L-Moves and Markov theorems
Sofia Lambropoulou
Abstract
Given a knot theory (virtual, singular, knots in a 3-manifold etc.)
there are deep relations between the diagrammatic knot equivalence in
this theory, the braid structures and a corresponding braid equivalence.
The L-moves between braids, due to their fundamental nature, may be
adapted to any diagrammatic situation in order to formulate a corresponding braid equivalence. In this short paper we discuss and compare
various diagrammatic set-ups and results therein, in order to draw the
underlying logic relating diagrammatic isotopy, braid structures, Markov
theorems and L-move analogues. Finally, we apply our conclusions to
singular braids.
Keywords: Markov theorem, L-moves, conjugation, commuting, stabilization,
virtuals, singulars, welded braids, mixed braids, links in 3–manifolds.
Mathematics Subject Classification 2000: 57M25, 57M27, 57N10.
1
Markov equivalence and L-equivalence in S 3
The central problem in classical Knot Theory is the complete classification of
knots and links up to their isotopy in three-space. Reidemeister [20] translated
isotopy in space into an equivalence relation on knot diagrams in the plane,
generated by the three Reidemeister moves R1, R2, R3 of Fig. 2, together with
planar isotopy. From a different point of view, taking the closure of a braid,
that is, joining its corresponding endpoints with simple arcs yields an oriented
link, and by Alexander’s theorem [1] any oriented link in S 3 is isotopic to the
closure of a braid (not unique). The set of classical braids on n strands forms the
braid group Bn [2], whose generators σi are the elementary crossings between
the ith and (i + 1)st strand. The basic relation in Bn , σi σi+1 σi = σi+1 σi σi+1 ,
is a special case of the isotopy move R3. Markov’s theorem [19], improved by
Weinberg [22] with the two well-known moves (see also [4]), gives an equivalence
relation in the set of all braids B∞ :
Theorem 1 (Markov). Two oriented links in S 3 are isotopic if and only if
any two corresponding braids differ by braid relations and the moves:
(i) Conjugation in each Bn : σi−1 ασi ∼ α,
(ii) Bottom stabilization in B∞ : α ∼ ασn±1 , where α ∈ Bn .
L-Moves and Markov theorems
2
braid
L under
L over
isotopy
Figure 1: An L–move introduces a crossing
The Jones polynomial [9] was the first knot invariant that was constructed
using braids and the Markov theorem. (Whenever we say ‘knots’ we mean
both knots and links.) Among other proofs of the Markov theorem, in [16] we
introduced a new type of braid move, the L-move, and we proved the following
one-move theorem, sharpening the classical result.
Theorem 2 (cf. [16], Thm. 2.3). Two oriented links in S 3 are isotopic if
and only if any two corresponding braids differ by braid relations and L-moves.
A classical L-move consists in cutting a strand of the braid and taking the
two ends of the cut, the top end to the bottom of the braid and the bottom
end to the top of the braid both entirely over or both entirely under the braid,
creating at the same time a new pair of corresponding strands. By small braid
isotopies, an L-move is equivalent to adding an in-box crossing. So, the L-move
generalizes the stabilization move. Fig. 1 illustrates abstractly various instances
of L-moves. (Since the L-moves may adapt to other diagrammatic set-ups, the
black disc inside the braid box indicates other types of crossings, e.g. singular,
and the thicker strands may stand for a knot complement or a c.c.o. 3-manifold
or a handlebody, see below.) Closing the pair of strands of an L-move yields a
kink in the diagram, so it corresponds to planar isotopy or to the isotopy move
R1.
Theorems 1 and 2 imply that braid conjugation follows from the L-moves
(see also Fig. 5 below). Theorem 2 has two main advantages: First, there is only
one type of braid move. Second, the L-moves are very local and fundamental.
Thus, adapted every time to a given diagrammatic set-up (where diagrammatic
isotopy is given and the notion of a braid can be defined), they may provide a
uniform ground for formulating and proving a corresponding braid equivalence.
