Monotonic Simplification and Recognizing Exchange Reducibility

0.86 MB·English

by  William W. Menasco

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Monotonic Simplification and Recognizing Exchange Reducibility
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ul William W. Menasco ∗
J
University at Buffalo
7
Buffalo, New York 14214

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T Dedicated to Joan Birman on the occasion of her advancement to active retirement.
G
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h July 7, 2005
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a
m
[

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v Abstract
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In[BM4]theMarkovTheoremWithoutStabilization(MTWS)establishedtheexistence
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7 ofacalculusofbraidisotopiesthatcanbeusedtomovebetweenclosedbraidrepresentatives
0 of a given oriented link type without having to increase the braid index by stabilization.
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0 Although the calculus is extensive there are three key isotopies that were identified and
/ analyzed—destabilization, exchange moves and elementary braid preserving flypes. One
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t of the critical open problems left in the wake of the MTWS is the recognition problem—
a
m determiningwhenagivenclosedn-braidadmitsaspecifiedmoveofthecalculus. Inthisnote
: wegiveanalgorithmicsolutionto therecognitionproblemforthree isotopiesofthe MTWS
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i calculus—destabilization, exchange moves and braid preserving flypes. The algorithm is
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“directed” by a complexity measure that can be monotonic simplified by that application
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a of elementary moves.
1 Introduction.
1.1 Preliminaries.
Given anoriented linkX ⊂ S3 itisaclassical resultof Alexander[A]thatX can berepresented
as a closed n-braid. For expository purposes of this discussion it is convenient to translate
Alexander’s result into the following setting. Let S3 = IR3 ∪{∞} and give IR3 an open-book
decomposition, i.e. IR3\{z−axis} is fibered by a collection of half-planes fibers H = {H |0 ≤
θ
θ < 2π} where the boundary of H is the z-axis. Equivalently, we consider the cylindrical
θ
coordinates (r,θ,z) on IR3 and H is the set of points having its second coordinate θ.
θ
∗partially supported by NSFgrant #DMS 0306062
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2 William W. Menasco
AnorientedlinkX inIR3(⊂ S3)isaclosed n-braidifX ⊂ IR3\{z−axis}withX transversely
intersecting each fiber of H in n points. The braid index of X is the cardinality b(X) =
|X ∩H | = n which is invariant for all H ∈ H.
θ θ
axis
projection of X onto C
1
Figure 1:
Possibly after a small isotopy of X in IR3\{z−axis}, we can consider a regular projection
π : X → C given by π : (r,θ,z) 7→ (1,θ,z), where C = {(r,θ,z)|r = 1}. The projection
1 1
π(X) ⊂ C is isotopic to a standard projection that is constructed as follows. For n = b(X)
1
we first consider the n circles c = {(r,θ,z)|r = 1,z = k} for 1 ≤ k ≤ n. Next, we alter
k n
the projection of this trivial unlink of n components to construct the projection of π(X) by
having adjacent circles c and c cross via the addition of a positive or negative crossing at
i i+1
the needed angle to produce a projection of X. It is clear through an ambient isotopy of IR3
which preserves the fibersof H that we can always reposition X in IR3\{z−axis} so that π(X)
is a standard projection. (See Figure 1 and 2.)
c
i+2
c
i+1
c
i
The circles c on a portion of −1
i s s
i i+1
the cylinder C
1
Figure 2:
From a standard projection π(X) on C we can read off a cyclic word in the classical Artin
1
generators σ±1, 1 ≤ i ≤ n−1. Specifically, β(X) is the cyclic word that comes from recording
i
the angular occurrence of crossings where a positive (respectively negative) crossing between
the circles c and c contributes a σ (respectively σ−1) element to β(X), 1 ≤ i≤ n−1.
i i+1 i i
Monotonic Simplification 3
n n
n
i+1 i+1
i+1
i i i
1 1
1
s s −1 Reidemeist type II move s −1 s
i i i i
n n
n n
j+1 j+1
i+2 i+2 j j
i+1 i+1
i+1 i+1
i i
i i
1 1 1 1
s s s s s s s s s s
i i+1 i i+1 i i+1 i j j i
Reidemeister type III move
braid group relation for |i−j|>1
Figure 3: For type III moves there is also moves corresponding to σ−1σ±1σ±1 goes to
i i+1 i
σ±1σ±1σ−1, and σ σ−1σ−1 goes to σ−1 σ−1σ . For the braid group relations we also have
i+1 i i+1 i i+1 i i+1 i i+1
the moves σǫσδ goes to σδσδ where ǫ,δ ∈ {−1,1}.
i j i
We can alter π(X) (and, correspondingly, β(X)) while preserving the n-braid structure of
X, plus its standard projection characteristic, by judicious use of type-II and -III Reidemeister
moves and the braid group relation. (See Figure 3.)
