j. differential geometry
55 (2000) 325-354
ON TRANSVERSALLY SIMPLE KNOTS
JOAN S. BIRMAN & NANCY C. WRINKLE
Abstract
This paper studies knots that are transversal to the standard contact
structure in R3 , bringing techniques from topological knot theory to bear
on their transversal classification. We say that a transversal knot type T K is
transversally simple if it is determined by its topological knot type K and its
Bennequin number. The main theorem asserts that any T K whose associated K satisfies a condition that we call exchange reducibility is transversally
simple.
As a first application, we prove that the unlink is transversally simple,
extending the main theorem in [10]. As a second application we use a new
theorem of Menasco [17] to extend a result of Etnyre [11] to prove that
all iterated torus knots are transversally simple. We also give a formula
for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on
K in order to prove that any associated T K is transversally simple. We
also give examples of pairs of transversal knots that we conjecture are not
transversally simple.
1. Introduction
Let ξ be the standard contact structure in oriented 3-space R3 =
(ρ, θ, z), that is the kernel of α = ρ2 dθ + dz. An oriented knot K in
contact R3 is said to be a transversal knot if it is transversal to the
planes of this contact structure. In this paper, the term ‘transversal’
refers to this contact structure only. If the knot K is parametrized by
′ (t)
= −(ρ(t))2 for
(ρ(t), θ(t), z(t)), then K is transversal if and only if zθ′ (t)
Received February 10, 2000, and, in revised form, December 11, 2000. The first
author received partial support from the U.S. National Science Foundation under
Grants DMS-9705019 and DMS-9973232. The second author is a graduate student
in the Mathematics Department of Columbia University. She was partially supported
under the same grants.
325
326
joan s. birman & nancy c. wrinkle
every t. We will assume throughout that α > 0 for all t, pointing out
later how our arguments adapt to the case α < 0.
For the benefit of the reader who may be unfamiliar with the standard contact structure, Figure 1(a) illustrates typical 2-planes in this
structure in R3 , when z is fixed, and ρ and θ vary. The structure is
radially symmetric. It is also invariant under translation of R3 parallel
to the z axis. Typical 2-planes are horizontal at points on the z axis and
twist clockwise (if the point of view is out towards increasing ρ from the
z axis,) as ρ → ∞.
There has been some discussion about whether the planes tend to
vertical as ρ → ∞ or to horizontal. If one looks at the limit of α, it
appears that the limit is a rotation of π/2. However, if one derives the
standard contact structure on S 3 from the Hopf fibration, as described
below, and wants to have this structure be consistent with the one
defined on R3 , it is necessary to take the limit to be a rotation up to
(but not through, as a rotation of more than π results in an overtwisted
structure) π. The resulting contact structures on R3 are equivalent,
through a contactomorphism that untwists the planes from π to π/2.
Thus we can work in the standard contact structure on S 3 , which has
horizontal planes in the limit, while using the contact form α, which
induces vertical planes in the limit. The details are below.
z
θ
(a)
(b)
Figure 1: The standard contact structure on R3 and the Hopf fibration
on S 3 .
The standard contact structure extends to S 3 and has an interesting
interpretation in terms of the geometry of S 3 . Let
S 3 = {(z1 , z2 ) = (ρ1 eiθ1 , ρ2 eiθ2 ) ∈ C2 / ρ21 + ρ22 = constant.}
Then ξ is the kernel of ρ21 dθ1 + ρ22 dθ2 . The field of 2-planes may be
on transversally simple knots
327
thought of as the field of hyperplanes which are orthogonal to the fibers
of the Hopf fibration π : S 3 → S 2 . See Figure 1(b) for a picture of
typical fibers. Identify the z axis in R3 with the core of one of the solid
tori. There is a fiber through each point in S 3 , and the 2-plane at a
point is orthogonal to the fiber through the point.
The (topological) type K of a knot K ⊂ R3 is its equivalence class
under isotopy of the pair (K, R3 ). A sharper notion of equivalence is its
′ (t)
transversal knot type T K, which requires that zθ′ (t)
+ (ρ(t))2 be positive
at every point of the deformed knot during every stage of the isotopy.
The difference between these two concepts is the central problem studied
in this paper.
A parametrized knot K ⊂ R3 is said to be represented as a closed
braid if ρ(t) > 0 and θ′ (t) > 0 for all t. See Figure 2(a). It was proved
by Bennequin in §23 of [1] that every transversal knot is transversally
isotopic to a transversal closed braid. This result allows us to apply
results obtained in the study of closed braid representatives of topological knots to the problem of understanding transversal isotopy. We carry
Bennequin’s approach one step further, initiating a comparative study of
the two equivalence relations: topological equivalence of two closed braid
representatives of the same transversal knot type, via closed braids,
and transversal equivalence of the same two closed braids. Transversal
equivalence is of course more restrictive than topological equivalence.
Topological equivalence of closed braid representatives of the same
knot has been the subject of extensive investigations by the first author
and W. Menasco, who wrote a series of six papers with the common
title Studying links via closed braids. See, for example, [6] and [5]. See
also the related papers [3] and [2]. In this paper we will begin to apply
what was learned in the topological setting to the transversal problem. See also Vassiliev’s paper [21], where we first learned that closed
braid representations of knots were very natural in analysis; also our
own contributions in [9], where we began to understand that there were
deep connections between the analytic and the topological-algebraic approaches to knot theory.
A well-known invariant of a transversal knot type T K is its Bennequin number β(T K). It is not an invariant of K. We now define it
in a way that will allow us to compute it from a closed braid representative K of T K. The braid index n = n(K) of a closed braid K is the
linking number of K with the oriented z axis. A generic projection of K
onto the (ρ, θ) plane will be referred to as a closed braid projection. An
joan s. birman & nancy c. wrinkle
328
z axis
+
K
θ2
K
θ1
(b)
(c)
(a)
Figure 2: (a) Closed braid; (b) Example of a closed braid projection;
(c) Positive and negative crossings
example is given in Figure 2(b). The origin in the (ρ, θ) plane is indicated as a black dot; our closed braid rotates about the z axis in the
direction of increasing θ. The algebraic crossing number e = e(K) of the
closed braid is the sum of the signed crossings in a closed braid projection, using the sign conventions given in Figure 2(c). If the transversal
knot type T K is represented by a closed braid K, then its Bennequin
number β(T K) is:
β(T K) = e(K) − n(K).
Since e(K) − n(K) can take on infinitely many different values as K
ranges over the representatives of K, it follows that there exist infinitely
many transversal knot types for each topological knot type. It was
proved by Bennequin in [1] that e(K) − n(K) is bounded above by
−χ(F), where F is a spanning surface of minimal genus for K. Fuchs
and Tabachnikov gave a different upper bound in [13]. However, sharp
upper bounds are elusive and are only known in a few very special cases.
