Thanks to Andre de Carvalho and Toby Hall for their collaboration on the dynamics of the horseshoe map. Thanks to Michael Handel for some information on the case of four-strand braids.
Initial Definitions
Main Results
- Low-Dilatation Pseudo-Anosov Braids
- Dilatation and Hyperbolic Volume
- Sharkovskii’s Theorem and Horseshoe Dynamics
- Definitions
- Pseudo-Anosov Braids
- Forcing
All pseudo-Anos sequences with the lowest possible growth have extremely low tori mapping volumes. Consequently, we would like to find analogues of the Šarkovsky theorem in the pseudo-Anosov braid forcing setting.
Entropy
Markov Partitions
If you now fill in the original singularities and instead delete these three blue points, then the original mapβ relative to the blue points will actually be mapα instead. You can see this by folding the subintervals together in the natural way and looking at how the blue dots swirl around each other. In general, when we refer to the Markov matrix from now on, we will think in the context of Markov matrices for invariant train tracks.
Entropy
For a pseudo-Anosov differential called) M forf, P viaMi,j= #of components of intFs(p1, Ri)∩intf(Fu(p2, Rj)) forp1∈ intRi enp2∈int Rj (which will be well defined independently of the choice of p1 , p2) . Let f : Σg,k,r →Σg,k,r be a pseudo-Anosov diffeomorphism with an invariant train track τ and an associated ordering of the principal edges.
Entropy and Forcing
For a pseudo-Anosov mapping class group element, entropy is minimal on the pseudo-Anosov diffeomorphism, and is log of the dilation. We define the growth rate (or dilation) of a pseudo-Anosov mapping class group element as the dilation of the pseudo-Anosov representative diffeomorphism.
Example of Braids on 3 Strands
- Description of the Group
- Description of the Order
- Hall’s Implementation of the Bestvina-Handel Algorithm
- Mathematica
- Java/C++
- SnapPea
- KnotTwister
For example, we know from earlier that the growth of a pseudo-Anos braid can be calculated as the largest real eigenvalue of a Markov matrix. Another important calculation here is the calculation of the number of orbits of a certain size for a pseudo-Anos n-braid.
The Fundamental ˜ ψ 3 Braid
Every pseudo-Anosov braid (on any number of strands) is constrained by some pseudo-Anosov 3-braid. Each pseudo-Anosov braid (on any number of strands) is constrained by some power of ψ˜3.
Finding “Simple” Braids
Experimentally, one observes that many higher-string pseudo-Anosov braids are forced by some 3-string pseudo-Anosov braids. For a pseudo-Anosov concatenation with growth λ andg×g Markov matrixM, the sum|M|of the entries of M|M| -g+ 1≤λg.
More Cusps
- The Case of n Odd, n ≥ 7
- The Case n = 4k, k ≥ 2
- The Case n = 8k + 2, k ≥ 2
- The Case n = 8k + 6, k ≥ 1
Forn∈N, n= 4kandk≥2, an expression (in terms of the usual braid generators) for the automorphism ψn of Dn described in terms of the chosen train track map above is ψn = L2k+1n σ1−1σ2−1. Remember that in case of oddity, our train track map was essentially a 1k rotation clockwise.
The ˜ ψ n Sequence
- The Case of n Odd, n ≥ 5
- The Case n = 4k, k ≥ 2
- The Case n = 8k + 2, k ≥ 1
- The Case n = 8k + 6, k ≥ 1
Specifying some additional information about the peripheral edges in a natural way creates a fundamentally unique braid in Bn/(∂n2). Using this particular map of the train track, we can calculate the expressions for the corresponding braid. Forn∈N, nodd dhen≥5, an expression (in terms of ordinary braid generators) for the automorphism ψ˜n eDn described by the chosen train track map above is ψ˜n=L2nσ1σ2. For n ∈ N, n = 4k and k ≥ 2, an expression (in terms of ordinary braid generators) for the automorphism ψ˜n of Dn corresponding to the train track map chosen above is ψ˜n=L2k+ 1n σ1σ2.