We consider the equivalence classes under the moves in Figure 3. Specifically, X and X′
are braid isotopic if π(X) can be altered to produce π(X′) through a sequence of type-II and
-III moves, and braid group relations. We will let B (X) be notation for the equivalence class
n
of n-braids which are braid isotopic to X.
Next, let W = {σ1,···,σn−2} and U = {σ2,···,σn−1}. We now have a sequence of defini-
tions.
An n-braid X admits a destabilization if for π(X), its associated braid word β(X) is of the
form Wσ±1 , where W is a word using only the generators in the set W. (See Figure 4(a).) X
n−1
admits an exchange move if β(X) is of the form WU where W (respectively U) is a word using
only the generators in the set W (respectively U). Alternatively, if X admits an exchange move
then there is a braid isotopic X′ with β(X′) of the form W1σn−1W2σn−−11 where W1 and W2
are words using only the generators in W. (See Figure 4(b).) Finally, X admits an elementary
(braid preserving) flype if β(X) is of the form W σp W σ±1 where W and W are words in
1 n−1 2 n−1 1 2
W and p ∈ Z−{0}. In Figure 4(c) we allow for the strands to be weighted. Thus, the crossings
4 William W. Menasco
Figure 4:
can be seen as generalized crossings and the performance of the isotopy can products twisting
in these weighted strands.
We say that B (X) admits a destabilization, exchange move or elementary flype, respec-
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tively, if there exists a braid representative X′ ∈ B (X) which admits a destabilization, ex-
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change move or elementary flype, respectively.
A long standing problem (Problem 1.84 in [K]) is determining when an n-braid equivalence
class B (X) contains a braid X that admits either a destabilization, an exchange move, or
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an elementary flype. Our main result (Theorem 1) states that there is a simple algorithmic
method for making these determinations. To understand this algorithm we need to consider
our representation of X in H anew using ‘rectangular diagrams’.
1.2 Rectangular diagrams and main results.
A horizontal arc, h ⊂ C , is any arc having parametrization {(1,t,z )|t ∈ [θ ,θ ]}, where the
1 0 1 2
horizontal level, z , is a fixed constant and |θ − θ | < 2π. The angular support of h is the
0 1 2
interval [θ ,θ ]. Horizontal arcs inherit a natural orientation from the forward direction of the
1 2
θ coordinate. A vertical arc, v ⊂ H , is any arc having parameterization {(r(t),θ ,z(t))|0 ≤
θ0 0
t ≤ 1,r(0) = r(1) = 1; and r(t) > 1, dz > 0 for t ∈ (0,1)}, where r(t) and z(t) are real-valued
dt
functions that are continuous on [0,1] and differentiable on (0,1). The angular position of v is
θ . The vertical support of v is the interval [z(0),z(1)]. (We remark that the parametrization
0
Monotonic Simplification 5
of the the vertical arcs will not be used in assigning orientation to the vertical arcs.)
axis axis
(a)
standard projection arc presentation
transition between
standard projection
and arc presentation
(b)
Figure 5:
Let X be an oriented link type in S3. Xη ∈ X is an arc presentation if Xη = h ∪v ∪···∪
1 1
h ∪v where:
k k
1. each h , 1 ≤ i ≤ k, is an oriented horizontal arc having orientation agreeing with the
i
components of Xη,
2. each v , 1≤ i ≤ k, is a vertical arc,
i
3. h ∩v ⊂ ∂h ∩∂v for 1 ≤ i≤ k and j (modk) = {i,i−1},
i j i j
4. the horizontal level of each horizontal arc is distinct,
5. the vertical position of each vertical arc is distinct.
6. the orientation of the horizontal arcs will agree with the forward direction of the θ coor-
dinate, and the orientation of the vertical arcs is assigned so as to make the components
of Xη oriented.