We now explain the geometric meaning of β(T K). Choose a point
(z1 , z2 ) = (x1 + ix2 , x3 + ix4 ) ∈ T K ⊂ S 3 . Thinking of (z1 , z2 ) as a
point in R4 , let p = (x1 , x2 , x3 , x4 ). Let q = (−x2 , x1 , −x4 , x3 ) and
let r = (−x3 , x4 , x1 , −x2 ). Then r · p = r · q = p · q = 0. Then q
may be interpreted as the outward-drawn normal to the contact plane
at p, so that r lies in the unique contact plane at the point p ∈ S 3 .
Noting that a transversal knot is nowhere tangent to the contact plane,
it follows that for each point p on a transversal knot T K ⊂ S 3 the
vector r gives a well-defined direction for pushing T K off itself to a
related simple closed curve T K ′ . The Bennequin number β(T K) is the
on transversally simple knots
329
linking number Lk(T K, T K ′ ). See §16 of [1] for a proof that β(T K) is
invariant under transverse isotopy and that β(T K) = e(K) − n(K).
We say that a transversal knot is transversally simple if it is characterized up to transversal isotopy by its topological knot type and its
Bennequin number. In [10] Eliashberg proved that a transversal unknot
is transversally simple. More recently Etnyre [11] used Eliashberg’s techniques to prove that transversal positive torus knots are transversally
simple.
Our first main result, Theorem 1, asserts that if a knot type K is
exchange reducible (a condition we define in Section 2), then its maximum Bennequin number is realized by any closed braid representative
of minimum braid index. As an application, we are able to compute the
maximum Bennequin number for all iterated torus knots. See Corollary 3. Our second main result, Theorem 2, asserts that if T K is a
transversal knot type with associated topological knot type K, and if K
is exchange reducible, then T K is transversally simple. As an application, we prove in Corollary 2 that transversal iterated torus knots are
transversally simple. The two corollaries use new results of Menasco
[17], who proved (after an early version of this paper was circulated)
that iterated torus knots are exchange reducible. In Theorem 3 we
establish the existence of knot types that are not exchange reducible.
Here is an outline of the paper. Section 2 contains our main results.
In it we will define the concept of an exchange reducible knot and prove
Theorems 1 and 2. In Section 3 we discuss examples, applications and
possible generalizations.
Acknowlegements. We thank Oliver Dasbach, William Menasco
and John Etnyre for conversations and helpful suggestions relating to
the work in this paper. We are especially grateful to Menasco for the
manuscript [17]. In an early version of this paper, we conjectured that
iterated torus knots might be exchange reducible. We explained our
conjecture to him, and a few days later he had a proof! We also thank
Wlodek Kuperberg for sharing his beautiful sketch of the Hopf fibration
(Figure 1(b)) with us. Finally, we thank William Gibson, who noticed
our formula for the Bennequin number of iterated torus knots in an
earlier version of this paper and pointed out to us in a private conversation that it could be related to the upper bound which was given by
Bennequin in [1], by the formula in Corollary 3, part (2).
330
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joan s. birman & nancy c. wrinkle
Remarks on techniques
This subsection contains a discussion of the techniques used in the
manuscripts [4], [5], [7] and [17], tools which form the foundation on
which the results of this paper are based. We compare these techniques
to those used in the manuscripts [1], [10] and [11], although Bennequin’s
paper rightfully belongs in both sets. This description and comparison
is of great interest, but is not essential for the reading of this paper or
the digestion of its arguments.
We concern ourselves with two known foliations of an orientable
surface F associated to K: the characteristic foliation ξF from contact
geometry and the topological foliation from braid theory. The characteristic foliation of F is the line field ξ ∩ T F, given by the intersection of
the planes of the contact structure with the planes of the tangent space
of the surface, which is then integrated to a singular foliation of F. The
topological foliation is the foliation of F which is induced by intersecting
the foliation of R3 minus the z axis (see Figure 3), with the surface F.
z
θ
Figure 3: Half-planes in the braid structure on R3 .
The foliation of R3 minus the z-axis by half-planes is called the standard
braid structure on R3 − the z-axis. The surface used in [10] and [5] was
a spanning surface for K; in [4] it was a 2-sphere which intersects K
twice; in [7] it was a torus in the complement of K; in [11] and [17] it
is a torus T ⊂ S 3 on which K is embedded. Menasco also considers
the foliation of a meridian disc in the solid torus which T bounds. The
characteristic foliation of a surface (associated to a transversal knot or
on transversally simple knots
331
to another transversal or Legendrian curve,) is a tool of study in contact
geometry. It was the main tool in the manuscripts [1], [10] and [11].
In topological knot theory, one studies the topological foliation of
F defined above. The review article [2] may be useful to the reader
who is unfamiliar with this area. The study of the topological foliations
has produced many results, for example the classification of knots that
are closed 3-braids [6] and a recognition algorithm for the unknot [3].
Braid theory was also a major tool in the work in [1], but it appears
that Bennequin’s detailed study of the foliation is based entirely on the
characteristic foliation, as it occurs for knots in R3 and S 3 . To the
best of our knowledge this paper contains the first application of braid
foliations to the study of transversal knots.
We note some similarities between the two foliations: The characteristic foliation is oriented and the braid foliation is orientable. (The
orientation is ignored, but a dual orientation, determined by an associated flow, plays an equivalent role.) The foliations can be made to agree
in the limiting case, as ρ → ∞ (see the comments above Figure 1).
After an appropriate isotopy of F both foliations have no leaves that
are simple closed curves. Also, their singularities are finite in number,
each being either an elliptic point or a hyperbolic point (the hyperbolic
point corresponding to a saddle-point tangency of F with the 2-planes
of the structure). The signs of the singularities of each foliation are determined by identical considerations: the surface is naturally oriented
by the assigned orientation on the knot. If at a singularity the orientation of the surface agrees (resp. disagrees) with the orientation of the
foliation, then the singularity is positive (resp. negative). See Figure 4.
p
h
K
Figure 4: A positive elliptic singularity (p) and a positive hyperbolic
singularity (h).
joan s. birman & nancy c. wrinkle
332
In both foliations the hyperbolic singularities are 4-pronged singularities. If s is a hyperbolic singular point, then the four branches of
the singular leaf through s end at either elliptic points or at a point on
K. (The condition that no singular leaves of the characteristic foliation connect hyperbolic points is a genericness assumption appearing in
the literature on Legendrian and transversal knots). The three possible
cases are illustrated in Figure 5. In that figure the elliptic points are
depicted as circles surrounding ± signs (the sign of the elliptic singularity) and the hyperbolic singularities are depicted as black dots. Two
of the four branches of the singular leaf end at positive elliptic points.
The other two end at either two negative elliptic points, or one negative
elliptic point and one point on K, or two points on K.
-
-
type bb
type ab
K
type aa
+
+
-
+
+
+
+
K
K
Figure 5: The three types of hyperbolic singularities.
There are also differences between the two foliations. In the braid
foliation, elliptic points always correspond to punctures of the surface by
the z axis. In the characteristic foliation, elliptic points on the surface
may or may not correspond to punctures by the z-axis. That is, there
may be elliptic points not corresponding to punctures, and there might
be punctures not corresponding to elliptic points. Here is an example. In
the braid foliation, if there is a piece of the surface along the boundary,
foliated by a single positive pair of elliptic and hyperbolic singularities,
then the only possible embedding for that piece is shown in Figure 4.
