- Email: [email protected]
On the maximal Thurston–Bennequin number of knots and links in spatial graphs
Topology and its Applications 180 (2015) 132–141
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On the maximal Thurston–Bennequin number of knots and links in spatial graphs Toshifumi Tanaka Department of Mathematics Education, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193 Gifu, Japan
a r t i c l e
i n f o
Article history: Received 17 May 2014 Received in revised form 19 November 2014 Accepted 19 November 2014 Available online xxxx MSC: 57M27 57M50
a b s t r a c t It is known that there does not exist a Legendrian embedding of K4 such that all its cycles realize the maximal Thurston–Bennequin numbers that only consist of odd numbers. This paper represents an infinite family of spatial graphs that are Legendrian embeddings of K4 such that, for each of the Legendrian embeddings, all its cycles but one realize the maximal Thurston–Bennequin numbers that consist of odd numbers and its one cycle realizes the maximal Thurston–Bennequin number equal to zero. © 2014 Elsevier B.V. All rights reserved.
Keywords: Knot Spatial graph Legendrian link Legendrian graph
1. Introduction Legendrian graphs are spatial graphs in R3 with the standard contact structure, that are everywhere tangent to contact 2-planes. Legendrian graphs have appeared in the study of contact manifolds [2,5]. In a recent paper, O’Donnol and Pavelescu [12] have investigated Legendrian graphs and shown that a graph admits a Legendrian embedding with all its cycles trivial unknots if and only if it does not contain K4 as a minor. There seems to be no attempt to classify spatial graphs via Legendrian link theory yet. In this paper, we study a spatial graph by considering the set of the maximal Thurston–Bennequin numbers of its cycles, derived from a classical invariant of Legendrian knots and links. A contact structure on the 3-space R3 = {(x, y, z) | x, y, z ∈ R} is a global differential 1-form ξ such that ξ ∧ dξ = 0 everywhere on R3 . We say that a contact structure on R3 is standard if it is given by the E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2014.11.011 0166-8641/© 2014 Elsevier B.V. All rights reserved.
T. Tanaka / Topology and its Applications 180 (2015) 132–141
133
Fig. 1. A front diagram.
differential 1-form dz − ydx. The 3-space endowed with a contact structure dz − ydx is called the standard contact 3-space. A Legendrian link is a smooth embedding of disjoint circles in the standard contact 3-space such that its tangent vector lies in the contact 2-plane, which is the kernel of the standard contact structure, at each point. The front diagram of a Legendrian link is its projection onto the (x, z)-plane. Generically, the only singularities of a front diagram are cusps and transverse double points [13]. (See Fig. 1(a).) We assume that all front diagrams are generic. For example, Fig. 1(b) shows a generic front diagram of a Legendrian knot which is ambient isotopic to the figure eight knot. We obtain a link diagram of the same topological type from a front diagram by rounding the cusps and making the strand with smaller slope overcross at each double point as in Fig. 1(c). For a front diagram F of an oriented Legendrian link, let c(F ) and w(F ) be the number of left cusps of F and the writhe of a link diagram obtained from F as above. The Thurston–Bennequin number is defined as tb(F ) = w(F ) − c(F ). A Legendrian isotopy between Legendrian links J0 and J1 is an ambient isotopy between J0 and J1 with each level Legendrian. The Thurston–Bennequin number is known to be a Legendrian isotopy invariant of Legendrian links. For an oriented link L, we denote by TB(L) the maximal value of tb over all Legendrian links which are ambient isotopic to L. The link invariant TB(L) is called the maximal Thurston–Bennequin number of L. Let L be an oriented link and D a diagram of L. The Kauffman polynomial [6] F(a,z) (L) ∈ Z[a± , z ± ] is defined as a−w(D) (a,z) (D), where (a,z) (D) is a regular isotopy invariant with properties as follows. (i) (ii) (iii)
(a,z) ()
(a,z) (
(a,z) (
= 1; )=a )+
(a,z) (
(a,z) (
) and
) = z(
) = a−1
(a,z) (
(a,z) (
)+
(a,z) (
(a,z) (
);
)).
In the late of 1990’s, an upper bound for the maximal Thurston–Bennequin number in terms of the Kauffman polynomial (Theorem 1.1) was given by Fuchs and Tabachnikov [4,14]. We call the bound the Kauffman bound. Let f ∈ Z[x± , y ± ] be a Laurent polynomial and write f = i fi (y)xi where fi (y) are polynomials in y ±1 . We denote the largest exponent of x in f by max-degx f . Theorem 1.1. Let K be an oriented link in R3 . Then −max-dega F(a,z) (L) − 1 ≥ TB(L). It is known that the Kauffman bound is sharp for any positive link and any alternating link [3,10,11,15,16], and Kálmán has shown that the bound is sharp for all +adequate links [7]. (For example, all positive links and alternating links are +adequate.) In general, the Kauffman bound is not necessarily sharp. For example, many negative torus knots do not have the sharpness [11]. A spatial graph of a finite graph G is an embedding of G into R3 . A Legendrian graph is a spatial graph consisting of edges and vertices such that each edge is everywhere tangent to the 2-plane field which is the
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T. Tanaka / Topology and its Applications 180 (2015) 132–141
Fig. 2. Reidemeister moves.