For a given arc presentation Xη = h ∪ v ∪ ··· ∪ h ∪ v there is a cyclic order to the
1 1 k k
horizontal levels of the h′s, as determined by their occurrence on the axis, and a cyclic order to
i
the angular position of the v′s, as determined by their occurrence in H. It is clear that given
j
two arc presentations with identical cyclic order for horizontal levels and angular positions
6 William W. Menasco
there is an ambient isotopy between the two presentations that corresponds to re-scaling of the
horizontal levels and angular positions along with the horizontal and angular support of the
arcs in the presentations. Thus, we will think of two arc presentations as being equivalent if
the cyclic orders (up to a change of indexing) of their horizontal levels and vertical positions
are equivalent.
We define the complexity of the arc presentation Xη as being C(Xη) = k, i.e. the number
of horizontal (or vertical) arcs.
Given a closed n-braid X and a corresponding standard projection π(X) we can easily
produce a (not necessarily unique) arc presentation Xη as illustrated in transition of Figure
5. Clearly, there is also the transition from an arc presentation Xη to an n-braid X with a
standard projection π(X). We will use the notation X−N→Xη or Xη−B→X it indicate these two
presentation transitions.
Figure 6: Illustration (a) corresponds to horizontal exchange moves and (b) corresponds to
vertical exchange moves. In each of the illustrated sequences dotted arcs are used to indicate
that the adjoining vertical arcs (for (a)) and horizontal arcs (for (b)) have two possible ways of
attaching themselves to the labeled solid arc.
We next define elementary moves on an arc presentation Xη. To setup these moves we let
Xη = h ∪v ∪···∪h ∪v with: z the horizontal position for h ; [θ ,θ ] the angular support
1 1 k k i i i1 i2
for h ; θ the angular position for v ; and [z ,z ] the vertical support for v (where 1 ≤ i ≤ k
i i i i1 i2 i
in all statements).
Horizontal exchange move—Let hj and hj′ be two horizontal arcs that are, first, consecutive
and, second, nested. Namely, first, when the points (0,0,z ), 1 ≤ i ≤ k, are viewed on the
i
z-axis (0,0,zj) and (0,0,zj′) are consecutive in the cyclic order of the horizontal levels. And
second, either [θj1,θj2]⊂ [θj1′,θj2′], or [θj1′,θj2′] ⊂ [θj1,θj2], or [θj1′,θj2′]∩[θj1,θj2] = ∅, i.e. nested.
Monotonic Simplification 7
Then we can replace Xη with X′η = h ∪···∪v′ ∪h′ ∪v′ ∪···∪v′ ∪h′ ∪v′ ∪···∪h ∪v
1 j−1 j j j′−1 j′ j′ k k
where for the corresponding horizontal positions we have zj′ = zj′ and zj′′ = zj, and the vertical
support for v′ , v′, v′ and v′ are adjusted in a corresponding manner. (See Figure 6(a).)
j−1 j j′−1 j′
Vertical exchange move—Let vj and vj′ be two vertical arcs that are again consecutive and
nested. Namely,theangularpositionsθj andθj′ areconsecutiveinthecyclicorderofthevertical
arcs of Xη. And, either [zj1,zj2] ⊂ [zj1′,zj2′], or [zj1′,zj2′] ⊂ [zj1,zj2], or [zj1′,zj2′]∩[zj1,zj2] = ∅,
i.e. nested. Then we can replace Xη with X′η = h ∪···∪h′ ∪v′ ∪h′ ∪···∪h′ ∪v′ ∪
1 j−1 j j j′−1 j′
h′j′ ∪···∪hk ∪vk where for the corresponding angular positions we have θj′ = θj′ and θj′′ = θj,
and the angular support for h′ , h′, h′ and h′ are adjusted in a corresponding manner.
j−1 j j′−1 j′
(See Figure 6(b).)
Horizontal simplification—Let h and h be two vertical arcs that are consecutive (as defined
j j+1
in the horizontal exchange move). Then we can replace Xη with X′η = h ∪ ··· ∪ v′ ∪
1 j−1
h′ ∪v′ ···∪h ∪v where: the horizontal position of h′ is z ; the angular support of h′ is
j j+1 k k j j j
[θj1,θj2]∪[θ(j+1)1,θ(j+1)2]; the angular position of vj′−1 (respectively vj′) is θj−1 (respectively
θ ); and the vertical support of v′ and v′ are adjusted correspondingly.
j j−1 j
Vertical simplification—Let v and v be two vertical arcs that are consecutive (as defined in
j j+1
theverticalexchangemove). ThenwecanreplaceXη withX′η = h ∪···∪h′∪v′∪h′ ···∪h ∪
1 j j j+2 k
v where: theangularposition ofv′ is θ ; thevertical supportof v′ is [z ,z ]∪[z ,z ];
k j j j j1 j2 (j+1)1 (j+1)2
the horizontal position of h′ (respectively h′ ) is z (respectively z ); and the angular
j j+2 j1 (j+1)1
support of h′ and h′ are adjusted correspondingly.
j j+2
Given an arc presentation Xη we notice that for any sequence of elementary moves applied
to Xη, the complexity measure C(Xη) is non-increasing. That is, any sequence of elementary
moves which includes the uses of either horizontal or vertical simplification will be monotonic
simplification.
