Recall that the definition of braid forcing order is intended to generalize the idea of the period forcing order of interval maps in the classical Sharkovskii Theorem. Consequently, we would like to find analogues of the Sharkovskii Theorem in the pseudo-Anosov braid forcing setting.
Theorem 8.26(Carvalho, Hall). Ωp,qºΩp0,q0 iff (p−p1)−1q+1 ≥
(p0−1)q0+1
p0−1 andq≤q 0
.
Note that our expression (p−1)qq+1 for the surgery coefficient corresponding to Ωp,qis quite similar
to the expression (p−p1)−1q+1 of the theorem. This suggests an approach for computing the more general order with respect toMmagic.
Conjecture 8.27. After some simple coordinate change on Mmagic,the braid forcing order on the
set of braids corresponding to positive surgeries onMmagic is essentially given by the usual order on
surgery coefficients (along with perhaps some auxiliary condition, i.e.,q≤q0).
We observed earlier that the ψn and ˜ψn braids were all minimal. These were braids where
the surgery coefficients were very close to 1. Considering now surgeries of fixed denominator, the horseshoe theory applies in fact to show that the Ωp,2are all minimal. More generally,
Conjecture 8.28. Pseudo-Anosov braids with mapping tori resulting from non-trivial surgery on
Mmagic are all minimal.
We suspect however that not all minimal pseudo-Anosov braids arise from surgeries on Mmagic.
For instance, we suspect the ˜θn braids defined earlier are potentially minimal as well, although we
have not computed this yet.
Last, observe that if we fix q, the sequences Ωp,q will be linearly ordered (i.e., Ωp,q ºΩp0,q iff p≤p0). The above concepts thus imply the existence of many linearly ordered sequences of minimal braids in the forcing partial order. Some related ideas may be found in [28], [29].
Chapter 9
Mapping Tori with More Cusps
9.1
Construction of the
θ
nSequence
Having observed that many low-growth pseudo-Anosov braids have mapping tori isometric to surg- eries onMmagic, and moreover having analyzed such braids to some extent, we would now like to
ask
Question 9.1. What are the lowest growth pseudo-Anosov n−braids whose mapping tori are not isometric to surgeries onMmagic?
Using the strategy from Chapter 3, i.e., looking at braids forced by powers of the fundamental braid ˜ψ3,we stumbled across the sequence ˜θn.
Definition 9.2. Define the n−strand braid sequence θ˜n via θn = L3nσ1−1σ2−1σ−n−14σn−−15 for n =
4 + 3k, k∈N.
Conjecture 9.3. The sequence(˜θn),defined for n= 4 + 3k, k∈N, attains minimal growth among
pseudo-Anosovn−braids with mapping tori not isometric toMmagic.
Definingan to be the growth ofθn, we observe that a= limn→∞ann exists and appears experi-
mentally to satisfy 30≤a≤40.
Fact 9.4. The sequenceθ˜n is astounding when defined.
We have not found any other pseudo-Anosovn−braids in the quotient of the braid group by its center attaining the exact same growth as ψn or ˜ψn in the non-exceptional cases. In other words,
these two braids appear to usually be the unique minimizers in the quotient Bn/(∂2n) for pseudo-
Anosov growth. However, we do not observe a similar sort of phenomenon with the ˜θn braids. In
fact, asn grows, there appear to be many pseudo-Anosov n−braids attaining the same growth as ˜
θn, with the number of such braids growing at least linearly. For instance, we note that often the
related braidsL3
nσ−11σ−21σ−i+11σi−1are pseudo-Anosovn−braids of the same growth as ˜θnfor varying
Recall now that when nwas odd the braids ˜ψn were of horseshoe type [22], although this was
not true in the case ofneven. Moreover, in spite of this, the ψn were not in fact horseshoe in the
case ofnodd either. This leads us to
Question 9.5. Are there pseudo-Anosovn−braids of horseshoe type attaining the same growth rates as theθ˜n for some subsequence of N?
We are not sure whether the ˜θn are horseshoe themselves, but nonetheless the resolution to this
question brings us to the naturalθn sequence.
Definition 9.6. We define the braidθn, forn= 4 + 3k andk∈N,to be the horseshoe braid with
horseshoe code10k+110k10k.
Lemma 9.7. The braidθn is pseudo-Anosov with growth rate equal to that ofθ˜n.
We conclude this section by taking a look at invariant train-tracks for the θn. Consider the
train-tracks of the diagram below.
By putting a single puncture in each valence 1 vertex of the above train-track diagram, one in fact gets an invariant train-track for the corresponding braidθn.We denote the invariant train-track