#### Special representations forn-bridge links

Discrete Comput Geom
6i try
A. Cavicchioli a n d B. Ruini 0
0 Dipartimento di Matematica, Universith di Modena , Via Campi 213/1],41100 Modena , Italy
We describe a simple algorithm to obtain a catalogue of 3-bridge links by computer, depending upon 6-tuples of positive integers. This permits us to represent the genus two 3-manifolds by standardly constructed graphs with colored edges. Finally, we prove some results about the topological structure of these manifolds and extend the combinatorial representation to n-bridge links.
1. Introduction
* This work was performed under the auspices of the GNSAGA of the CNR (National Research
Council) of Italy and financiallysupported by the Ministero della Universit~e della Ricerca Scientifica
e Tecnoiogica of Italy within the project "Geometria Reale e Complessa."
A link L has bridoe index n if L has an n-bridge diagram but no m-bridge diagram
for any m < n.
In 1956 Schubert succeeded in solving a major problem in mathematics when
he found a special way to represent 2-bridge knots and links (see [25] and [26]).
His representation begins with a partial normal form in which the links are
characterized by a pair of integers, say (p, q). More precisely, he showed that any
2-bridge link diagram can be reduced, using the Reidemeister moves, either to a
diagram with no crossing, or else to a diagram K(p, q) of the type illustrated in
Fig. 1, called the Schubert normalform. To construct K(p, q) join each " e n d " (0,/)
to (1, i + q), i e Z2p, without introducing any new crossing. We assume, of course,
that 1 _ q < p. Schubert found necessary and sufficient conditions for K(p, q) to
have the same link type as K(p', q'). Thus, he "solved the knot problem" in this
special case. Notice that 1-bridge links are necessarily unknotted circles, so in
characterizing the cases which give nontrivial knots and links he also showed
when the bridge index is precisely 2. Finally, it is easily seen that K(p, q) is a
diagram of a knot if and only if (p, q) = 1 and p is odd. In this case Schubert
proved that the twofold cover of S a branched over K(p, q) is the lens space fl_(p,q)
(compare, for example, with [1], [10], and [16]). Moreover, Hodgson has shown
that 2-bridge knots are the only knots whose twofold branched covers are lens
spaces (see [12]).
In this paper we give partial generalizations of Schubert's work in the case n = 3
(0,0)
(
1, 0
)
(o,2p-:)
(o,2~-2)
:
(o,p+ :)
(o,:)
(o,2)
(o,p- :)
(o,p)
Bo
(a)
and obtain some results about the case of general n. More precisely, we show that
each 3-bridge link can be represented by a special type of diagram L(p, q, k, h, t, s)
which is characterized by six integers. Our representation is not unique as
L(
13, 12,4, 11, 1, 10
) and L ( l l , 1,0, 11,0, 1) are equivalent to the torus knot
T(
11, 3
) (see (
3
) of Corollary 4 and the proof of (
3
) of Proposition 6). At present
we have not found any general criterion for solving the knot problem in our case.
Furthermore, we do not know whether L(p, q, k, h, t, s) represents a knot or a link.
Our result and the graphical construction given in [6] yield a combinatorial
representation of 3-manifolds of Heegaard genus 2. Indeed, they are twofold cyclic
coverings of 3 branched along 3-bridge links (see [1]). This allows us to generate
a catalogue of all 3-bridge links (and hence their twofold cyclic covers) by
computer. Finally, we study the topological structure of the named manifolds and
extend some results to the case of n-bridge links.
2. Link Moves
Here we describe a simple algorithm to construct all 3-bridge links, starting from
the Schubert normal form. More precisely, we prove that any 3-bridge link can
be obtained from a 2-bridge link K(p, q) by means of a cut-and-paste move
depending upon four integers (k, h, t, s). Thus any 3-bridge link admits a
representation of type L(p, q, k, h, t, s). For this, let Bo, BI be the bridges of K(p, q),
1 < q < p, and let e~be the arc of K(p, q) with ends (0, i), (I, i + q) for any i t Z2p.
The cut-and-paste move of type I on K(p, q) is defined by the following rules:
(
1
) Cut h consecutive arcs ek, ek+t . . . . . ek+h-1 of K(p, q) by using a new bridge,
B 2 say, which is disjoint and parallel to Bo and Bx. Here we assume that
2 < h < 2p and 0 < k < p - 1. The bridge B 2 has exactly h - 1
undercrossings numbered as shown in Fig. 2(a).