One would hope that for a closed n-braid X which admits, respectively, a destabilization,
exchange moves, or elementary flype, there exists a sequence of elementary moves to the arc
presentation Xη (coming from X−N→Xη) such that for the resulted arc presentation X′η, the
closed n-braid X′ coming from X′η−B→X′ admits, respectively, a destabilization, exchange
move or elementary flype (as seen from the standard projection π(X′)). Unfortunately, this
is too good to be true. In order to produce a X′ that admits the assumed isotopy it may be
necessary to increase the number of arcs in the arc presentation. At first glance this seems to
disturb our ability to maintain the characteristic of monotonic simplification. However, it is
possible to control the manner in which we introduce additional arcs in the arc presentation to
maintain monotonic simplification using an altered complexity measure. In particular, we have
the following move for introducing additional arcs in an arc presentation.
Horizontal shear—Foragiven arcpresentation Xη = h ∪v ∪···∪h ∪v letI beacollection of
1 1 k k
8 William W. Menasco
Figure 7: The left illustration is Xη and the right is Xη. We can recover Xη from Xη through
I I
a sequence of horizontal (or vertical) simplifications. The number of additional arcs allowed
when we shear is undetermined.
disjointclosed angularintervaloftheform[θ ,θ +ǫ],0 ≤ k ≤ lsuchthatforθ ∈ I wehavethat
k k
H contains no vertical arc of Xη. Suppose h be a horizontal arc whose horizontal support
θ i
contains intersect I. Then a horizontal shear of Xη along I is new shear arc presentation
Xη that is related to Xη via the alterations illustrated in Figure 7. In particular, Xη can be
I
obtained from Xη via a sequence of horizontal simplifications that occur inside the angular
I
interval I. There is no limit on the number of arcs of Xη, but the growth in arcs can only
I
occur inside the interval I. We allow shearing on any or all of the horizontal arcs of Xη whose
angular support intersects the intervals of I.
Given a horizontal shear Xη of Xη let k′ be the number of horizontal arcs of Xη whose
I I
angular support does not intersect I. Let k′′ be the number of point of intersection between
horizontalarcsand∂I. Wedefinethecomplexity measureC(Xη) = k′+1k′′. Thereadershould
I 2
notice that C(Xη) = C(Xη). (We will abuse notation using C for both complexities. Thus, the
I
reader is required to refer to the context in determining which complexity is being referenced.)
Since the complexity measure for Xη does not measure the total number of arcs of Xη we
I I
can treat Xη as the equivalence class of all presentations coming from the alteration depicted
I
in Figure 7. For our purposes we will allow I to be: a single interval {[θ ,θ +ǫ]} when we are
0 0
discussing whether X admits a destabilization; two disjoint intervals {[θ ,θ +ǫ ],[θ ,θ +ǫ ]}
1 1 1 2 2 2
when we are discussing whether X admits an exchange move; and three disjoint intervals
{[θ ,θ +ǫ ],[θ ,θ +ǫ ],[θ ,θ +ǫ ]} when we are discussing whether X admits an elementary
1 1 1 2 2 2 3 3 3
flype. Once the number of intervals in I has been specified it is easily observed that due to the
ability to rescale the angular support of horizontal arcs and the intervals of I given Xη there
are combinatorially only a finite number of possible Xη.
I
Next, we adapt the elementary moves on arc presentations to moves on Xη so as to have
I
them be monotonic simplifications with respect to C(XIη). First, if the horizontal arcs hj,hj′ of
our original horizontal exchange move (respectively, vertical arcs vj,vj′ of the vertical exchange
Monotonic Simplification 9
move) are contained in a single interval of either [0,2π]\I or I, then we allow the performance
of a horizontal (respectively, vertical) exchange move. Second, in a similar fashion if h ,h
j j+1
(respectively, v ,v ) are contained in a single interval of either [0,2π]\I or I then we again
j j+1
allow theperformanceof a horizontal (respectively, vertical) simplification. Inboth cases, if the
move occurs in an interval of I the complexity of our resulting Xη is unchanged. For horizontal
I
or vertical exchange moves, independent of what interval the move occurs in, the complexity
measure is unchanged. For the simplification moves the occur in intervals of [0,2π] \ I the
η
complexity of theresultingX decreases. Thus,weachieve monotonic simplification inouruses
I
ofourpreviouselementary moves byrestrictingtheirapplication totheintervals ([0,2π]\I)∪I.