Moreover, due to the fact that an L-move has a nice algebraic expression (by
pulling it out of the braid box, cf. [16], Remark 2.2), these geometric analogues
of the Markov theorem can be turned into algebraic statements, according to
the braid structures of the given situation (which depend on the diagrammatic
equivalence) and the manifold where the knots live. Indeed, in [16] we first
proved a Relative Version of Theorem 2 ([16], Thm. 4.7) concerning a fixed
braided portion in the knot. Using this we proved L-move analogues of the
Markov theorem for links in knot complements and for links in an arbitrary
c.c.o. 3-manifold (cf. [16], Thms. 5.5 and 5.10). These results were turned into
L-Moves and Markov theorems
3
algebraic statements in [17], Thms. 4 and 5. The Relative Version of Theorem 2
lead also to an L-move equivalence as well as to algebraic analogues of Markov’s
theorem for knots in a handlebody (cf. [8], Thms. 3, 4 and 5). In a further
development, similar results were proved in [14] for virtual knots (cf. Thms. 2
and 3), for flat virtuals (cf. Thms. 4 and 5), for welded knots (cf. Thms. 6 and
7) and for unrestricted virtuals (cf. Thm. 8).
In this paper we shall discuss and compare various diagrammatic set-ups
and results therein, in order to draw the underlying logic relating diagrammatic
isotopy and braid structures, and to be able to guess statements for Markov
theorems and L-move analogues. Finally, we apply our conclusions to singular
knots and singular braids.
We begin by revising the two moves of Theorem 1. Conjugation is clearly
b
related to an isotopy moves R2 occuring in the back side of a closed braid β.
Opening βb with a cutting line, this line will either put both crossings of the
move at the same side of the braid (and then the initial R2 move becomes a
braid cancellation, due to the fact that σi has an inverse) or it will separate
the two crossings, one at the top and one at the bottom of the braid, yielding
a conjugate of β. Even simpler, a cutting line of a closed braid may cut before
or after a crossing σi , yielding the braid equivalence move: ασi ∼ σi α. This
move is here equivalent to conjugation. Regarding the bottom stabilization we
just note that the braid move α1 α2 ∼ α1 σn±1 α2 together with move (i) implies
bottom stabilization. Stabilization is related to the isotopy move R1, since the
closure of a braid with a stabilization move is a knot with a kink at that place.
2
Other diagrammatic set-ups
Before discussing braid equivalence in more general settings let us first recall
briefly the various diagrammatic set-ups we consider in this paper.
Virtuals, Flat Virtuals, Welded and Singulars. Virtual knot theory was
introduced by Kauffman [12] and it is an extension of classical diagrammatic
knot theory. In this extension one adds a virtual crossing that is neither an
over-crossing nor an under-crossing. Virtual isotopy generalizes the ordinary
Reidemeister moves for classical links and is generated by planar isotopy and
the moves R1, R2, R3, V1, V2, V3 and V4 of Fig. 2, the last one being the
key move in the theory. Moves like F1 and F2 with two real crossings and one
virtual crossing inside the shaded disc are forbidden in virtual knot theory. The
virtual braid group V Bn is generated by the classical crossings and the virtual
crossings vi , satisfying vi2 = 1 and relations that are the braid versions of the
virtual isotopy moves (except, of course, R1 and V1).
A flat virtual knot is like a virtual knot but without the extra under/over
structure at the real crossings. Instead we have shadow crossings [21], the flat
crossings. The study of flat virtual knots and links was initiated in [12]. The
isotopy moves and the two forbidden moves in virtual knot theory are completely
analogous for the flat virtual setting. In the flat virtual braid group on n strands
L-Moves and Markov theorems
S1
F1
4
R1
V1
R2
V2
R3
V3
V4
S2
F2
F3
Figure 2: Diagrammatic moves for virtual, welded, singular and classical knots
the generators are the virtual crossings vi and the flat crossings ci , such that
c2i = 1. The mixed relation vi ci+1 vi = vi+1 ci vi+1 is not symmetric.
