Encuesta aplicada a los funcionarios de la empresa pratt transporte y logística

In the case in whichh is periodic and reducible we can still compute g(O),n(O),

χ(O)andpin polynomial time as before. To findN we use anh–maximal multicurve γ, which we may assume to be O(`(p))–bounded by Theorem 3.4.8. We crush S alongγ and examine each piece of the induced mapping classhγ∈Mod+(Sγ) in turn.

Suppose thatS′⊆Sγ is obtained by taking the orbit of one component of Sγ under

hγ. Leth′∈Mod+(S′)be the mapping class induced by hγ andT′ the triangulation

ofS′ induced byTγ. Using the results of Section 3.4, we can construct a pathp′from

T′ _toh′_(T′_{)such that `(p</sub>′<sub>) ≤`(p). The previous section allows us to compute the} signature of the orbifoldS′/h′ by using p′. Now ifx∈S′/h′ andord(x) < ∞thenx

lifts to a pointy∈S/h andord(y) =ord(x). Repeating this for all possibleS′ allows us to compute the orders of many of the irregular points ofO.

To deal with any omitted points, we note that there is also a representative c∈γ such that φ(c) =c. We may perform the previous procedure usingc; crushing S alongc to obtainSc with induced homeomorphism φc. Now ify is an irregular point

ofO which is omitted by the previous construction theny must either: • lie on one of the components ofc, or

• be contained in an annular region, bounded by two of the components of c. Regardless, the order ofy must be 2. Therefore by comparing the product of the list

of orbifold point orders found andp we can determine exactly how many of points of order two we are missing. Finally, asr(x) ≠0, the rotation numbers of these points

must be 1

2 and so we can compute N.

To bound the complexity of this construction we use the results of Section 3.4. We can computeTγ and pγ in O(poly(`(p))) operations by Theorem 3.4.2. From

this for each choice ofS′⊆Sγ we can construct p′ inO(poly(`(p)))operations. Now

as each such path has length at most `(p) we can compute the quotient orbifold S′/hγ in O(poly(`(p)))operations by the previous section. Finally determining how

many orbifold points of order two we are missing by comparingp and the product of the list of orbifold point orders found can also be done inO(poly(`(p)))operations. All together this shows that we can computeN inO(poly(`(p)))operations.

Therefore, using the standard path forh∈X∗ and given a O(`(h))–bounded h–maximal multicurve γ we see that this procedure takes polynomial time. From which we determine that:

Lemma 5.3.1. Deciding whether two periodic, reducible wordsg, h∈X∗ correspond

to conjugate mapping classes is a problem in NP∩co-NP. Together with Corollary 5.2.1 this shows that:

Corollary 5.3.2. Deciding whether two periodic words g, h ∈ X∗ correspond to

conjugate mapping classes is a problem in NP∩co-NP.

For a pointx∈S we define its rotation number (with respect toh) to be the rotation number of its image in theS/h quotient orbifold.

Lemma 5.3.3. Periodic mapping classes h and g are π–conjugate if and only if:

• for each marked point v we have that:

– v and π(v) have the same rotation number (with respect to h and g

respectively), and

– π(hk(v)) =gk(π(v)) for everyk∈Z

whenever these maps are defined.

Proof. The forward direction of this lemma holds trivially. For the reverse direction, we first note that without loss of generality we may assume that π is actually a permutation of the vertices ofS. If it is not then we consider each of the possible extensions of π in turn.

As g and h are conjugate there is a mapping class f such that h = f−1gf. If f∣V ≠ π then consider ϕ∶S/h →S/g, the homeomorphism induced by f. This

homeomorphism respects the orders and rotation numbers of points and satisfies the

lifting criterion: it maps the subgroup π1(S) ≤π1(S/h) to the subgroup π1(S) ≤

π1(S/g). We can modifyϕby precomposing it with another homeomorphismψ∶S/h→

S/h. If we takeψ to be a homeomorphism that swaps two of the marked points with the same rotation number then ψpreserves the subgroup π1(S) ≤π1(S/h). Hence

ϕ○ψalso satisfies the lifting criterion and so lifts to an alternate mapping classf′ which, like f, conjugatesg toh but whose action on V is permuted byψ.

Therefore, asπ sends marked points to marked points with the same rotation number and π(hk(v)) = gk(π(v)) for every k ∈ Z, modifications of this form are

sufficient to adjustf such thatf∣V =π. Henceh andg areπ–conjugate.

The rotation numbers of the marked points can be determined in polynomial time from the polygonal decomposition of S given by the multiarc w. Thus this additional criterion can also be tested in polynomial time.