On the other hand, in the characteristic foliation, if there is a piece of
the surface, also along the boundary, also foliated by a positive elliptichyperbolic pair, then the corresponding embedding may or may not be
the one shown in Figure 4. The embeddings will coincide if the tangent
to the surface at the z axis is horizontal.
In work on the braid foliation one uses certain properties that appear
to have been ignored in work based upon the characteristic foliation.
For example, the work on braid foliations makes much of the distinction between the three types of hyperbolic singularities which we just
illustrated in Figure 5, calling them types bb, ab and aa. The resulting
on transversally simple knots
333
combinatorics play a major role in the study of braid foliations. It seems
to us that the distinction between bb, ab and aa singularities can also be
made in the situation of the characteristic foliation, but that this has
not been done.
In the braid foliation the elliptic points have a natural cyclic order on
the z axis, if we are considering the ambient space as S 3 and the braid
axis as one of the core circles of the Hopf fibration, and the hyperbolic
points have a natural cyclic order in 0 ≤ θ ≤ 2π. These orderings do not
seem useful in the contact setting. On the other hand, the characteristic
foliation is invariant under rotation by θ and translation by z, so the
interesting parameter seems to be the coordinate ρ.
An essential tool in manipulating and simplifying the characteristic
foliation is the Giroux Elimination Lemma ([14], [10]), which allows one
to ‘cancel’ pairs of same sign singularities. In topological knot theory
different modifications have been introduced that are the braid foliation
analogue of isotopies of the Giroux Elimination Lemma, see [5] and also
[2]. They are called ab exchange moves and bb exchange moves, and
they use pairs of Giroux-like cancellations, but on a much larger scale.
2. Exchange reducibility and transversal simplicity
Our initial goal is to motivate and define the concept of exchange
reducibility. Let K be a topological knot type and let K be a closed nbraid representative of K. We consider the following three modifications
of K
• Our first modification is braid isotopy, that is, an isotopy in the
complement of the braid axis. In [18] it is proved that isotopy
classes of closed n-braids are in one-to-one correspondence with
conjugacy classes in the braid group Bn . Since the conjugacy
problem in the braid group is a solved problem, each conjugacy
class can then be replaced by a unique representative that can be
assumed to be transversal. Braid isotopy preserves the Bennequin
number since it preserves both braid index and algebraic crossing
number.
• Our second move is destabilization. See Figure 6(a). The box labeled P contains an arbitrary (n − 1)-braid, and the label n − 2 on
the braid strand denotes n − 2 parallel braid strands. The destabilization move reduces braid index from n to n − 1 by removing
joan s. birman & nancy c. wrinkle
334
a ‘trivial loop’. If the trivial loop contains a positive crossing, the
move is called a positive or + destabilization. Positive destabilization reduces algebraic crossing number and preserves the Bennequin number. Negative (−) destabilization increases the Bennequin number by 2.
• Our third move is the exchange move. See Figure 6(b). In general the exchange move changes conjugacy class and so cannot
be replaced by braid isotopy. The exchange move preserves both
braid index and algebraic crossing number, hence preserves the
Bennequin number.
n-2
1
P
P
Q
P
n-2
n-2
P
n-2
(a)
Q
n-2
1
1
n-2
(b)
Figure 6: (a) positive destabilization and (b) The exchange move
To motivate our definition of exchange reducibility, we recall the
following theorem, proved by the first author and W. Menasco:
Theorem A ([5], with a simplified proof in [2]). Let K be a closed
n-braid representative of the m-component unlink. Then K may be
simplified to the trivial m-braid representative, i.e., a union of m disjoint round planar circles, by a finite sequence of the following three
changes: braid isotopies, positive and negative destabilizations, and exchange moves.
Motivated by Theorem A, we introduce the following definition:
Definition. A knot type K is said to be exchange reducible if an arbitrary closed braid representative K of arbitrary braid index n can be
changed to an arbitrary closed braid representative of minimum braid index nmin (K) by a finite sequence of braid isotopies, exchange moves and
±-destabilizations. Note that this implies that any two minimal braid
index representatives are either identical or are exchange-equivalent, i.e.,
are related by a finite sequence of braid isotopies and exchange moves.
on transversally simple knots
335
Our first result is:
Theorem 1.
If K is an exchange reducible knot type, then the
maximum Bennequin number of K is realized by any closed braid representative of braid index nmin (K).
The proof of Theorem 1 begins with a lemma. In what follows,
we understand “transversal isotopy” to mean a topological isotopy that
preserves the condition α = ρ2 dθ + dz > 0 at every point of the knot
and at every stage of the isotopy.
Lemma 1. If a transversal closed braid is modified by one of the
following isotopies, then the isotopy can be replaced by a transversal
isotopy:
(1) Braid isotopy.
(2) Positive stabilization or positive destabilization.
(3) An exchange move.
Proof of Lemma 1.
Proof of (1). Since the braid strands involved in the isotopy will
be ≫ ǫ away from the z-axis at each stage (so avoiding −ρ2 = 0), any
isotopy will be transversal if we keep the strands involved ”relatively
flat” (dz/dθ ∼ 0) at each stage. Since everything is happening locally
there is space to flatten the strands involved without changing the braid.
Proof of (2). See Figure 7(a). Consider a single trivial loop around
the z axis, with a positive crossing. We have dθ > 0 along the entire
p
p
positive destabilization
negative destabilization
Figure 7: Destabilization, with a singularity at s, where dθ = 0 and
ρ = 0.
length of the loop since we are working with a closed braid. For a
positive crossing we have dz ≥ 0 throughout the loop as well. Therefore
the inequality dz/dθ > −ρ2 is true for all non-zero real values of ρ.
Crossing the z axis to destabilize the braid results in at least one singular
point, where dθ = 0, but if we continue to keep dz ≥ 0 then in the limit,
joan s. birman & nancy c. wrinkle
336
as −ρ2 → 0 from the negative real numbers, dz/dθ goes to ∞ through
the positives. Therefore dz/dθ = −ρ2 at any stage in the isotopy.
Proof of (3). The sequence of pictures in Figure 8 shows that an
exchange move can be replaced by a sequence of the following moves:
isotopy in the complement of the z axis, positive stabilization, isotopy
again, and finally positive destabilization. Claim 3 then follows from
Claims 1 and 2. q.e.d.
Q
Braid isotopy
Q
P
P
Braid isotopy
Braid isotopy
Q
+ Stabilization
P
Braid isotopy
P
Q
Q
P
Q
Braid isotopy
P
Q
P
Q
P
+ Destabilization
Figure 8: An exchange move corresponds to a sequence consisting of
braid isotopies, a single positive stabilization and a single positive destabilization.
Remark 1. Observe that the argument given to prove (2) simply
doesn’t work for negative destabilization. See figure 7(b). The singularity in this destabilization is a point at which dz/dθ = −ρ2 in the limit.