Welded braids were introduced in [6]. Welded knots satisfy the same isotopy
relations as the virtuals, but for welded knots one of the two forbidden moves
of Fig. 2 is allowed, the move F1, which contains an over arc and one virtual
crossing. The welded braid group on n strands is a quotient of the virtual braid
group, so it can be presented with the same generators and relations, with the
additional relations: vi σi+1 σi = σi+1 σi vi+1 (F1).
Singular knots are related to Vassiliev’s theory of knot invariants. In singular knot theory isotopy is generated by planar isotopy, by the moves R1, R2,
R3, S1, S2 and by the moves F1 and F2, where in the shaded disc a singular
crossing is placed. Move F3 and move R1 with a singular crossing placed in
the shaded disc are forbidden in this theory, see [11]. The singular crossings
τi generate together with the real crossings σi the singular braid monoid SBn
[3, 5], satisfying relations analogous to the braid versions of the singular isotopy
relations.
b be a
Knots in Handlebodies, Knot Complements, 3-manifolds. Let B
b Then
closed braid and let L be an oriented link in the knot complement S 3 \B.
b S L, with the manifold
L may be represented uniquely in S 3 by a mixed link B
L-Moves and Markov theorems
5
...
...
...
...
...
...
...
...
...
...
...
...
...
uo
u
B
Figure 3: Mixed links, a geometric mixed braid and an algebraic mixed braid
1
i m
... ...
a
=
i
1 m 1
n
1
...
j
= ...
j
...
j+1 n
...
Figure 4: The mixed braid generators ai and σj
b remaining fixed throughout. Analogous set-up we have for links in
subbraid B
a handlebody
S of genus m. Here a link L is represented uniquely by a mixed
tangle Im L, where Im is the identity braid on m (infinitely extended) strands.
Fig. 3 illustrates a link in the complement of the Hopf link and the same link
in a handlebody of genus 3. Moreover, two links are isotopic in the manifold
if and only if the corresponding mixed links are isotopic in S 3 (see [16, 8] for
details). Mixed link isotopy is generated by the ordinary Reidemeister moves
for the link part and by moves R2, R3 involving also the manifold part. For the
b resulting by doing surgery
case of a link L in the c.c.o. 3-manifold χ(S 3 , B),
b we have similar set-up and reasoning, only that we add in the isotopy
along B,
a handle-sliding move.
b S L may be isotoped to the closure of a geometric mixed braid
AS
mixed link B
S
B β. In the case of a handlebody the closure of a mixed braid Im β is realized
by closing only the link part, indicating at the same time whether the simple
closing arc will pass all way over or all way under the mixed braid. Geometric
mixed braids may be represented by algebraic mixed braids, in which the two
sets of strands
S are separated and the manifold part is the identity braid (the
braid B of B β has been combed away), cf. [17]. View Fig. 3 for a geometric
mixed braid in a handlebody and for an algebraic mixed braid. Thus, the braid
groups Bm,n , which contain braids on m + n strands with the first m strands
forming the identity braid Im , are the braid structures related to all the above
spaces, cf. [15]. They are generated by the classical crossings σj and by the
‘loops’ ai as illustrated in Fig. 4.
L-Moves and Markov theorems
3
6
Braid equivalences
Consider now a general setting. In [13] we gave a braiding algorithm, which
is applicable to all the categories in which braids are defined. Let Bn denote
the set of braids with n strands in a given diagrammatic knot theory and let
B∞ := ∪n Bn . Let also α, α1 , α2 ∈ Bn and let xi some braid generator in Bn ,
with xn ∈ Bn+1 . There are braid structures where some generators do not have
inverses (e.g. the singular crossings τi in singular knot theory), so move (i)
cannot appear in the braid statement. Yet, a cutting line is permitted to cut
before or after a braid generator xi . For this reason we will adopt the following
more general move, which is equivalent to conjugation if xi has an inverse.