(
2
) Join the "hanging" arc with vertex (0, k + / ) to the vertex (2, s + / ) , i t 7/h,
0 < s < h, without introducing any new crossing.
(
3
) Join the "hanging" arc with vertex (1, k + h + q - i - 1 ) to the vertex
(2, s + h + 0, i t 7/h, without introducing any new crossing.
Obviously, we operate in such a way that the resulting diagram is planar (see,
for instance, Fig. 3).
The cut-and-paste move of type II on K(p, q) is defined by the following rules:
(
1
) Cut h consecutive arcs ek, ek+ 1. . . . . ek+h-l of K(p, q) by using a new bridge,
B2 say, which is disjoint and parallel to Bo and B1. Here we assume that
2 < h < 2p, 0 < k < p - 1 , and that the arcs ep_l, ep belong to the set
{ek+i:itZh}. Now we require that the bridge B2 has exactly h - t - 1
(t > 0) undercrossings numbered as shown in Fig. 2(b).
(
2
) Join the vertex (0, p + / ) to (0, p - i - 1) by a new arc for any i t Z,. Then
it follows that 0 < t < t* = min{p - k, k + h - p} < [h/2], where [x]
denotes the integer part of x.
(
3
) If t < t*, then join the vertex (0, k + 0 to (2, s + i), i t 7/r_,_ k, the vertex
(
1
)1)
(
1,2
)
( I , ~ - I)
(I)2p - I)
(1,2v - 2~
(1~p+1)
0,o)
O,p)
.BI
(I,0)
(z,p)
B,
(0,0)
(0,2p- ~)
(0,2p-2)
(0,~)
(0,2)
(O,p+~)
(o,p-1)
(O,p)
(o,2p- 1)
(o,2p- 2)
(o,1)
(o,2)
(o,p+l)
(O,p,-i)
(0,0)
(o,p)
Bo
(
2,,1
)
(2,2.)
((22,~2h-I)
(2,2h - 2)
(2, h - I)
(2,h-F1)
(
2,0
)
(2,h)
B~
(a)
(
2,0
)
(2,h-t)
B2
(b)
(
2,1
)
(
2,2
)
(2,2h - 2t - 1)
( 2 , 2 h - 2 t - 2 )
T h e o r e m 1. Any 3-bridge link in 5 3 can be represented by a 3-bridoe diagram
L(p, q, k, h, t, s) obtained from the Schubert normal form of a 2-bridge link K(p, q)
by a cut-and-paste move. Moreover, the 6-tuple o f inteoers satisfies the following
conditions:
O < t < m i n { p - k , k + h - p } ,
Spedal Representations for n-Bridge Links
(a) (b)
Fig. 3. (a) K(
7, 3
). (b) Link obtained from K(
7, 3
)by a cut-and-paste move I, k = 3, h = 5, and s = 2,
(I)
Proof. Let L b e a d i a g r a m o f a 3 - b r i d g e link a n d let B t d e n o t e the b r i d g e s o f L
d r a w n in the p l a n e z = 0, i t 7/3. L e t us define the f o l l o w i n g d a t a :
T h e link L is c o m p l e t e l y d e t e r m i n e d (up t o i s o t o p y ) b y t h e 6 - t u p l e o f integers
(~tOl, flo, ill, 9,o, 9'1, 9,2). I n d e e d , b y p l a n a r i t y we o b t a i n t h e f o l l o w i n g r e l a t i o n s :
~to2 = 2 / / 0 - ~tol ,
~q2 = 2fll - ~tol,
Further, we can always assume that the inequalities, listed below, hold:
80 E min{fl,, f12},
O < T i < f l l
1,
Substituting (I) in the inverse relations of (III) yields
(II)
(III)
(IV)
(V)
Otol = h - 2t,
8o=P - t ,
8 t = h - t ,
N o w we prove that the inequalities hold. Indeed, (I) and (II) imply that 0q2 = 2fl2 - ~o2 ~ 2/3o - 0to2 = ~ol, hence ~q2 -> Ctol. Thus, it follows that 2~12 _> 2ill = Ctl2 q- tZ01 >_ 2~01 as claimed (use (I) too).
We have to show that the 6-tuple (p, q, k, h, t, s), given by (IV), satisfies the
inequalities of the statement. Obviously, we have that p = f12 > 1 b y definition (
2
).