Indeed, a negative destabilization can’t be modified to one which is
transversal, because the Bennequin number (an invariant of transversal
knot type) changes under negative destabilization.
Remark 2. If we had chosen α < 0 along the knot, we would consider negative stabilizations and destabilizations as transversal isotopies
and would use those instead of positive stabilizations and destabilizations in the exchange sequence. All the other moves translate to the
negative setting without change.
Proof of Theorem 1. Let K be an arbitrary closed braid representative of the exchange reducible knot type K. Let K0 be a minimum
on transversally simple knots
337
braid index representative of K, obtained from K by the sequence described in the definition of exchange reducibility. We must prove that
the transversal knot T K0 associated to K0 has maximum Bennequin
number for the knot type K. Note that in general K0 is not unique,
however it will not matter, for if K0′ is a different closed braid representative of minimal braid index, then K0 and K0′ are related by a
sequence of braid isotopies and exchange moves, both of which preserve
both braid index and algebraic crossing number, so β(K0′ ) = β(K0 ).
We obtain K0 from K by a sequence of braid isotopies, exchange
moves, and ±-destabilizations. Braid isotopy, exchange moves and positive destabilization preserve β(T K), but negative destabilization increases the Bennequin number by 2, so the sequence taking K to K0
changes the Bennequin number from β(T K) = c to β(T K0 ) = c + 2p,
where p is the number of negative destabilizations in the sequence. The
question then is whether c + 2p is maximal for the knot type K. If c + 2p
is less than maximal, then there exists some other closed braid representative K ′ of the knot type K with maximum β(T K ′ ) > β(T K0 ).
Since K0 has minimum braid index for the knot type K, it must be
that n(K ′ ) ≥ n(K0 ). If n(K ′ ) = n(K0 ), then the two braids are equivalent by a sequence of Bennequin number preserving exchange equivalences, so suppose instead that n(K ′ ) > n(K0 ). Then, since K ′ is
a closed braid representative of the exchange reducible knot type K,
there must exist a sequence of braid isotopies, exchange moves, and
±-destabilizations taking K ′ to a minimum braid index braid representative K0′ . Since β(T K ′ ) is assumed to be maximum, and none of
the moves in the sequence taking K ′ to K0′ reduce Bennequin number,
it must be that β(T K0′ ) = β(T K ′ ). But since K0′ and K0 are both
minimum braid index representatives of K, they must be equivalent by
a sequence of Bennequin number preserving exchange moves and isotopies. Thus β(T K0 ) = β(T K0′ ). q.e.d.
Our next result is:
Theorem 2. If T K is a transversal knot type with associated topological knot type K, where K is exchange reducible, then T K is transversally simple.
The proof of Theorem 2 begins with two lemmas. Our first lemma
had been noticed long ago by the first author and Menasco, who have
had a long collaboration on the study of closed braid representatives of
knots and links. However, it had never been used in any of their papers.
It is therefore new to this paper, although we are indebted to Menasco
338
joan s. birman & nancy c. wrinkle
for his part in its formulation. A contact-theory analogue of Lemma 3,
below, appears as Lemma 3.8 of [11].
Lemma 2. Using exchange moves and isotopy in the complement
of the braid axis, one may slide a trivial loop on a closed braid from one
location to another on the braid.
Proof of Lemma 2.
See Figure 9. It shows that, using braid
isotopy and exchange moves, we can slide a trivial negative loop past
any crossing to any place we wish on the braid. The argument for sliding
a positive trivial loop around the braid is identical. q.e.d.
braid isotopy
exchange move
braid isotopy
Figure 9: An exchange move that allows a negative trivial loop to slide
along a braid.
Lemma 3. Let K1 and K2 be closed n-braids that are exchangeequivalent. Let L1 and L2 be (n + 1)-braids that are obtained from K1
and K2 by either negative stabilization on both or positive stabilization
on both. Then L1 and L2 are exchange-equivalent.
Proof of Lemma 3. We already know there is a way to deform K1
to K2 , using exchange equivalence. Each braid isotopy may be broken
up into a sequence of isotopies, each of which only involves local changes
on some well-defined part of the braid. (For example, the defining relations in the braid group are appropriate local moves on cyclic braids).
Similarly, exchange moves have local support. It may happen that the
trivial loop which we added interferes with the support of one of the
isotopies or exchange moves. If so, then by Lemma 2 we may use exchange equivalence to slide it out of the way. It follows that we may
deform L1 to L2 by exchange equivalence. q.e.d.
Proof of Theorem 2. We are given an arbitrary representative of
the transversal knot type T K. Let K be the associated topological knot
type. By the transversal Alexander’s theorem [1] we may modify our
representative transversally to a transversal closed n-braid K = T K
on transversally simple knots
339
that represents the transversal knot type T K and the topological knot
type K. By the definition of exchange reducibility, we may then find a
finite sequence of closed braids
K = K1 → K2 → · · · → Km−1 → Km ,
all representing K, such that each Ki+1 is obtained from Ki by braid
isotopy, a positive or negative destabilization or an exchange move,
and such that Km is a representative of minimum braid index nmin =
nmin (K) for the knot type K. The knots K1 , . . . , Km in the sequence
will all have the topological knot type K.
In general K will have more than one closed braid representative of
minimum braid index. Let M0 (K) = {M0,1 , M0,2 . . . } be the set of all
minimum braid index representatives of K, up to braid isotopy. Clearly
Km ∈ M0 . By [1], each M0,i ∈ M0 may be assumed to be a transversal
closed braid.
By Theorem 1 each M0,i has maximal Bennequin number for all
knots that represent K. In general this Bennequin number will not be
the same as the Bennequin number of the original transversal knot type
T K. By Lemma 1 the moves that relate any two M0,i , M0,j ∈ M0 may
be assumed to be transversal. After all these modifications the closed
braids in the set M0 will be characterized, up to braid isotopy, by their
topological knot type K, their braid index nmin (K) and their Bennequin
number βmax (K).
If the transversal knot type T K had Bennequin number βmax (K),
it would necessarily follow that T K is characterized up to transversal
isotopy by its ordinary knot type and its Bennequin number. Thus we
have proved the theorem in the special case of transversal knots that
have maximum Bennequin number.
We next define new sets M1 , M2 , . . . of transversal knots, inductively. Each Ms is a collection of conjugacy classes of closed (nmin (K)+
s)-braids. We assume, inductively, that the braids in Ms all have topological knot type K, braid index nmin (K) + s and Bennequin number
βmax (K) − 2s. Also, their conjugacy classes differ at most by exchange
moves. Also, the collection of conjugacy classes of (nmin (K) + s)-braids
in the set Ms is completely determined by the collection of conjugacy
classes of braids in the set M0 . We now define the set Ms+1 by choosing an arbitrary closed braid Mi,s in Ms and adding a trivial negative
loop. Of course, there is no unique way to do this, but by Lemma 2 we
can choose one such trivial loop and use exchange moves to slide it completely around the closed braid Mi,s . Each time we use the exchange
340
joan s. birman & nancy c. wrinkle
move of Lemma 2, we will obtain a new conjugacy class, which we then
add to the collection Ms+1 . The set Ms+1 is defined to be the collection
of all conjugacy classes of closed braids obtained by adding trivial loops
in every possible way to each Mi,s ∈ Ms . The closed braids in Ms+1
are equivalent under braid isotopy and exchange moves. They all have
topological knot type K, braid index nmin (K) + s + 1, and Bennequin
number βmax (K) − 2(s + 1). The collection of closed braids in the set
Ms+1 is completely determined by the collection of closed braids in Ms ,
and so by the closed braids in M0 .