Figure 8: Figure (a) illustrates a shear-horizontal exchange and (b) illustrates a shear-vertical
simplification. Again, in each of the illustrated sequences dotted arcs are used to indicate that
the adjoining vertical arcs (for (a)) and horizontal arcs (for (b)) have two possible ways of
attaching themselves to the labeled solid arc.
We now add two new elementary moves that utilizes the intervals of I.
Shear horizontal exchange move—We refer to Figure 8(a). Let hj and hj′ be two horizontal
η
arcs of X that are consecutive and nested with respect to I. That is, first, for an interval
I
I ⊂ [0,2π]\I when we consider all of the horizontal arcs of Xη the horizontal levels of h and
I j
hj′ are consecutive in the ordering along A. Second, the angular support of hj restricted to
an interval I ⊂ [0,2π]\I is contained inside the angular support of hj′ restricted to I. Then
we can interchange the horizontal level of these to arcs. This is achieved by first applying a
horizontal shear to the portion of hj and hj′ that is contained in I. Thus, when we consider
the resulting arc presentations an original horizontal exchange move is realizable. We abuse
notation referring to the resulted arc presentation as Xη. Notice that C(Xη) remains constant.
I I
Shear vertical simplification—We refer to Figure 8(b). Let v be a vertical arc of Xη that are
j I
consecutive with respect to I. That is, for an interval I ⊂ [0,2π] \I there is no vertical arc
whose angular position is between v and I. We can then push v into I. Again, we abuse
j j
notation by referring to the resulted arc presentation as Xη. Notice that C(Xη) is decreased
I I
by a count of one.
We are now in a position to state our main results.
10 William W. Menasco
Theorem 1 Let X be a closed n-braid such that B (X) admits, respectively, a destabilization,
n
exchange move or elementary flype. Consider any arc presentation coming from the a pre-
sentation transition X−N→Xη. Then there exists a set of intervals I and a sequence of arc
presentations
Xη = X0→X1→···→Xl = X′η
I I
such that:
1. If B (X) admits, respectively, a destabilization, exchange move or elementary flype then
n
I has, respectively, one, two or three intervals.
2. Xi+1 is obtained from Xi via one of the elementary moves—horizontal exchange move,
vertical exchange move, horizontal simplification, vertical simplification, shear horizontal
exchange move, and shear vertical simplification. All of these moves are with respect to
the intervals of I.
3. C(Xi+1) ≤ C(Xi) for 0 ≤ i ≤ l. In particular, if Xi→Xi+1 corresponds to a horizontal,
vertical or shear horizontal exchange move then C(Xi) = C(Xi+1). If it corresponds to
a horizontal, vertical or shear vertical simplification then C(Xi+1) < C(Xi). Thus, our
sequence is monotonic simplification.
4. The closed n-braid obtained from the presentation transition X′η−B→X′ admits, respec-
I
tively, a destabilization, exchange move or elementary flype (as seen from the standard
projection π(X′)).
Itshould bereadily evident thatgiven an arcpresentation Xη onecan generate only finitely
many equivalent arc presentations through any sequence of horizontal and vertical exchange
moves. As previously remarked combinatorially there are only finitely many possible choices
for I (for one, two or three intervals). Thus, as equivalence classes there are only finitely
many Xη. Finally, since the number of arcs of Xη in [0,2π]\I is fixed and bounded by the
I I
number of arcs of Xη, and since our elementary moves on the arcs in [0,2π]\I is monotonic
simplification then there are only a finite number of possible X′η that can be produced by any
I
sequence of elementary moves. The production of such a finite set is easily seen as algorithmic.
Moreover, given an arc presentation from the cyclic order information of the horizontal levels
and the vertical positions it can readily be determined whether either a horizontal or vertical
simplification can be applied. Therefore, Theorem 1 implies that following corollary.
Corollary 2 There exists an algorithm for deciding whether an closed n-braid is braid isotopic
to one that admits either a destabilization, exchange move, or elementary flype.