Since 7o -< 72 < f12 + 70 - 1 by (II) a n d p = f12 by (IV), the integer q = ])2 -- 70 "~- 1
verifies the inequalities 1 _< q _< p - 1 as requested. Further, (II) also gives
k = 7o -< f l o - 1 _< f12 - 1 = p - 1, hence0_< k_< p - 1. Sincecq2 _> fl~ b y ( V ) , w e
have that h = cq2 >_ fll >-2 by definition (
2
). Relations (I) and the inequality
9o l -< 2flo imply that
h = ~12 = 2fll - ~ol ~ 2fll + 2 f l o - 2~ol = ~o2 + a12 = 2fl2 = 2p,
hence it follows that 2 < h < 2p.
N o w we p r o v e that
0_< t_< m i n { p - k, k + h - p}.
Obviously, the inequality t _> 0 holds as t = fll - ~ol by (IV) and fit >_ ~tol by (V).
Thus, we have to show that t _< p - k and t _< k + h - p. F o r the first one, (I)
implies that t = fll -- ~tOl -< flo + fll -- Otol - 70 = (~to2 + ~t12)/2 - 7o = f12 - 70 =
p - k as flo > 7o and f12 = P. Hence, we have t _< p -- k as requested.
The assumption that the arc with end (0, flo) undercrosses BI yields the
inequality
70 + ~ot
>/~o
This and (I) imply that
Then w e have
70 "~ 0~12 - - fl0 ~> 0~12 -- 0~01 = 2fll - - 2 g o l .
t = f l l - ~ t o t _ < T o + 0 q z - f l o - f l l + a t o l
= k + h - P
as k = 7o, h = 0t12 by (IV), and p = f12 = (~to2 + ~q2)/2 = flo + fll - ~tOl by (I).
Thus we have proved that
t _ min{p - k, k + h - p}
as claimed.
N o w we have to show that l < s < h - t . Obviously, s = 7 1 + 1 > 1 as
Yl > O. Further, (II) gives Yl < f l l - 1 and hence by (I) and (IV) we obtain
s = y l + l _ < f l l = ( 2 f l l - 0 t o 1 ) - ( f l l - ~ t o l ) = h - t as 2 f l l - ~ o 1 = a 1 2 = h and
t = fll -- ~ T h u s the p r o o f is completed. [ ]
It can be easily verified that the link L(p, q, k, h, t, s) is again a 2-bridge link
w h e n e v e r o n e o f t h e b l l o w i n g c a s e s h o l d s :
k + h - l > p ,
k + h + q - p - l = t - s ,
k + h + q - l > ~ ,
k + h + q - l > 2 p ,
p - k = h - s ,
k + h + q - 2 p - l = t - s ,
2 p - q - k = s - 1 ,
k + h - l > 2 p ,
k + h + t - 2 p = s .
(
1
)
(
2
)
(
3
)
(
4
)
(
5
)
However, this fact surely happens in m o r e subtle ways but we d o not k n o w the
general answer.
The next result describes the connected s u m of 2-bridge knots in terms o f o u r
representation.
Proposition 2. Let K(po q~) be the Schubert normal form of a 2-bridoe knot for
i = 1, 2. Then the connected sum K(pl, ql) 4e K(p 2, q2) is equivalent to the 3-bridge
knot L(p, q, k, h, t, s), where
P = P2,
Finally, we complete the section with some results about a certain class of
3-bridge knots, introduced by Morikawa in [18] and [19]. He defined a normal
form K(m, n; r) to study a class of 3-bridge knots, where m, n are coprime odd
integers, 0 < In[ < m, and r e Z. Now we give a description of this normal form
in terms of our representation. This yield some consequences about the knot type
of L(p, q, k, h, t, s). First we sketch the construction of K(m, n; r) as given in [18]
and [19]. Let K(m, n) be the Schubert diagram of a 2-bridge knot. Then K(m, n)
is the union of two (oriented) bridges AB and EF, whose interiors are contained
in the upper half-space of R 3, and of two (oriented) underpasses BE and FA in
the plane R 2 x 0. Let A' and B' be the nearest points in p(AB) to A and B,
respectively. Here p: R 3 --* R2 x 0 denotes the standard projection. Then the union
of the (oriented) arcs AA', A'B, BB', and B'A bounds a closed 2-cell of the plane
R2 x 0, called a parallel area (see [18]). Let C and D be points in the interior of
the parallel area near A and B, respectively.
