Recall from earlier the nice property of the0(ψn) and0( ˜ψn).
Theorem 9.8. The mapping tori of the0(ψn)and0( ˜ψn) are all isomorphic toMmagic.
We observe now a similar nice property for the0(θn,♣1n).First consider the link pictured below.
Definition 9.9. Define the manifoldM˜hip to be the complement in S3 of the link pictured above.
We conjectured earlier the magic manifold Mmagic should attain minimal volume among ori-
entable hyperbolic 3−manifolds of 3 cusps [27].
Conjecture 9.10. The manifoldM˜hipattains minimal volume among orientable hyperbolic3−manifolds
of 4cusps.
Theorem 9.11. The mapping tori of the0(θn,♣1n)are all isometric to a hyperbolic4−cusp3−manifold
of the same volume asM˜hip.
From this result, we make the following definition
Definition 9.12. Define thehip manifold, denotedMhip,to be the manifold formed as the mapping
These considerations above lead to the following corollary
Corollary 9.13. The mapping tori of theθn are all isometric to Dehn surgeries on the hyperbolic
4−cusp3−manifoldMhip having the same volume asM˜hip,possibly M˜hip itself.
Analogous to the way in which we proceeded fromMmagicto Mhip,we would now like to ask
Question 9.14. What are the lowest growth pseudo-Anosovn−braids not corresponding to surgery on an orientable hyperbolic4−cusp3−manifold of the same volume asMhip?
This seems like a challenging question in general to answer. Recall that among the ψn braids
there was an infinite subsequence of horseshoe braids. Moreover, the θn sequence above, having
the same growth as the ˜θn sequence, consisted entirely of horseshoe braids. This suggests trying to
answer the above question in the setting of horseshoe braids for some infinite subset of Nat first. The number of horseshoen−braids is a finite, relatively small number for relatively lown,in stark contrast with the case of all pseudo-Anosov n−braids, so we would expect this to be quite a bit more tractable computationally.
Recall that, in the case of n= 2k+ 3,our braidsψn had horseshoe codes of the form 10k+110k.
Next, by definition, our θn braids were defined by horseshoe codes for n = 3k + 4 via θn =
10k+110k10k. This critical observation suggests attempting to generalize these horseshoe codes in
some way. Two reasonable possibilities for the next stage of generality are horseshoe codes of the form 10k+110k10k10k and 10k+110k+110k10k. It is the latter expression we prefer to focus on for
now.
Definition 9.15. Define the braid †n, forn = 4k+ 6 and k ∈ N, to be the horseshoe braid with horseshoe code10k+110k+110k10k.
We observe a corresponding set of invariant train-tracks ♣2
In the diagrams above, one gets the †n braids by placing a single puncture in each singularity
of valence 1 in the train-track graph and performing a suitable train-track map. We point out at this stage that the train-track maps for the invariant train-tracks given in the cases ofθnand†n are
again essentially rotations of the train-track, just like for theψn and ˜ψn,except with an additional
bit of twist. If one generalizes the construction further, i.e., a sequence beyond the θn and †n, it
is possible that the additional bit of twist could get larger and larger. We’re not quite sure at this stage exactly how large the extra twist could conceivably get for the more complex sequences.
We observe experimentally that the braids 0(†n,♣2n) have very low growth (possibly minimal)
among pseudo-Anosovn−braids with 5−cusp mapping tori. Such braids are necessarily not derived from Dehn surgery on a 4−cusp hyperbolic 3−manifold. We expect the braids0(†n,♣2n) to have
relatively low growth among braids not coming from Dehn surgery on a minimal volume orientable 4−cusp hyperbolic 3−manifold. Of course the braids†n,as well as ones formed by puncturing an appropriate subset of the non-punctured singularites, would have the same growth on fewer number of strands. We are not sure yet, however, whether any such braids correspond to surgeries onMhip
or other conjectured minimal volume orientable 4−cusp hyperbolic 3−manifolds.
In analogy with ˜Mhip,we would like to produce a corresponding manifold for the †n sequence.
Definition 9.16. Define the manifoldM˜cool to be the complement inS3 of the link pictured above.
Conjecture 9.17. The manifoldM˜coolattains minimal volume among orientable hyperbolic5−manifolds
of 4cusps.
Theorem 9.18. The mapping tori of the0(†n,♣2
n)are all isometric to a hyperbolic5−cusp3−manifold
of the same volume asM˜cool.
From this result, we make the following definition
Definition 9.19. Define the cool manifold, denoted Mcool, to be the manifold formed as the
mapping torus of any0(†n,♣2n).
The considerations above lead to the following corollary
Corollary 9.20. The mapping tori of the†n are all isometric to Dehn surgeries on the hyperbolic