Geometric intersection of curves on punctured disks

Abstract

We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.2010 Mathematics Subject Classification. 57N16, 57N37,57N05.

Dynnikov and train track transition matrices of pseudo-Anosov braids

“…These matrices are much easier to compute than computing train-track transition matrices (Section 3 gives an example to contrast the computation of a Dynnikov matrix with that of the train track transition matrix of a given pseudo-Anosov isotopy class on D 4 ). Roughly speaking, a Dynnikov matrix is an integer matrix which describes the action of a given pseudo-Anosov isotopy class in a neighbourhood of its invariant unstable measured foliation in terms of Dynnikov's coordinates [6,17,[20][21][22] on the space of projective measured foliations on D n . The dilatation of a given a pseudo-Anosov isotopy class equals the spectral radius of its Dynnikov matrix.…”

Integral laminations on nonorientable surfaces

“…Here we shall give the analogy of the Dynnikov coordinate system [1][2][3] on a finitely punctured disk that has several useful applications such as giving an efficient method for the solution of the word problem of the n -braid group [1], computing the geometric intersection number of integral laminations [9], and counting the number of components they contain [11].…”

“…The components in each S i and S ′ i are glued together in a unique way up to isotopy, and hence L is constructed uniquely. 2 Next we give a list of properties that an integral lamination L ∈ L k,n satisfies in terms of its triangle coordinates as in [9], and then we construct a new coordinate system from the triangle coordinates that describes integral laminations in a unique way. In particular, we shall generalize the Dynnikov coordinate system [1-3, 5, 9-11] for N k,n .…”

Counting components of an integral lamination

“…The action of MCG(D n ) on L n can be calculated using the update rules of Theorem 3 below (see for example [4,8,3,7,10]), which describe how Dynnikov coordinates transform under the action of the Artin generators and their inverses. In this theorem statement we again use the notation x + to denote max(x, 0).…”

“…Algorithm 9 below computes the number of components of an integral lamination L ∈ L n . We assume that n > 3, since otherwise the number of components is given by Step 1 Replace (a; b) with (a; b) ∈ Z 2n given by (10). Set Y = 0 and input the pair ((a; b), Y ) to Step 2.…”