For n ∈ N, n = 8k+ 6 and k ≥ 1, an expression (in terms of the usual braid generators) for the automorphism ψ˜n of Dn corresponding to the chosen train track map described above is ψ˜n =L6k+ 5n σ1σ2.
Exceptional Cases
- The ψ 5 Braid
- The ψ 6 Braid
- The ψ 10 Braid
- The ˜ ψ 3 Braid
- The ˜ ψ 4 Braid
- The ˜ ψ 6 Braid
- The ς 6 Braid
- The ˜ ς 6 Braid
As for conventional braid group generators, the braid ψ6 corresponding to the train track map chosen above is ψ6=σ5σ4σ3σ2σ1σ5σ4σ3σ5σ4. In terms of conventional braid group generators, the braid apsi˜ 4 corresponding to the train track map chosen above is ψ5=σ3σ2σ1−1. In terms of conventional braid group generators, the braid ς6 corresponding to the train track map chosen above is ς6=σ5σ4σ3σ2σ1σ5σ4.
In terms of the usual braid group generators, the braid ς˜6 corresponding to the chosen train track map above is ς˜6=σ5σ4σ3σ2σ1σ5σ4σ3σ2σ3σ3σ4σ5.
Astounding Sequences and the Sequential Value
In conjunction with a convenient notion of "very small expansion sequence," this analysis leads to Benson Farb's conjecture. All pseudo-Anosov braids of very small dilation must be the product of a finite-order element with something that has supports contained in a subsurface of universally finite complexity.
Sparsity
- The Case of n Odd, n ≥ 5
- The Case n = 4k, k ≥ 2
- The Case n = 8k + 2, k ≥ 1
- The Case n = 8k + 6, k ≥ 1
Two of the paths of length 2k are obtained by traversing each path of length k twice. Two of the paths of length 4k−2 are obtained by traversing each path of length 2k−1 twice. Two of the paths of length 8k−2 are obtained by traversing each path of length 4k−1 twice.
Two of the paths of length 8k+2 are obtained by traversing every path of length 4k+1 twice.
The ψ n Sequence
- The Case of n Odd, n ≥ 7
- The Case n = 4k, k ≥ 2
- The Case n = 8k + 2, k ≥ 2
- The Case n = 8k + 6, k ≥ 1
However, these two paths give rise to the same ψn−invariant sets as the corresponding paths of length k. However, these two paths give rise to the same ψn−invariant sets as the corresponding paths of length 2k−1. However, these two paths give rise to the same ψn−invariant sets as the corresponding paths of length 4k−1.
However, these two paths give rise to the same ψn−invariant quantities as the corresponding paths with length 4k+1.
Exceptional Cases
Similarly, one could consider Dehn operations on a link which simply consist of Dehn operations on a subset of components in the link. The Dehn surgery construction is central to the study of 3−manifolds, as can be seen from the following result. Any closed, oriented, connected 3−manifold M is realized by (integral) Dehn surgery on a link Lin the3−sphereS3.
We conclude that it is possible for different toral diffeomorphisms to give rise to the same manifold under the Dehn operation.
Dehn Surgery Realization of Mapping Tori
Hence, sticking a solid torus in the cusp corresponding to the new central puncture yields the mapping tori of theψn and ˜ψn. This gives us the following corollary. The mapping tori of the(ψn)and(˜ψn)order braids are all isometric to sprouts obtained by Dehn surgery on a single point of Mmagic, with the exception of n= 6. The mapping tori for the exceptionalψ6,ψ˜ 6 is actually isometric to Mmagic. Otherwise, the surgery coefficients are for the ψn order. The mapping torus of ψn will be isometric to the complement in S3 of the link pictured below, obtained by forming the usual braid closure of ψn and adding an additional component around all the braid strands.
Now by performing a Rolfsen twist of order 1 on the bottom component of the diagram, the operation coefficient on the transverse component changes from 1k to 1k+ 1 = k+1k. This gives our final diagram.