(iii) Commuting in Bn : αxi ∼ xi α.
In most diagrammatic situations commutings (even conjugations) are permitted.
For example, in the virtual or welded or flat virtual knot theory all conjugations
are allowed in the braid equivalence, while in the singular knot theory all commutings are allowed (conjugation only for real crossings). Similarly, in the knot
theory of the solid torus all conjugations are allowed. But there are also situations where a braid generator, even if it has an inverse, cannot be commuted,
because the cutting line of the closure cannot cut anywhere. For example, the
loop generators ai in a handlebody of genus g ≥ 2 (see below). Then, such
commutings will not appear in a braid equivalence of the theory: neither in an
L-equivalence nor in an algebraic equivalence. Furthermore, for the same reason (not commuting of some braid generators) we will adopt the more general
stabilization:
(iv) Stabilization in B∞ : α1 α2 ∼ α1 x±1
n α2 .
Stabilization by a real crossing may occur in all the above-mentioned knot theories. This is because the isotopy move R1 is permitted in all these theories.
Similarly, stabilization by a virtual crossing may occur in virtual, flat virtual
and welded braids, corresponding to the isotopy move V1. Stabilization by a
singular crossing is not permitted in singular braids (due to the absence of the
corresponding move in the theory), neither does stabilization by the loops ai in
handlebodies, knot complements and c.c.o. 3-manifolds make sense.
Surprisingly, conjugation (not commuting) by a braid generator may sometimes be achieved simply by L-moves within the braid, without having to change
the position of the cutting line. For example, for a real crossing in classical and
singular knot theory, as well as in handlebodies, knot complements and c.c.o.
3-manifolds, for the loop generator in the solid torus or for a virtual crossing in
virtual and flat virtual knot theory. Then, these conjugations will not appear in
the L-move statement of the theory, but they will appear in the algebraic braid
equivalence. As a versatile example, Fig. 5 illustrates abstractly real conjugation as a composition of L-moves. The black disc inside the braid box indicates
the possible existence of singular crossings. The thicker braid may represent
a handlebody, a knot complement or a c.c.o. 3-manifold. This diagrammatic
L-Moves and Markov theorems
1
...
m 1 j j+1 n
...
...
Lo
...
B
B
...
1
...
...
...
m 1
7
n
1
~
...
B
m 1
n
1
...
n
m 1
undo
Lo
...
B
...
Figure 5: Conjugation by a real crossing as a composition of L–moves
o
u
k
k
Figure 6: Causing conjugation by a loop ai
proof is essentially due to Häring-Oldenburg and it will also work for the loop
generator in the solid torus as well as for virtual crossings and virtual L-moves.
It will not work for a loop generator ai in the three types of spaces we consider
neither for real crossings in virtual knot theory (and certainly not for singular
crossings).
Another comment is due. As we said, the basic difference between the handlebody and a knot complement lies in the way we form the closure of a mixed
braid. The choices involved in the handlebody correspond to crossing a hypothetical arc k at the point at infinity (in Fig. 6 at the back side of the braid). In
the case of classical links or links in knot complements or c.c.o. 3-manifolds this
is allowed. In the case of a handlebody it is not allowed. On the braid group
level, crossing the arc k corresponds to conjugation by a loop ai , see [8, 17].
That’s why this conjugation is not permitted for handlebodies.
Regarding the L-moves, in all diagrammatic set-ups we consider here, an
L-move can be easily shown to follow from stabilization and conjugation. We
shall, now, explain how to look for the different types of L-moves needed in a
given knot theory. The interpretation of an L-move as introducing an in-box
crossing is very important in this search and this, in turn, is related to the types
of kinks allowed in the diagrammatic isotopy. (This interpretation of an L-move
is also essential for moving from a geometric L-move statement to an algebraic
statement of the braid equivalence.)