Now the knot K(m, n; 0) is constructed from K(m, n) by the following rules:
(
1
) Add a new underpass CD in the interior of the parallel area.
(
2
) Substitute the bridge AB of K(m, n) with two disjoint parallel overpasses AD
and CD.
To obtain K(m, n; r), r ~ O, from K(m, n; 0) we perform [r[ twists around the arcs
AF and CD. More precisely, if r > 0 (resp. r < 0), then we move the top vertex A
(resp. C) so that it passes through the bridge CD (resp. AD) [rl-times.
Now we can prove the following result:
Proposition 3. Let K(m, n) be the Schubert normal form of a 2-bridoe knot, where
m, n are odd coprime integers and 0 < Inl < m. Let (~t,[3) be the unique pair of
inteoers which satisfies the followino conditions:
mfl - n~t =
0 < l f l l < ~ t < m , u even.
Then the Morikawa knot K(m, n; O) is equivalent to the 3-bridge normal diagram
L(p,q, k , h , t , s ) , w h e r e p = h = m + ~/2,q = n + (fl + 1)/2 - 1, andk = s = t = 1.
Further, the Morikawa knot K(m, n; r), r ~ 0, has the same knot type as
L(p, q, k, h, t, s), where
{z
p = m + ~ + 2 1 r l - 1 ,
q = p - 1 ,
Applying Propositions 2-4 of [18] yields the following:
Corollary 4.
(
1
) L(4g + 2, 2g + 1, 1, 4g + 2, 1, 1) and L(4g + 3, 4g + 2, 2, 4g + 3, 1, 2g + 2)
are equivalent to the Schubert knots K(8g + 1, 6g + 1) and K(8g + 3, 2g + 1),
respectively, for any g > 1.
(
2
) L(4g + 4, 2g + 1, 1, 4g + 4, 1, 1) and L(4g + 5, 4g + 4, 2, 4g + 5, 1, 2g + 3)
have the same knot types as the Schubert knots K(8g + 5, 2g + 1) and
K(8g + 7, 6g + 5), respectively, for any g > O.In particular, L(
4, 1, 1, 4, 1, 1
)
K(
5, 1
) is the torus knot T(
5, 2
) of type (
5, 2
).
(
3
) L ( 3 g + 2 , 3 g + l , 2 , 3 g + 2 , 1 , 3 g + l ) and L ( 3 g + 4 , 3 g + 3 , 4 , 3 g + 2 , 1,3g+ 1)
are equivalent to the torus knots T(3g + 1, 3) and T(3g + 2, 3), respectively,
for any g >_ 1.
3. Twofold Branched Covers
In this section we obtain a combinatorial representation of all closed orientable
3-manifolds of Heegaard genus 2 by means of special colored graphs, which are
standardly constructed from the 6-tuples of integers (p, q, k, h, t, s). First we recall
some definitions and results about the representation of manifolds by means of
edge-colored graphs, called crystallizations (see [7] and [22]). For basic graph
theory we refer, for example, to [8]. We use the term graph instead of multigraph,
i.e., loops are forbidden but multiple edges are allowed. An edge-coloration on a
graph G = (V(G), E(G)) is a map c: E(G) ~ An = {0, 1. . . . . n} such that c(e) ~ c(f)
for any two adjacent edges e, f of G. An (n + 1)-colored graph is a pair (G, c)
formed by a regular graph G of degree n + 1 together with an edge-coloration
c: E(G) --* An. An n-pseudocomplex (see [11]) IGI can be associated to (G, c) by the
following rules:
(
1
) Take an n-simplex ~(v) for each vertex v e V(G) and label its vertices by An.
(
2
) If v, w ~ V(G) are joined by an /-colored edge, i e An, then identify the
(n - 1)-faces of a~(v) and an(w) opposite to vertices labeled by i.
We identify IGI with the underlying polyhedron and say that (G, c) represents IGI
and every PL homeomorphic space. The colored graph (G, c) is said to be
contracted if IG I has exactly n + 1 vertices. A crystallization of a closed connected
n-manifold M is a contracted (n + 1)-colored graph which represents M. It was
proved in [22] that each dosed connected PL n-manifold admits a crystallization.
