Narrow Results

A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parametrization of this family of knots by a four-tuple of integers, which they call Schubert's normal form. This paper presents an algorithm for constructing a genus 1 doubly-pointed Heegaard diagram compatible with K, given a Schubert's normal form for K. The construction, coupled with results of Ozsváth and Szabó, provides a practical way to compute knot Floer homology groups for (1,1) knots. The construction uses train tracks, and its method is inspired by the work of Goda, Matsuda, and Morifuji.

In this paper, we describe the equivalence classes of simple arcs between the two punctures on a 2-punctured torus ${\mathrm{\Sigma }}_{1,2}$ up to isotopy by using the given four generators $g_1,g_2,g_3$ and $g_4$. Actually, we show that a class of simple arcs is represented by an ordered sequence of four integers. Also, we introduce an algorithm to check whether or not an ordered sequence of four integers represents a class of simple arcs in ${\mathrm{\Sigma }}_{1,2}.$ We want to point out that this result classifies the $(1,1)$-positions.

We investigate the structure of the characteristic polynomial det(xI - T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial invariants of [F], one of them being the product of the other two, and all three being divisors of det(xI - T). The degrees of the new polynomials are invariants of [F] and we give simple formulas for computing them by a counting argument from an invariant train-track. We give examples of genus 2 pseudo-Anosov maps having the same dilatation, and use our invariants to distinguish them.