In general negative destabilizations will occur in the chain K1 →
Km . Our plan is to change the order of the moves in the sequence
K1 → Km , pushing all the negative destabilizations to the right until
we obtain a new sequence, made up of two subsequences:
K = K1⋆ → K2⋆ → · · · → Kr⋆ = K0′ → · · · → Ks′ = Kp ,
where Kp has minimum braid index nmin (K). The first subsequence S1 ,
will be K = K1⋆ → K2⋆ → · · · → Kr⋆ , where every Ki⋆ is a transversal representative of T K and the connecting moves are braid isotopy,
positive destabilizations and exchange moves. The second subsequence,
′
S2 , is Kr⋆ = K0′ → · · · → Kq′ , where every Ki+1
is obtained from Ki′
by braid isotopy and a single negative destabilization. Also, Kq′ has
minimum braid index nmin (K).
To achieve the modified sequence, assume that Ki → Ki+1 is the first
negative destabilization. If the negative trivial loop does not interfere
with the moves leading from Ki+1 to Km , just renumber terms so that
the negative destabilization becomes Km and every other Kj , j > i
becomes Kj−1 . But if it does interfere, we need to slide it out of the
way to remove the obstruction. We use Lemma 2 to do that, adding
exchange moves as required.
So we may assume that we have our two subsequences S1 and S2 .
The braids in S1 are all transversally isotopic and so they all have the
same Bennequin number and they all represent T K. The braids Ki′ ∈
′ ) = β(K ′ ) + 2 for each
S2 all have the same knot type, but β(Ki+1
i
i = 1, . . . , s − 1. With each negative destabilization and braid isotopy,
the Bennequin number increases by 2 and the braid index decreases by 1.
Each braid represents the same knot type K but a different transversal
knot type T K.
Our concern now is with S2 , i.e., Kr⋆ = K0′ → · · · → Ks′ , where every
′
Ki+1 is obtained from Ki′ by a single negative destabilization and braid
on transversally simple knots
341
isotopy. The number of negative destabilizations in subsequence S2 is
exactly one-half the difference between the Bennequin number β(T K) of
the original transversal knot T K and the Bennequin number βmax (K).
Let us now fix on any particular minimum braid index representative of K as a minimum braid index closed braid representative of the
transversal knot type that realizes βmax (K). It will not matter which we
choose, because all belong to the set M0 and so are exchange-equivalent.
We may then take the final braid Kr⋆ in S1 , which is the same as the
initial braid K0′ in S2 , as our representative of T K, because it realizes
the minimal braid index for T K and by our construction, any other
such representative is related to the one we have chosen by transversal
isotopy. We may also proceed back up the sequence S2 from Ks′ to a
new representative that is obtained from K0′ by adding s negative trivial
loops, one at a time. By repeated application of Lemma 3 we know that
choosing any other element of M0 will take us to an exchange-equivalent
element of Ms . In this way we arrive in the set Ms , which also contains
Kr⋆ , and which is characterized by K and β. The proof of Theorem 2 is
complete. q.e.d.
3. Examples, applications and possible generalizations
In this section we discuss examples which illustrate Theorems 1
and 2.
3.1
The unlink and the unknot
Theorem A, quoted earlier in this manuscript, asserts that the mcomponent unlink, for m ≥ 1, is exchange reducible. In considering
a link transversally, it should be mentioned that we are assuming each
of the components of the link satisfy the same inequality α > 0. We
also need to define the Bennequin number properly for this transversal
link. The natural way to do so, suggested by Oliver Dasbach, is by the
following method. For a crossing involving two different components of
the link, assign ±1/2 to each component depending on the sign of the
crossing. Assign ±1 to each crossing consisting of strands from the same
component, as in the case of a knot. Then the Bennequin number of
each component is the difference between the algebraic crossing number
e (a sum of ±1’s and ±1/2’s) and n, the braid index of that component. Define the Bennequin number of the link to be the collection of
the Bennequin numbers of the components of the link. The following
342
joan s. birman & nancy c. wrinkle
corollary is an immediate consequence of Theorems 1 and A:
Corollary 1. The m-component unlink, m ≥ 1, is transversally
simple. In particular, the unknot is transversally simple.
Note that Corollary 1 gives a new proof of a theorem of Eliashberg
[10].
3.2
Torus knots and iterated torus knots
In the manuscript [11] J. Etnyre proved that positive torus knots are
transversally simple. His proof failed for negative torus knots, but he
conjectured that the assertion was true for all torus knots and possibly
also for all iterated torus knots. In an early draft of this manuscript
we conjectured that torus knots and iterated torus knots ought to be
exchange reducible, and sketched our reasons. Happily, the conjecture
is now a fact, established by W. Menasco in [17]. Two corollaries follow.
To state and prove our first corollary, we need to fix our conventions for
the description of torus knots and iterated torus knots.
Definition. Let U be the unit circle in the plane z = 0, and let
N (U ) be a solid torus of revolution with U as its core circle. Let λ0
be a longitude for U , i.e., λ0 is a circle in the plane z = 0 which lies
on ∂N (U ), so that U and λ0 are concentric circles in the plane z = 0.
See Figure 10. A torus knot of type e(p, q), where e = ±, on ∂N (U ),
denoted Ke(p,q) , is the closed p-braid (σ1 σ2 · · · σp−1 )eq on ∂N (U ), where
σ1 , . . . , σp−1 are elementary braid generators of the braid group Bp .
Note that Ke(p,q) intersects the curve λ0 in q points, and note that the
algebraic crossing number of its natural closed braid projection on the
plane z = 0 is e(p − 1)q. The knot Ke(p,q) also has a second natural
closed braid representation, with the unknotted circle U as braid axis
and the closed q-braid (σ1 σ2 · · · σq−1 )ep as closed braid representative.
Since p and q are coprime integers, one of these closed braids will have
smaller braid index than the other, and without loss of generality we will
assume in the pages which follow that we have chosen p to be smaller
than q, so that Ke(p,q) has braid index p.
Definition. We next define what we mean by an e(s, t)-cable on a
knot X in 3-space. Let X be an arbitrary oriented knot in oriented S 3 ,
and let N (X) be a solid torus neighborhood of X in 3-space. A longitude
λ for X is a simple closed curve on ∂N (X) which is homologous to X
in N (X) and null-homologous in S 3 \ X. Let f : N (U ) → N (X) be a
on transversally simple knots
343
U
λ0
Figure 10: The standard solid torus N (U ), with K+(2,3) ⊂ ∂N (K)
homeomorphism which maps λ0 to λ. Then f (Ke(s,t) ) is an e(s, t)-cable
about X.