In particular, a simple algorithm to construct a crystallization (called 2-symmetric)
of the twofold covering of 53 branched over a bridge-presentation of a link L
exists (for details, see [6]). Figures 7 and 8 show the construction for K(
5, 3
) and
L(
5, 2, 3, 4, 0, 3
).
Now let G(p, q, k, h, t, s) denote the 2-symmetric crystallization of the twofold
branched covering over the 3-bridge diagram L(p, q, k, h, t, s). As a direct
consequence of [6] and Theorem 1 we obtain the following result:
Proposition 5. Let M be a closed connected orientable 3-manifold of Heegaard
genus <_2. Then a 6-tuple of integers exists such that G(p, q, k, h, t, s) is a
crystallization of M.
Thus the set of 6-tuples of integers satisfying the conditions of Theorem 1 gives
a complete catalogue of all 3-manifolds of Heegaard genus < 2. Let M(p, q, k, h, t, s)
be the twofold covering o f S 3 branched over the 3-bridge diagram L(p, q, k, h, t, s).
- ~
xf
s I
+
9
s J
I
~- - ~ I
I
11 ~' 9
s
9
IL I
"~'--. ........
(a)
(b)
The following p r o p o s i t i o n describes the topological structure of the manifolds
M(p, q, k, h, t, s) for several 6-tuples of integers.
P r o p o s i t i o n 6.
Proof. (1.1) The link L(2q - 1, q, 0, 2q - 1, 0, q) has exactly q unlinked
components, which are K(2q - 1, q) and q - 1 unknots.
(1.2) Let G = G(p, q, 0, 2, 0, 1) be the 2-symmetric crystallization of the twofold
branched covering M = M(p, q, 0, 2, 0, 1). Let C and C' denote the {0, 1}- and
{2, 3}-colored cycles of G with vertex sets {(
2, 0
), (
2, 1
), (
2, 2
), (
2, 3
)} and {(0, 1), (
0, 5
),
(
0, 9
), (
1, 2
), (
1, 8
), (
2, 2
)}, respectively. The subgraph 0 of G determined by C and
C' is a (
3, 5
)-dipole in the sense of [7]. Let G' be the colored graph obtained from
G by canceling the dipole 0 (for details see [7]). Then G' also represents the
manifold M. N o w it is very easy to see that M is the lens space of the statement
since G' has genus 1.
(1.3)--(1.6) are direct consequences of Corollary 5.
(2.1) and (2.2) follow from Corollary 5 and Theorem 3.8 of [19].
(2.3) Let G be the 2-symmetric crystallization of
M = M(4g + 2r + 3, 4g + 2r + 2, 2r, 4g + 5, 1, 20 + 3).
As shown in [7], a finite presentation of the fundamental group H~(M) of M can
be directly obtained from G by the following rules:
The generators are the connected components, except one (arbitrarily
chosen), of the subgraph Gto' 1~= (V(G), c-1({0, 1})), where c: E(G) --, A 3 =
{0, 1, 2, 3} is the coloring of G.
(//) The relators are obtained by walking along the {2, 3}-colored cycles of G,
giving the exponent + 1 (resp. - 1) to each generator that is met if 0 (resp.
1) is the color of the edge by which it is arrived at. Furthermore, one relator
(arbitrarily chosen) may be dropped too.
N o w let x, y, z be the {0, 1}-colored cycles of G. By construction, they correspond
to the bridges Bo, B1, B2 of L in this order. As written above, x, y, z generate the
group HI(M ). To determine the relators of I-II(M) we consider the {2, 3}-colored
cycles, except one, of G. Walking along these cycles yields the finite presentation
I - I I ( M ) = ( x , y, 2: ( x y - l ) 2 r - l ( g y - 1)2 = 1, X Z - t ( x g - l x y - f g y - 1 ) 2 g + l
-----1, y = 1)
'~ (X, Z: X 2 r - I = Z -2, X Z - I ( x z - I x z ) 2g+1 = 1).
Moreover, this presentation is geometric, i.e., it arises from a Heegaard diagram
of M. This also means that the presentation corresponds to a spine of M as shown
in [41. Hence M must be a Seifert fibered space over the 2-sphere with three
exceptional fibers as III(M) admits a relation of type x" = zm(see Theorem 3.1 of
[21], p. 485). N o w we simplify the above presentation. Since z 2 is central in the
group III(M), the second relation is equivalent to
X Z - I(XZ- I X Z - 1Z2)2g+ 1 = 1,
(XZ-1)4$+3Z 4r
= 1,
hence
1-It(M) = ( x , z: x 2 r - 1 = z - 2 , ( x z - 1)4g+3z4g+2 = 1>.
