I am trying to create several presentations for the basic set of surface mapping tori.

For example when $ gcd (p, q) = 1 $, then $ M = $ the T.$ (p, q) $ Torus knot outside in $ S ^ 3 $ is a map torus of a genus $ g = frac {pq-p-q + 1} {2} $ Surface with a boundary component, S, which is mapped onto itself $ phi $, a card of order $ pq $.

The suspension flow of this mapping has closed all webs and actually results in a Seifert fiber of $ M $. With the Seifert fiber we can express the basic group quite well in terms of the two singular fibers and the regular fiber. $ pi_1 (M) =$. Here $ z $ is clearly a third-party generator, but is central and includes the class of regular fibers, so somehow nice. The others $ x $ and $ y $ are the classes of the singular order fibers $ q $ and $ p $ respectively.

Since $ M $ is a mapping torus that we also get $ pi_1 (M) = <a_1, …, a_g, b_1, …, b_g, t mathrel {|} ta_it ^ {- 1} = phi _ * (a_i), $ $ tb_it ^ {- 1} = phi _ * (b_i)> = pi_1 (S) rtimes _ { phi_ *} mathbb {Z} $.

The Dehn Twist presentation by $ phi $ is something that can be worked out, and such $ phi _ * $ could actually be specified explicitly.

In order not to go too far, we could instead consider a surface S of the genus $ g geq 2 $ with a boundary component and a pseudo-Anosov monodromy, $ phi $, that gives us a hyperbolic knot outside, M, in $ S ^ 3 $. We get again $ pi_1 (M) = <a_1, …, a_g, b_1, …, b_g, t mathrel {|} ta_it ^ {- 1} = phi _ * (a_i), $ $ tb_it ^ {- 1} = phi _ * (b_i)> = pi_1 (S) rtimes _ { phi_ *} mathbb {Z} $.

Now we have a pseudo-Anosov suspension flow and a collection of closed, singular orbits that correspond to the singularities of the stable and unstable singular leaves. So there should be a representation of the basic group in relation to the closed singular orbits, right?

I'm assuming that I can look at Knot Info's list of hyperbolic node monodromes related to strain rotations to generate examples that I can use to calculate $ phi _ * $ expressly. Then I could fill that out $ ta_it ^ {- 1} = phi _ * (a_i) $ Relators. However, I would like a more general approach since I am interested in all pseudo Anosov monodromes, including narrow surfaces, not just the fiber node monodromy cards in $ S ^ 3 $. Apparently the singular courses are a candidate for a general approach.

I tried to search Thurston's Work on Surfaces (FLP) and I couldn't see such information. Has that been worked out somewhere? If so, then this is probably a reference request. If not, does anyone know of any obstacles to creating the base group with these loops?