Definition. Let {ei (pi , qi ), i = 1, . . . , r} be a choice of signs ei = ±
and coprime positive integers (pi , qi ), ordered so that for each i we have
pi , qi > 0. An iterated torus knot K(r) of type (e1 (p1 , q1 ), . . . , er (pr , qr )),
is defined inductively by:
• K(1) a torus knot of type e1 (p1 , q1 ), i.e., a type e(p1 , q1 ) cable on
the unknot U . Note that, by our conventions, p1 < q1 .
• K(i) is an ei (pi , qi ) cable about K(i − 1). We place no restrictions
on the relative magnitudes of pi and qi when i > 1.
Here is one of the simplest non-trivial examples of an iterated torus
knot. Let K(1) be the positive trefoil, a torus knot of type (2, 3). See
Figure 11(a) and (b). Note that in the left sketch the core circle is our
unit circle U , while in the right sketch the core circle is the knot K(1).
The iterated torus knot K(2) = K(2,3),−(3,4)) is the −(3, 4) cable about
K(1). See Figure 12.
In [17], it was shown that the braid foliation machinery used for the
torus in [7] could be adapted to the situation in which the knot is on
the surface of the torus. The main result of that paper is the following
theorem.
Theorem. ([17], Theorem 1) Oriented iterated torus knots are exchange reducible.
344
joan s. birman & nancy c. wrinkle
U
(a)
λ
1
K(1)
λ0
(b)
Figure 11: (a) the torus knot K+(2,3) . (b)the solid torus neighborhood
N (K(1)) of K(1), with core circle K(1) and longitude λ1 marked.
U
λ0
(a)
λ1
K(2)
(b)
Figure 12: (a) the torus knot of type -(3,4) embedded in ∂N (U ), (b)
the iterated torus knot K(2) = K+(2,3),−(3,4)
on transversally simple knots
345
Combining Menasco’s Theorem with Theorem 2, we have the following immediate corollary:
Corollary 2. Iterated torus knots are transversally simple.
Our next contribution to the theory of iterated torus knots requires
that we know the braid index of an iterated torus knot. The formula
is implicit in the work of Schubert [20], but does not appear explicitly
there.
Lemma 4. Let K(r) = Ke1 (p1 ,q1 ),...,er (pr ,qr ) be an r-times iterated
torus knot. Then the braid index of K(r) is p1 p2 · · · pr .
Proof. We begin with the case r = 1. By hypothesis p1 < q1 , also
the torus knot Ke1 (p1 ,q1 ) is represented by a p1 -braid (σ1 σ2 · · · σp1 −1 )e1 q1 .
By the formula given in [15] for the HOMFLY polynomial of torus knots,
together with the Morton-Franks-Williams braid index inequality (discussed in detail in [15]), it follows that this knot cannot be represented
as a closed m-braid for any m < p1 .
Passing to the general case, Theorem 21.5 of [20] tells us that the
torus knot Ke1 (p1 ,q1 ) and the array of integers e1 (p2 , q2 ), . . . , er (pr , qr )
form a complete system of invariants of the iterated torus knot K(r) =
Ke1 (p1 ,q1 ),...,er (pr ,qr ) . Lemma 23.4 of [20] tells us that, having chosen a
p1 -braid representative for Ke1 (p1 ,q1 ) , there is a natural p1 p2 · · · pr -braid
representative of K(r). This representative is the only one on this number of strings, up to isotopy in the complement of the braid axis. Theorem 23.1 of [20] then asserts that K(r) also cannot be represented as
a closed braid with fewer strands. That is, its braid index is p1 p2 · · · pr .
q.e.d.
Remark. The iterated torus knot Kr has two natural closed braid
representatives. The first is a p1 p2 · · · pr - braid which has the core circle
U ′ of the unknotted solid torus S 3 \ N (U ) as braid axis. The second
is a q1 p2 · · · pr -braid which has the core circle U of the unknotted solid
torus N (U ) as braid axis. In the case r = 1, the second choice gives a
closed braid which is reducible in braid index, i.e., it has q1 − p1 trivial
loops. From this it follows that if r > 1 it will have (q1 − p1 )p2 · · · pr
trivial loops, thus the second closed braid representation is reducible to
the first.
We are now ready to state our second corollary about iterated torus
knots. Let χ be the Euler characteristic of an oriented surface of minimum genus bounded by K(r).
joan s. birman & nancy c. wrinkle
346
Corollary 3. Let K(r) = Ke1 (p1 ,q1 ),...,er (pr ,qr ) be an iterated torus
knot, where p1 < q1 . Then the maximum Bennequin number of K(r) =
Ke1 (p1 ,q1 ),...,er (pr ,qr ) is given by the following two equivalent formulas:
(1) βmax (K(r))
= ar − p1 p2 · · · pr , where
r
ar = i=1 ei qi (pi − 1)pi+1 pi+2 . . . pr .
(2) βmax
(K(r)) = −χ − d, where
d = ri=1 (1 − ei )(pi − 1)qi pi+1 pi+2 . . . pr .
(3) Moreover, the upper bound in the inequality βmax (K(r)) ≤ −χ is
achieved if and only if all of the e′i s are positive.
Proof. We begin with the proof of (1). By Lemma 4 the braid
index of K(r) is p1 p2 · · · pr . Therefore βmax (K(r)) = ar − p1 p2 · · · pr ,
where ar is the algebraic crossing number of the unique p1 p2 · · · pr -braid
representative of K(r). To compute ar we proceed inductively. If r = 1
then K(1) is a type e1 (p1 , q1 ) torus knot, which is represented by the
closed p1 -braid (σ1 σ2 . . . σp−1 )e1 q1 . Its algebraic crossing number is a1 =
e1 (p1 − 1)q1 .
The knot K(i) is an ei (pi , qi ) cable on K(i−1). Note that Kei (pi ,qi ) ⊂
∂N (K0 ), also Kei (pi ,qi ) is a pi -braid, also Kei (pi ,qi ) ∩ λ0 consists of qi
points. We shall think of the projection of the braided solid torus
N (K(i − 1)), which is a p1 p2 · · · pi−1 -braid, as being divided into three
parts. The reader may find it helpful to consult Figures 13 (a), (b), (c)
as we examine the contributions to ai from each part.
K(i)
K(i-1)
K(i)
λ i-1
λ i-1
λi-1
K(i)
λ i-1
(a)
(b)
Figure 13: Iterated torus knots.
(c)
K(i)
on transversally simple knots
347
(a) The first part of N (K(i − 1)) is the trivial p1 p2 · · · pi−1 -braid.
The longitude λi−1 is parallel to the core circle of N (K(i − 1))
in this part. The surface ∂N (K(i − 1)) contains on one of its
p1 p2 · · · pi−1 cylindrical branches the image under f of the braided
part of Ker (pr ,qr ) . See Figure 13(a), which shows the braid when
ei (pi , qi ) = −(3, 4). This part of K(i) contributes ei (pi −1)qi to ai .