N o w we prove that M is h o m e o m o r p h i c to the Seifert manifold
Z = (o 0 o: 0(2r - 1, 1X2, 1X4e + 3, - ( 2 f f + 1))).
F o r this, it suffices to show that the fibered manifolds have isomorphic
fundamental groups (use T h e o r e m 6, p. 97, and Section 5.4, p. 99, of [20]). By Section
5.3, p. 90, of [20], it is well k n o w n that H t ( Z ) can be presented by
(ql, q2, qa, h: qel"-th = 1, q~h = 1, q3ag+3h - ( 2 g + l ) = 1,
q l q 2 q a = 1, [qj, hi = l ( j = 1, 2, 3)).
Deleting the generators qa = q2 lq~-t and h = q2 2 yields
I'It(X;) = ( q t , q2: ql2"-1 = q2, q2,g+2 = (qtq2),g+3).
Setting x = ql and z = q2 t implies that I-It(E) is isomorphic to I I t ( M ) as required.
(2.4) W e proceed as in case (2.3). T h e n we obtain the finite presentation
H I ( M ) = ( x , y, z: ( x y - l ) 2 " z x - l z y
- 1 = 1, ( x z - l x y - l z y - 1 ) 2 g z y
- 1 = 1, x = i )
( y , z: z 2 = y2,+1, ( z - l y - l z y - 1 ) 2 a z y - i
= 1).
N o w the result follows by using arguments similar to the previous case.
(
3
) It suffices to n o t e that L(p, 1, 0, p, 0, 1) is equivalent to the torus link of type
(p, 3). T h e n the result is a direct consequence of T h e o r e m 3.8 of [17] and L e m m a
1.1 of [14]. [ ]
Remark. P r o p o s i t i o n 6(2.3) and (2.4) gives an alternative p r o o f of T h e o r e m 4 of
[19] by using the t h e o r y of 3-manifold spines. This also allows us to obtain simple
geometric presentations of the fundamental g r o u p of the considered manifolds.
4. n-Bridge Links
In this section we define a cut-and-paste m o v e on an n-bridge link depending
u p o n a 5-tuple of integers (k, h, t, s, r). F o r r = 0, we essentially obtain again the
m o v e of type I/II described in Section 2. T h e case r # 0 yields a new cut-and-paste
m o v e of type III. T h e n we prove that any n-bridge link L in 5 a, n > 4, can be
represented by a special type of diagram obtained from a 3-bridge diagram
L(p, q, k, h, t, s) by a sequence of (ki, hi, ti, si, ri)-moves, for i = 1, 2. . . . . n - 3. Thus
any n-bridge link can be standardly constructed from a set of 5 n - 9 integers
(i,~,O)
(i,.,1)
(i~,2~
(i,,,, 2/~i. - 1)
(i~, 2B~_ - 2)
(i,, I)
.(i,,2)
(i~, 2~, - 1) 9"
(6,2~11 - 2) ...
(i~,~i, - t)
9.- (i,,,~i~ - 1)
(i,2fli - 1)
(i, 2~i - 2)
(i,/3i + 1)
(i, I)
(!,2) ,,
i
(i, ~ - 1)
(i.0)
(i, ~)
Bi
(i, O)
(i,A)
Bi
(ix,O)
(i:, ~, )
Bit
(a)
(.,0)
(,,h - t -Irl)
B .
(b)
which satisfy certain inequalities (n > 3). N o w applying the construction of [3] to
our n-bridge diagrams yields a c o m b i n a t o r i a l representation of closed 3-manifolds
by special edge-colored graphs (also c o m p a r e [ i ] , [10], [151 and [16-1).
Let L be a d i a g r a m of an n-bridge link. Let Bi denote the bridges of L d r a w n
in the plane z = 0 and let fit - 1 be the n u m b e r of undercrossings of By We denote
by Bh, Bt2. . . . . Bi. the bridges of L which are connected to Bt by some arcs of L
and assume the n u m b e r i n g of vertices of Bij as shown in Fig. 9(a).