In handlebodies, knot complements and c.c.o. 3-manifolds there is only one
type of L-move, applied only on the link subbraid, not the manifold subbraid:
L-Moves and Markov theorems
8
Figure 7: The three main types of virtual L-moves
introducing a real in-box crossing and the two strands running over or under
the rest of the braid, as in Fig. 1. In the singular domain, where a real kink is
allowed (move R1) but not a singular one, there is also only one type of L-move:
introducing a real in-box crossing, see Fig. 1. The strands of the L-move may
run both over or both under the rest of the braid, but they may not form any
singular crossings with the other strands, as, then, the closure will not contract
to a kink.
In virtuals, where both real and virtual kinks are permitted in the isotopy
(moves R1 and V1) we have L-moves introducing a real or a virtual crossing.
Because of the presence of the forbidden moves (moves F1 and F2): (a) the
strands of an L-move cross the other strands of the braid only virtually. (b)
We have a third type of L-move, the threaded L-move, coming from a ‘trapped’
virtual kink. (c) L-moves facing the right may be different from L-moves facing
the left, cf. [14]. In Fig. 7 we illustrate the three main types of virtual L-moves.
Obvious analogues of these L-moves are needed for the flat virtuals and for
welded braids. In the case of welded, where there is only one forbidden move,
the threaded type of L-moves follows from the others. For details see [14].
Conclusions
• For the L-move theorem for virtuals [14] we need the moves:
(i) Real conjugation in V Bn ,
(ii) Right virtual L-moves,
(iii) Right real L-moves,
(iv) Right and left under-threaded L-moves.
For an algebraic statement ([14], see also [10]) replace moves (ii) and (iii) by
virtual conjugation and right virtual and real stabilization in V B∞ . The statements for flat virtual braid equivalence are completely analogous, whilst for
welded, move (iv) is not needed.
• For the L-move theorem for handlebodies, knot complements, c.c.o. 3-manifolds
[8, 16] we need L-moves that do not touch the manifold subbraid. For c.c.o.
3-manifolds we need to add braid sliding moves. For an algebraic statement
[8, 17] replace the L-moves by real conjugation and real stabilization. For knot
L-Moves and Markov theorems
9
complements add twisted loop conjugation, a move resulting from combing the
loop ai through the manifold subbraid. For c.c.o. 3-manifolds add twisted loop
conjugation as well as braid sliding moves.
• We conclude with the case of singulars. In the discussions above we guessed
the L-moves needed for a singular braid equivalence: L-moves with the two
strands crossing the rest of the braid only with real crossings, all over or all
under. Moreover, such L-moves can achieve conjugation (and thus commuting)
by a real crossing but they cannot achieve commuting by a singular crossing,
which is obviously allowed in the theory. So, we have the following.
Theorem 3 (L-move theorem for singulars). Two oriented singular links
are isotopic if and only if any two corresponding singular braids differ by braid
relations in SB∞ and a finite sequence of the following moves:
(i) Singular commuting: τi α ∼ ατi ,
(ii) Real L-moves.
Proof. An L-move between singular braids follows from real conjugation and
real stabilization (see this by pulling the strands of the L-move out of the braid
box and using real conjugation). Conversely, stabililization by a real crossing is
a special case of an L-move, and real conjugation can be realized by L-moves
(recall Fig. 5). So, the two moves in our statement generate in SB∞ the same
equivalence relation as the moves:
(i) Singular commuting: τi α ∼ ατi ,
(ii) Real conjugation: σi−1 ασi ∼ α,
(iii) Real stabilization: α1 α2 ∼ α1 σn±1 α2 .
These algebraic moves give rise to the Markov equivalence for singulars and this
is proved by Gemein in [7].
Remark 1. All the issues treated in this paper are expanded, discussed thoroughly and proved in detail in the author’s monograph [18].
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L-Moves and Markov theorems
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S. Lambropoulou: Department of Mathematics, National Technical
University of Athens, Zografou campus, GR-157 80 Athens, Greece.
E-mail:
[email protected]
URL: http://www.math.ntua.gr/˜sofia/