Note that there are qi points where f (Kei (pi ,qi ) ) intersects λi−1 .
(b) The second part of N (K(i − 1)) contains all of the braiding in
K(i − 1), and so also in N (K(i − 1)). In Figure 13(b) we have
illustrated a single crossing in K(i−1) and the associated segments
of N (K(i − 1)). We show a single crossing of λi−1 (as a thick line)
over K(i−1). The single signed crossing contributes p2i crossings to
K(i), so the total contribution from all of the crossings in K(i−1)
will be ai−1 p2i . The illustration shows the case pi = 3.
(c) The third part of N (K(i − 1)) is again the trivial p1 p2 · · · pi−1 braid. It contains corrections to the linking number of λi−1 with
K(i − 1) which result from the fact that a curve which is everywhere ‘parallel’ to the core circle will have linking number ai−1 ,
not 0, with K(i − 1). To correct for this, we must allow the
projected image of λi−1 to loop around ∂N (K(i − 1)) exactly
−ai−1 times, so that its total linking number with K(i − 1) is
zero. See the left sketch in Figure 13(c), which shows the 3 positive loops which occur if ai−1 = −3. We have already introduced
qi intersections between f (Kei (pi ,qi ) ) and λi−1 , and therefore we
must avoid any additional intersections which might arise from
the −ai−1 loops. See the right sketch in Figure 13(c). When
λi−1 wraps around N (K(i − 1)) the additional −ai−1 times, the
pi -braid f (Kei (pi ,qi ) must follow. Each loop in λi−1 introduces
(pi − 1) + (pi − 2) + · · · + 2 + 1 = pi (p2i −1) crossings per half-twist in
f (Kei (pi ,qi ) ). Since there is a full twist to go around the positive
loop this number is doubled to pi (pi − 1). We have shown the
12 crossings in K(i) which come from a single loop in λi−1 when
pi = 4. The total contribution is −ai−1 pi (pi − 1).
Adding up all these contributions we obtain
ai = ei (pi − 1)qi + ai−1 (pi )2 − ai−1 (p2i − pi ) = ei (pi − 1)qi + ai−1 pi .
Summing the various terms to compute ar , we have proved part (1) of
the Corollary.
joan s. birman & nancy c. wrinkle
348
The proof of (2) will follow from that of (1) if we can show that:
χ = p1 p2 · · · pr − d − ar ,
where d = ri=1 (1 − ei )(pi − 1)qi pi+1 pi+2 . . . pr . To see this, we must
find a natural surface of minimum genus bounded by K(r) and compute its Euler characteristic. By Theorem 12, Lemma 12.1 and Theorem
22 of [20], a surface of minimum genus bounded by K(r) may be constructed by Seifert’s algorithm, explained in Chapter 5 of [19], from a
representative of K(r) which has minimal braid index. We constructed
such a representative in our proof of Part (1). To compute its Euler
characteristic, use the fact that χ is the number of Seifert circles minus
the number of unsigned crossings (Exercises 2 and 10 on pages 119 and
121 of [19]). By a theorem of Yamada [22] the number of Seifert circles
is the same as the braid index, i.e., p1 p2 · · · pr in our situation. The
number of unsigned crossings is br , where bi = (pi − 1)qi + bi−1 pi and
b1 = (p1 − 1)q1 and χ = p1 p2 ·
· · pr − br . Adding up the contributions
r
′
from all the bi s
we get br =
i=1 (pi − 1)qi pi+1 pi+2 . . . pr which can
r
be rewritten as
(1
−
e
+
e
)(pi − 1)qi pi+1 pi+2 . . . pr . Separating
i
i
i=1
terms:
br =
r
(1 − ei )(pi − 1)qi pi+1 pi+2 . . . pr
i=1
r
+
(ei )(pi − 1)qi pi+1 pi+2 . . . pr = d + ar .
i=1
The claimed formula for χ follows.
To prove (3), observe that the only case when βmax = −χ exactly
occurs when d = 0, i.e., the sum
r
(1 − ei )(pi − 1)qi pi+1 pi+2 . . . pr = 0.
i=1
That is, all the ei ’s are +1.
3.3
q.e.d.
Knots that are not exchange reducible
A very naive conjecture would be that all knots are exchange reducible,
however that is far from the truth. We begin with a simple example. In
the manuscript [6] Birman and Menasco studied knots that are represented by closed 3-braids, up to braid isotopy, and identified the proper
on transversally simple knots
349
subset of those knots whose minimum braid index is 3 (i.e., not 2 or 1).
They prove that the knots that have minimum braid index representatives of braid index 3 fall into two groups: those that have a unique
such representative (up to braid isotopy) and infinitely many examples
that have exactly two distinct representatives, the two being related by
3-braid flypes. A flype is the knot type preserving isotopy shown in
Figure 14. Notice that the flype is classified as positive or negative depending on the sign of the isolated crossing. After staring at the figure,
it should become clear to the reader that the closed braids in a ‘flype
pair’ have the same topological knot type. We say that a braid representative admits a flype if it is conjugate to a braid that has the special
form illustrated in Figure 14.
R
R
P
Q
Q
R
P
Q
P
Positive flype
Q
R
P
Negative flype
Figure 14: Closed 3-braids that are related by a flype.
Theorem 3 [6]. The infinite sequence of knots of braid index 3
in [6], each of which has two closed 3-braid representatives, related by
flypes, are examples of knots that are not exchange reducible.
Proof. It is proved in [6] that for all but an exceptional set of P, Q, R
the closed braids in a flype pair are in distinct conjugacy classes. Assume
from now on that a ‘flype pair’ means one of these non-exceptional
pairs. Since conjugacy classes are in one-to-one correspondence with
braid isotopy equivalence classes, it follows that the braids in a flype
pair are not related by braid isotopy. On the other hand, it is proved
in [6] that when the braid index is ≤ 3 the exchange move can always
be replaced by braid isotopy, so the braids in a flype pair cannot be
exchange equivalent. q.e.d.
On closer inspection, it turns out that a positive flype can be replaced by a sequence of braid isotopies and positive stabilizations and
destabilizations, which shows that it is a transversal isotopy. See Figure 15.
joan s. birman & nancy c. wrinkle
R
R
R
P
R
Braid isotopy
P
Braid isotopy
R
Q
R
P
Braid isotopy
R
Q
P
P
Braid isotopy
+ Stabilization
Q
Q
Braid isotopy
P
Q
Q
P
Q
P
R
Q
350
+ Destabilization
Figure 15: A positive flype can be replaced by a sequence of transversal
isotopies.
The figure is a generalization of Figure 8 (proving that exchange
moves are transversal), because flypes are generalizations of exchange
moves. We replace one of the σn±1 with the braid word we label R. A
negative flype also has a replacement sequence similar to the one pictured in Figure 15, but the stabilizations and destabilizations required
are negative. Therefore the negative flype sequence cannot be replaced
by a transversal isotopy using these methods. There may well be some
other transversal isotopy that can replace a negative flype, but we did
not find one. Thus we are lead to the following conjecture:
Conjecture. Any transversal knot type whose associated topological knot type K has a minimum braid index representative that admits
a negative flype is not transversally simple.