Let ej,, be the arc of L with ends (i, j) and (it, j + q,) for i = 0, 1, . . . , 2fit - 1 and
l = 1, 2. . . . . m. By definition, the arc ej.~ joins the bridge B~ to Bt,. Let us consider
h consecutive arcs dl, d2 . . . . . dh, 2 < h < 2flo, of L which connect Bt to Bit, starting
from d 1 = ek. h 0 ~ k < flo -- 1.
N o w we define a (k, h, t, s, r)-move on L. If r = 0, then operate on dl, d 2 . . . . . d h
by a cut-and-paste m o v e of type I / l I as shown in Section 2. Here we insert a new
bridge, B, say, which is disjoint and parallel to B t and B~,. If r # 0, then the m o v e
of type I I I is defined by the following rules:
(
1
) C u t the arcs dl, d 2 . . . . . d h by a new bridge B, as before. Let x p yj denote
the ends of d~ incident with B~, Bt,, respectively. Suppose t h a t an index at,
1 < ~ < h, exists such that x~ = (i, 0). T h e n we require that B, has exactly
h - t - Irl - 1, t > 0, undercrossings n u m b e r e d as shown in Fig. 9(b).
(
2
) A d d t undercrossings d'l, d~ . . . . . d't (resp. s d~ . . . . . d'[) between the vertices
(i, fit) and (i, fli - 1) (resp. (i, fit) and (i, fit + 1)). T h e n change the n u m b e r i n g
on the bridge obtained from B i as follows. Call ar the arc with end xr for
any ~ = 1, 2. . . . . ~t-- 1, and set a,+~ = drr, a~+~+t+t = d rtt for any ~/= 1,
2. . . . . t. Finally, let a,+t+l be the arc with end x~ and a~+~+2~+ 1 the one
with end xr for any y = 0, 1. . . . . h - ~t - 1.
(
3
) (r > 0) Join the hanging arc a v with the vertex y, for any v = 1, 2. . . . . r,
1 < r < h. T h e n connect the 2(h - t - r) remaining arcs with a free end to
vertices of B, consecutively so that a,+v+~ (resp. Yu) is joined to (n, s + v)
( r e s p . ( n , s - / z - 1 ) ) f o r 0 < v < h - 2 t + r - l a n d r + l < # < h .
(3') (r < 0) Join the hanging arc ah+2t+ ~-~ with the vertex Y h - , - l for any v = 1,
2. . . . , - r , - h < r < 0. Then connect the 2(h - t + r) remaining arcs with
a free end to vertices of B., consecutively, so that a, + 1 (resp. y~ +1) is joined
t o ( n , s + v ) ( r e s p . ( n , s - # - l ) ) f o r 0 < v < h - 2 t + r - - 1 a n d 0 < # <
h + r - 1. Observe that there is a unique w a y (up to isotopy) to do these
constructions in order to preserve the planarity and without introducing
a n y new crossing (see, for example, Fig. 10 for the case r = 0 and Fig. 11
for r ~ 0).
Proposition 7. With the above notation, any n-brid#e link in 5 3, n _> 4, can
be obtained f r o m an (n - 1)-bridye link by a (k, h, t, s, r)-move, where
0 < k < f l o - 1 ,
2 < h <
2flo,
0 < t < min{fl o - k, k + h - flo},
l < s < h - t - l r l ,
0 _ < l r [ < h .
Proof. Assume that flo < fll and that the vertex (0, flo) is joined to the bridge B1.
Let (0, fl) and (1, ~) denote the ends of the first arc, with respect to the numbering,
which connects B o and B t. T h e inequalities 1 < fl _< flo hold b y a possible rotation
of Bo t h r o u g h an angle n. N o w we construct an (n - 1)-bridge link L' from which
L is o b t a i n e d by a suitable cut-and-paste move. F o r this, take out the bridge B1
(a)
(b)
Fk. le. (a) L(
7, 1, 4, 7, 1, 2
) and (b) rink obtained from it by a (
3, 6, 1, 1, 0
)-move.
from L. T h e n there are 2fll h a n g i n g arcs, n a m e d xl, x2 . . . . . xh,, Yl, Y2, " , Yh2,
where xj (resp. Yi) is (resp. is not) j o i n e d to Bo. H e r e we h a v e h I > 1 a n d
h2 = 2fll - hi > 1. By hypothesis, the vertex (0, flo) belongs to a n arc x , for s o m e
~t, 1 < ~t < h 1. W e h a v e to consider three cases.
