MONODROMY MAPS OF FIBERED 2-BRIDGE KNOTS AS ELEMENTS IN AUTOMORPHISM OF FREE GROUPS HIROSHI GODA AND MASAAKI SUZUKI A BSTRACT. In this note, we study a monodromy map of a fibered 2-bridge knot. We show the monodromy map of a fibered 2-bridge knot as an element in automorphism of a free group.
1. I NTRODUCTION Let be a compact oriented connected surface of genus nents. Let be the mapping class group of
Namely,
Diff
with boundary compo-
orientation preserving diffeomorphism such that id isotopy fixing
pointwise
Set , where is a point in . acts on , then Aut is a group homomorphism. It is known (eg. [12]) that this map is injective, furthermore Aut
where is an element in , which is parallel to ignoring the orientation. If the boundary of is non-empty, i.e., , any boundary-fixing diffeomorphism must preserve the orientation. a Seifert surface of . The exterior of Let be an oriented link in the 3-sphere , and , say , is Int where means a regular neighborhood of in and . If is a fiber bundle over the circle with fiber Int is its interior. Set , then is called a fibered link. It is known that is a minimal genus Seifert surface in such case. Thus, if is a fibered link, is with an identification according by which is attached to . This map is to a diffeomorphism called a monodromy of . Note that we may regard as an element of . In this note, we focus on fibered 2-bridge knot. We study the monodromy maps of a fibered 2-bridge knot as an element of Aut . Torus knots are fibered, and torus knots of type odd are 2-bridge. We study these knots in Section 3. General case is discussed in Section 4. We put a list of the monodromies of fibered 2-bridge knots with up to crossings in Section 5.
Date: September 30, 2013. 2000 Mathematics Subject Classification. Primary 57M27, Secondary 57M25. Key words and phrases. 2-bridge knot, fibered knot, monodromy. This work was supported by JSPS KAKENHI Grant Numbers 24540068, 24740035. 1
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HIROSHI GODA AND MASAAKI SUZUKI
The authors thank Professor Kunio Murasugi for his helpful comments. The authors also thank Professor Makoto Sakuma for informing the contents of the paper [8]. 2. F IBERED 2- BRIDGE
LINKS
Let be a 2-bridge link in Schubert’s notation. Here, and are coprime integers, and is odd. It is known that and are equivalent if and only if and mod , and that gives the mirror image of . Consider a subtractive continued fraction expansion of (see [5]):
..
.
where and The length of this expansion is . Then is the boundary of the surface obtained by plumbing bands in a row, the -th band has half-twists (righthanded if and left-handed if ). If some is odd, the expansion is said to be of odd type. Otherwise, it is said to be even type. Any fraction is presented by both odd type and even type. In this paper, an expansion always means a subtractive one. We note that the following equality:
..
.
In what follows, we suppose that an expansion is even type. Then, if the length of an expansion is even (odd resp.), becomes a knot (2-component link resp.) and the genus of is equal to ( resp.) by [9]. Further, we suppose , then . It is known that is fibered if and only if ([2], [10]), namely is obtained from a disk by Hopf plumbings. The monodromy of the right-handed Hopf band (left-handed Hopf band resp.) is right Dehn twist (left Dehn twist resp.), and it is known that where is ( resp.) when ( the monodromy of is resp.), see Proposition 2 in [4]. Thus it is not difficult to treat the monodromy maps of fibered 2-bridge knots. In particular, if all are the same sign, becomes a torus knot of type , which we study in the next section.
Example 2.1. The knot is in Schubert’s notation. Since lowing continued fraction expansion:
has the fol-
the knot can be regarded as the knot obtained by plumbings of the right-handed Hopf band, the left-handed Hopf band, the right-handed Hopf band, and the left-handed Hopf band, successively. See Figure 1.
MONODROMIES OF FIBERED 2-BRIDGE KNOTS
3
F IGURE 1 3. TORUS
KNOTS
In this section, we study the monodromy of the torus knots of type ( odd). The torus knot of type has a projection as illustrated in Figure 2, and we denote has genus 2, then we may take a set by the knot. Its minimal genus Seifert surface , say , as in the right-hand side figure in Figure 2. of generators of Note that is on the knot . Let Int , and we denote by the meridian . Note that and disk of the solid torus containing . Further, set intersects one point. Let be this point. One can view the monodromy as the automorphism of obtained by pushing generators (with basepoint on ) off the side of through and onto the side of where the basepoint travels along the meridian. We draw the product manifold in as illustrated in Figure 3. Here we as . Then we can consider the set of generators of regard ( resp.) corresponding to , where is the point in ( resp.), corresponding to ( resp.). As illustrated in Figure 3, we denote the set by ( resp.).
a1
a4
a3
a2
* F IGURE 2 Since :the torus knot of type is a fibered knot and is the minimal genus Seifert surface of , Int is homeomorphic to , so that we may . Now, we move a generator suppose that toward by an isotopy, in . Note that the starting and ending by this isotopy. points of are in and move on so that arrive at are similar, so we will demonstrate the cases of The movement of and . as in It is not difficult to see there is an isotopy that we have the left-hand side figure in Figure 4. Denote by this isotopy. We suppose that this
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HIROSHI GODA AND MASAAKI SUZUKI
a2
a1
a1
a3 a2
a3
a4
a4
F IGURE 3
Æ
is in . Moreover, we may see by the next isotopy, say , in as illustrated in the right-hand side figure of Figure 4. Let be in respectively, then we can see that . Since is a simple closed curve on , it can be presented by the set of generator as an element of , so that we have . By the same argument, we have : , , . This means that the monodromy map of may have the presentation as an element of Aut( ) as follows:
a4 a2
a2
a4
F IGURE 4
By the same argument, we have: Theorem 3.1. Let be the torus knot of type and the minimal genus Seifert surface. Let be a set of generators of . Then the monodromy
MONODROMIES OF FIBERED 2-BRIDGE KNOTS
5
is given as follows:
4. T HE
MONODROMIES OF FIBERED
2- BRIDGE
KNOTS
Let be a 2-bridge knot of type , which has the following subtractive continued fraction:
..
even
.