A simple example which illustrates our conjecture is shown in Figure 16.
The essential difficulty we encountered in our attempts to prove or
disprove this conjecture is that the only effective invariants of transversal
knot type that are known to us at this writing are the topological knot
type and the Bennequin number, but they do not distinguish these
examples.
on transversally simple knots
351
Figure 16: A simple example which illustrates our conjecture
3.4
Knots with infinitely many transversally equivalent
closed braid representatives, all of minimal braid index
At this writing the only known examples of transversally simple knots
are iterated torus knots. By Theorem 24.4 of [20] iterated torus knots
have unique closed braid representatives of minimum braid index, and
it follows from this and Theorem 1 that they have unique representatives of maximum Bennequin number. It seems unlikely to us that all
transversally simple knots have unique closed braid representatives of
minimum braid index, and we now explain our reasons.
The exchange move was defined in Figure 6(b) of §2. It seems quite
harmless, being nothing more than a special example of a Reidemeister move of type II. It also seems unlikely to produce infinitely many
examples of anything, however that is exactly what happens when we
combine it with braid isotopy. See Figure 17 with n = 4. Proceeding
from the right to the left and following the arrows, we see how braid
isotopy and exchange moves can be used to produce infinitely many
examples of closed braids which are transversally equivalent. It is not
difficult to choose the braids R and S in Figure 17 so that the resulting
closed braids are all knots, and also so that they actually have braid
index 4, and also so that they are in infinite many distinct braid isotopy classes (using an invariant of Fiedler [12] to distinguish the braid
isotopy classes). We omit details because, at this writing, we do not
know whether the knots in question are exchange reducible, so we cannot say whether they all realize the maximum Bennequin number for
their associated knot type.
joan s. birman & nancy c. wrinkle
352
S
R
exchange
move
R
S
R
braid
isotopy
S
R
exchange
move
R
S
braid
isotopy
S
Figure 17: The exchange move and braid isotopy can lead to infinitely
many distinct closed n-braid representatives of a single knot type.
3.5
Generalizing the concept of ‘exchange reducibility’
Some remarks are in order on the concept of ‘exchange reducibility’.
Define two closed braids A ∈ Bn , A′ ∈ Bm to be Markov-equivalent if the
knot types defined by the closed braids coincide. Markov’s well-known
theorem (see [16]) asserts that Markov-equivalence is generated by braid
isotopy, ±-stabilization, and ±-destabilization. However, when studying
this equivalence relation one encounters the very difficult matter that
±-stabilization is sensitive to the exact spot on the closed braid at which
one attaches the trivial loop. On the other hand, Lemma 2 shows that
exchange moves are the obstruction to moving a trivial loop from one
spot on a knot to another. Therefore if we allow exchange moves in
addition to braid isotopy, ±-stabilization, and ±-destabilization, one
might hope to avoid the need for stabilization. That is the idea behind
the definition of exchange reducibility, and behind the proof of Theorem
A. However, that hope is much too naive, as was shown by the examples
in §3.3.
A way to approach the problem of transversal knots is to augment
the definition of exchange reducible by allowing additional ‘moves’. In
their series of papers Studying knots via closed braids I-VI, the first
author and Menasco have been working on generalizing the main result
in Theorem A to all knots and links. In the forthcoming manuscript
on transversally simple knots
353
[8], a general version of the ‘Markov theorem without stabilization’ is
proved. The theorem states that for each braid index n a finite set of
new moves suffices to reduce any closed braid representative of any knot
or link to minimum braid index ‘without stabilization’. These moves
include not only exchange moves and positive and negative flypes, but
more generally handle moves and G-flypes. Handle moves can always
be realized transversally. The simplest example of a G-flype is the 3braid flype that is pictured in Figure 14, with weights assigned to the
strands. This will change it to an m-braid flype, for any m. But other
examples exist, and they are much more complicated. We note, because
it is relevant to the discussion at hand, that any positive G-flype can be
realized by a transversal isotopy. The sequence in Figure 15 is a proof
of the simplest case. Awaiting the completion of [8], we leave these
matters for future investigations.
References
[1] D. Bennequin, Entrelacements et équations de Pfaff, Astérisque 107-8 (1983) 87–
161; English version: Russian Math. Surveys 44 (1989) 1–65. Section numbers in
the two versions agree.
[2] J. S. Birman & E. Finkelstein, Studying surfaces via closed braids, J. Knot Theory
Ramifications 7 (1998) 267–334.
[3] J. S. Birman & M. Hirsch, A new algorithm for recognizing the unknot, Geom.
Topology 2 (1998) 175–220.
[4] J. S. Birman & W. Menasco, Studying links via closed braids. IV: Composite and
split links, Invent. Math. 102 (1990) 115–139.
[5]
, Studying links via closed braids. V: The unlink, Trans. Amer. Math. Soc.
329 (1992) 585–606.
[6]
, Studying links via closed braids. III: Classifying links which are closed
3-braids, Pacific J. Math. 161 (1993) 25–113.
[7]
, Special positions for essential tori in link complements, Topology 33 No.
3 (1994) 525–556. See also Erratum, Topology 37 (1998) 225.
[8]
, Stabilization in the braid groups, manuscript in preparation.
[9] J. S. Birman & N. C. Wrinkle, Holonomic and Legendrian parametrizations of
knots, J. Knot Theory Ramifications, 9 (2000) 293–309.
[10] Y. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds,
Topological Methods in Modern Math. (1991) 171–193.
354
joan s. birman & nancy c. wrinkle
[11] J. Etnyre, Transversal torus knots, Geom. Topology 3 (1999) 253–268.
[12] T. Fiedler A small state sum for knots, Topology 32 (1993) 281–294.
[13] D. Fuchs & S. Tabachnikov, Invariants of Legendrian and transverse knots in the
standard contact space, Topology 36 (1997) 1025–1053.
[14] E. Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991)
637–677.
[15] V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann.
Math. 126 (1987) 335–388.
[16] A. A. Markov, Über die freie Äquivalenz der geschlossenen Zöpfe, Recueil Math.
Moscou 1 (1935) 73–78.
[17] W. Menasco, On iterated torus knots and transversal knots,
http://www.math.buffalo.edu/ menasco/index.html
Preprint:
[18] H. Morton, Infinitely many fibered knots having the same Alexander polynomial,
Topology 17 (1978) 101–104.
[19] D. Rolfsen, Knots and links, Publish or Perish 1976, reprinted 1990.
[20] H. Schubert, Knoten und Vollringe, Acta Math. 90 (1953) 131–226.
[21] V. Vassiliev, Holonomic links and Smale principles for multisingularities, J. Knot
Theory Ramifications 6 (1997) 115–123.
[22] S. Yamada, The minimum number of Seifert circles equals the braid index of a
link, Invent. Math. 89 (1987) 347–356.
Barnard College of Columbia University
Columbia University