Case 1: h a < h 2. W e c o n s t r u c t L' b y the following rules:
(
1
) A d d t = (h2 - hl)/2 = fll - hi u n d e r c r o s s i n g s b e t w e e n the vertices (0, flo)
a n d (0, flo - 1) (resp. (0, flo) a n d (0, flo + 1)), n a m e d d'l, d ' 2 , . . . , d'~ (resp. d~,
d~ . . . . . d~'). N o w there are exactly h 1 + 2t h a n g i n g arcs c o m i n g out the
bridge Bb derived f r o m B o. W e r e n u m b e r the vertices o f Bb as follows. A n y
vertex (0, j) o f B o preserves the s a m e c o o r d i n a t e s for j e 7/po. Let (0, flo + i)
a n d (0, flo + t + / ) be the vertices incident to d'i a n d d~', respectively, for e a c h
i = 1, 2. . . . . t. Finally, the vertex (0, flo) o f B o b e c o m e s (0, flo + t) in B~ a n d
(0,j) b e c o m e s (0, flo + 2t + j ) for flo + 1 <_j _< 2flo - 1.
(
2
) J o i n the free end o f x j (resp. x~) to yj (resp. y~) for each j = 1, 2 . . . . . ct - 1.
(
3
) J o i n the free end of d) (resp. d~) to yj+~_ i (resp. yj+~+~) for e a c h j = 1, 2 . . . . . t.
As usual, we p e r f o r m the c o n s t r u c t i o n preserving the p l a n a r i t y (see, for example,
Fig. 10). T h u s L is o b t a i n e d f r o m L ' by a (k', h', t', s', r')-move, where k ' = fl,
h' = h 2 = h t + 2t, t' = t = (h 2 - hl)/2, s' = ?, a n d r' = 0. It can b e easily verified
t h a t the 5-tuple o f integers satisfies the inequalities o f the statement.
Case 2: h 1 > h2, t = (hi - h2)/2 <- t*, where t* = min{flo - fl, fl + hi - flo}- T h e
a b o v e p r o o f can be easily adjusted in o r d e r t o s h o w t h a t L is o b t a i n e d f r o m L'
by a (k', h', t', r')-move, where k' = fl, h' = hi - 2t, t' = t = (h I - h2)/2, s' = ?, a n d
r t _ ~ _ O ,
Case 3 : hi > h2, t = ( h i - h2)/2 > t*. F o r convenience, we e x a m i n e the case
t * = f l o - f t . F o r t* = fl + h z - f l o , we can p r o c e e d in a similar way. I f t * =
flo - fl, t h e n we cut r = t - t* consecutive arcs d'l, d~ . . . . . d', o f L w i t h ends (
0, 1
),
(
0, 2
). . . . , (0, r), respectively. L e t aj (resp. bj) d e n o t e the e n d o f d~ w h i c h is (resp. is
not) i n c i d e n t t o the b r i d g e Bo for e a c h j = 1, 2, . . . , r. F u r t h e r , let a r § (resp. b,+i)
be the a r c with e n d xj (resp. Yi) for e a c h j = 1, 2 , . . . , h 1 (resp. i = 1, 2 . . . . . h2). N o w
there a r e e x a c t l y h I + h 2 + 2r h a n g i n g arcs, o f w h i c h hi + r c o m e o u t f r o m B o.
T o c o n s t r u c t L', w e c o n n e c t t h e free e n d s o f t h e arcs ai+~+ , a n d a ~ + , _ i _ 1 for e a c h
i e Z t a n d the o n e s o f a ~ § a n d bi for e a c h j = 1, 2 , . . . , f l l - f l o + f l ,
p r e s e r v i n g the p l a n a r i t y . T h e n L is o b t a i n e d f r o m L ' b y a (k', h', t', s', r ' ) - m o v e ,
w h e r e k' = fl - r (resp. fl) if t* = fl0 - fl (resp. t* = fl + ht - fl0), h' = hi - 2t + r,
t' = t, s' = 7, a n d r' = r (resp. - r ) if t* = fl0 - fl (resp. fl + hi - flo), as r e q u i r e d
(see, for e x a m p l e , Fig. 11). [ ]
Acknowledgment References
T h e a u t h o r s wish t o t h a n k t h e referee for his useful suggestions.
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