Then , where is the genus of . If the knot is fibered, . Let be the minimal genus Seifert surface of as illustrated in Figure 1. We take a set of generators of as follows. Set the base point, and as in Figure 5.
*
a2
a1
a 2i
a2i-1
F IGURE 5
as in Figure 5. (This Similarly, we define the generators and figure is the case of .) The loop starts the base point, goes along the untwisted parts of the Hopf bands corresponding to successively and the twisted part of the Hopf band corresponding to . Then it goes back to the base point via the untwisted parts of the Hopf bands corresponding to again. Since Int is the genus handlebody, we may take a set of generators of Int where is the loop that goes from the base point to upper side and goes through the disk and comes back to the base point. Here is one point and if . See Figure 6.
γ1
γ2
γ2g-1 F IGURE 6
γ2g
6
HIROSHI GODA AND MASAAKI SUZUKI
For example, in the case of the knot , the generators of can be seen in Figure 7. then we have and as in Figure 8 by the same argument in Section 3. Here ! " means ! is homotopic to " in Int .
*
a1
a3 a2
*
a4 F IGURE 7
γ4
γ3
γ2
γ1
a+4
a+2 F IGURE 8
In general, we have the following lemma as seen in Figure 9 where we draw the case of # and # . Since the parts of or which are not drawn in the figure run untwisted part of each Hopf band, they do not affect the result.
a2i-1
γ 2i-2
a+2i + a2i-1
a2i-1
γ2i-1
a2i-1
γ2i
a+
2i
c 2i-1 =1
a 2i
c 2i = 1 F IGURE 9
a+2i
γ 2i+1
MONODROMIES OF FIBERED 2-BRIDGE KNOTS
7
Lemma 4.1 (Section 5 in [8]). Let be a fibered 2-bridge knot of type and has the subtractive continued fraction . Set # , i.e., # . For , we have:
where
.
Theorem 4.2. Under the same notations as in Lemma 4.1, the monodromy of is given For , explicitly by the following. If , then
Proof. Just calculations using Lemma 4.1. Suppose . By Lemma 4.1, and . Then we have and , so that and . Suppose . We have via the same argument. Since and , we have . Because , . For , then . For , . Hence . So we have the conclusion via the presentation of
and . Similarly we have the case of .
¼
From Theorem 4.2, we obtain the following matrix ! as the transformation matrix : $ $ : !
!
!
!
¼
!
# # #
..
have:
! !
# # # #
), and ! !, where & is the
.
where !
#
# #
#
# # #
#
# # # # #
(
# # # # # # By Milnor [7], % # #
%&
identity matrix. By standard arguments of the linear algebra, we
Corollary 4.3. Let be a fibered 2-bridge knot as above. Then the coefficient of % of the # # . That of the term % ' of Alexander polynomial of is
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HIROSHI GODA AND MASAAKI SUZUKI
the Alexander polynomial of is given by the following:
'
'
# # #
#
# # #
#
# #
# #
Remark 4.4. In [6], Kanenobu gave an algorithm for calculating the Alexander polynomial of a 2-bridge link using the continued fraction expansion. Fukuhara [3] gave an explicit formula for the Alexander polynomial of a 2-bridge link using elementary number theoretical functions. 5. A PPENDIX :
LIST OF MONODROMIES
We put a list of the monodromies of fibered -bridge knots with up to crossings according to Theorem 4.2. A projection of each knot can be found in [1] and [11]. We use the Schubert’s notation in Table I [1]. Knot
Images of
,
,
MONODROMIES OF FIBERED 2-BRIDGE KNOTS
, , ,
, , , , , , ,
, , , , , ,
9
10
HIROSHI GODA AND MASAAKI SUZUKI
, , , ,
, , , , ,
, , , , , , , , , , , , ,
MONODROMIES OF FIBERED 2-BRIDGE KNOTS
11
,
,
R EFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
J. C.Cha and C. Livingston, KnotInfo, Table of knot invariants, http://www.indiana.edu/ knotinfo/. R. Crowell, Genus of alternating link types, Ann. of Math. (2) 69 (1959), 258–275. S. Fukuhara, Explicit formulae for two-bridge knot polynomials, J. Aust. Math. Soc. 78 (2005), 149–166. D. Gabai and W.H. Kazez, Pseudo-Anosov maps and surgery on fibred 2 -bridge knots, Topology Appl. 37 (1990), 93–100. A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), 225–246. T. Kanenobu, Alexander polynomials of two-bridge links, J. Austral. Math. Soc. Ser. A 36 (1984), 59–68. J. Milnor, Infinite cyclic coverings, 1968 Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), 115–133. K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), 143–167. K. Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248. K. Murasugi, Signatures and Alexander polynomials of two-bridge knots, C. R. Math. Rep. Acad. Sci. Canada 5 (1983), 133–136. D. Rolfsen, Knots and links, Mathematics Lecture Series, No. 7. Publish or Perish, Inc., Berkeley, Calif., 1976. H. Zieschang, E. Vogt and H. Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics, 835. Springer, Berlin, 1980.
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