Other Cases
Volume Concept
These two results in connection with our description of the mapping tori of theψnand ˜ψn give the following nice consequence. This finally explains why there is no natural pseudo-Anosov expansion of theψn to casesn= 3.4. Further understanding of the volume spectrum involves mapping the tori of extremely low-growth pseudo-Anosovn braids for a given if all have truly low volume.
Let P be a periodN-orbit of the horseshoe that is not of finite order braid type, with height q=q(P) = mn ∈(0.12]in the lowest terms.
Star-Shaped Train-Tracks
While generalized (n, m)-star train tracks can be particularly good to work with, it seems that valuable information about the set of pseudo-Anosov braids is lost by only looking at such train tracks. For example, it appears that truly low-growth behavior simply cannot be generated by such train tracks (ie, there are many low-growth pseudo-Anosov braids that do not have invariant generalized star-shaped train tracks). Note that the braids ψn and ˜ψn, nod and n≥ 7, have train tracks similar to generalized (2, m)-star train tracks.
Indeed, the train tracks for these braids described earlier are formed by simply adding a main edge and corresponding peripheral edge loop to a single spoke (ψn case) or deleting a main edge and corresponding peripheral edge loop from a single spoke (˜ψn . case ) ) of a general (2, m)-star train track.
The Ω p,q and ˜ Ω p,q Braids
Recall our conjecture earlier that lowest growth pseudo-Anosov braids with 3-cusp mapping tori are often formed by adding a puncture to the non-pierced singularities of lowest growth pseudo-Anosov braids with 2-cusp mapping tori. In the case where is an odd prime number, the presumably lowest growth pseudo-Anosov r−braids are the ψr and ˜ψr braids. Since the ψr braids are horseshoe, we consequently assume that the minimal growth pseudo-Anosov (r+ 1) − string braids of horseshoe type with 3−cusp mapping tori is the one formed by adding holes to the ψr braids.
Among the pseudo-Anos braids with toruses of the 2−cusp mapping, which are not isometric to positive operations on Mmagic, we previously assumed that the minimum sequence value was that obtained by the sequence ˜θn, which was experimental.
Forcing for M magic
Define J(n) as the largest integer, so that the lowest J(n) growth under pseudo-Anosov horseshoe n−braids can be achieved by braids of the form Ωp,q and Ω˜p,q. Pseudo-Anosov braids with mapping tori due to a non-trivial operation on Mmagic are all minimal. However, we suspect that not all minimal pseudo-Anosov braids arise from operations on Mmagic.
What are pseudo-Anosov n-braids of lowest growth whose mapping tori is not isometric to operations in Mmagic.
Surgeries on M hip and M cool
Theθn's mapping tori are all isometric to Dehn operations on the hyperbolic 4-cusp3 manifold Mhip with the same volume as M˜hip, possibly M˜hip itself. What are the lowest growth pseudo-Anosovn braids that do not correspond to an operation on an orientable hyperbolic 4-cusp3 manifold with the same volume as Mhip? The mapping tori of the0(†n,♣2n) are all isometric with respect to a hyperbolic 5−cusp3−manifold with the same volume as M˜cool.
The mappings tori of the†n are all isometric to Dehn operations on the hyperbolic 5−cusp3−manifoldMcool with the same volume as M˜cool, possibly M˜cool itself.
Minimally Twisted Chain Links and the Finite Cyclic Whitehead Link Covers
Indeed, when we consider orientable n -cusped hyperbolic 3-manifolds for a fixed n >10, finite cyclic covers of the Whitehead bond complement have lower volume than minimally twisted n -chain bonds. We note that the volumes of the finite cyclic covers of the complement of the Whitehead connection are simply consecutive multiples of the volume V8. It was observed by Agol that there must be an orientable hyperbolic 3-manifold that does not result from the Dehn operation on a finite cyclic cover of the complement of the Whitehead connection.
What (orientable) hyperbolic 3-manifolds can be constructed by surgery on a finite cyclic covering of the Whitehead link complement.
