. The thick round curve is the axis. It is known that every knot or link has an arc presentation [3, 4]. For a given knot or link L, the minimal number of arcs in all arc presentations of L is called the arc index of L, denoted by α(L).