kernel of the standard contact structure and non-tangent to each other at the vertices. We say that a spatial ˜ of an embedding of a finite graph G is a Legendrian embedding of G if G ˜ is a Legendrian graph. graph G O’Donnol and Pavelescu have investigated Legendrian graphs and extended classical invariants of Legendrian links to Legendrian graphs [12]. They have shown that if a graph G contains K4 as a minor, then there does not exist a Legendrian embedding of G such that all its cycles realize the maximal Thurston–Bennequin numbers that only consist of odd numbers. Then we consider the following problem which naturally arises from their result. Problem. Does there exist a Legendrian embedding of a graph G which contains K4 as a minor such that all its cycles realize the maximal Thurston–Bennequin numbers? In this paper, we show the following theorem. Theorem 1.2. There exists an infinite family of spatial graphs that are Legendrian embeddings of K4 such that, for each of the Legendrian embeddings, all its cycles but one realize the maximal Thurston–Bennequin numbers that consist of odd numbers and its one cycle realizes the maximal Thurston–Bennequin number equal to zero. In this paper, all knots and links are oriented unless otherwise stated. In Section 2, we shall recall some basic facts about spatial graphs. In Section 3, we shall prove Theorem 1.2. In Section 4, we shall consider which spatial graphs are ambient isotopic to Legendrian graphs such that all their cycles realize their maximal Thurston–Bennequin numbers. 2. Diagrams of spatial graphs In this section, we recall some basic definitions about spatial graphs. Two spatial graphs G1 and G2 are ambient isotopic if there exists an isotopy ht : R3 → R3 such that h0 = id and h1 (G1 ) = G2 . A diagram of a spatial graph is an image of an embedded graph under a projection to the 2-plane such that its singularities are a finite number of double points of edges equipped with over–under information. It is well-known that two diagrams represent isotopic embeddings of a graph if and only if they are related by a finite sequence of Reidemeister moves [17,18,8] depicted in Fig. 2.
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Fig. 3. A front diagram of the unknot with tb = −1.
Fig. 4. The standard smoothing of cycles.
Fig. 5. A cycle of a Legendrian embedding of K4 .
A subgraph of a spatial graph of a finite graph G is a spatial graph which is the image of a spatial embedding restricted on a subgraph of G. A cycle of a spatial graph G of a finite graph G is a subgraph of G which corresponds to a cycle of G. A spatial graph is said to contain a link if the link appears as cycles of G. A planar spatial graph is a spatial graph which is ambient isotopic to an embedding in the 2-plane. Note that any planar spatial graph only contains the unknot as its cycle. 3. Legendrian embeddings of K4 First we consider a trivial Legendrian embedding of K4 . By using the Kauffman bound, we know that the maximal Thurston–Bennequin number of the unknot is equal to −1 since the unknot has a front diagram with tb = −1 as in Fig. 3. A cycle in a Legendrian graph is a piecewise smooth Legendrian knot, that is, a simple closed curve that is everywhere tangent to the contact planes, but has finitely many points, at the vertices, where it may not be smooth. For a given cycle K of a Legendrian graph G, O’Donnol and Pavelescu have defined the standard smoothing of K as shown in Fig. 4 to obtain a (smooth) Legendrian knot, denoted by Kst . Then the Thurston–Bennequin number of K is defined as tb(K) = tb(Kst ). (See [12] for the detail.) The complete graph on 4 vertices K4 is described in Fig. 5(a). We take a Legendrian embedding of K4 as in Fig. 5(b). Then the cycle shown in Fig. 5(c) is the unknot and has the Thurston–Bennequin number −2. So it does not realize its maximal Thurston–Bennequin number. Proof of Theorem 1.2. We consider the following Legendrian graph (Fig. 6), denoted by G. Then we have seven cycles, denoted by L1 , L2 , . . . , L7 , of G with the front diagrams as in Fig. 7.
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Fig. 6. The Legendrian graph G.
Fig. 7. The cycles of G.
Notice that L1 , L2 and L3 are the unknot topologically and satisfy that TB(Li ) = tb(Li ) = −1 (i = 1, 2, 3). Thus they realize their maximal Thurston–Bennequin numbers. In the case of L4 , we know that tb(L4 ) = 0 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L4 as follows. F(a,z) (L4 ) = 2z − 3z 3 + z 5 a−9 + 2z 2 − 5z 4 + 2z 6 a−8 − z − z 3 + 4z 5 − 2z 7 a−7 + 3 − 6z 2 + 2z 4 − z 6 + z 8 a−6 − 5z − 9z 3 − 7z 5 − 3z 7 a−5 + 5 − 11z 2 + 10z 4 − 3z 6 + z 8 a−4 + −3z + 6z 3 − 2z 5 + z 7 a−3 + 1 + 3z 2 + 3z 4 a−2 − z − z 3 a−1 . By the Kauffman bound, we know that −max-dega F(a,z) (L4 ) − 1 = tb(L4 ) = TB(L4 ) (= 0). Thus L4 realizes its maximal Thurston–Bennequin number. In the case of L5 , we know that tb(L5 ) = 1 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L5 as follows. F(a,z) (L5 ) = − 2z − z 3 a−7 + 1 − 2z 2 + z 4 a−6 + −2z + 2z 3 a−5 + 1 − z 2 + z 4 a−4 + z 3 a−3 − 1 − z 2 a−2 .
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Fig. 8. The Legendrian graph Gn .
By the Kauffman bound, we know that −max-dega F(a,z) (L5 ) − 1 = tb(L5 ) = TB(L5 ) (= 1). Thus L5 realizes its maximal Thurston–Bennequin number. In the case of L6 , we know that tb(L6 ) = 5 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L6 as follows. F(a,z) (L6 ) = − z − z 3 a−15 − z 2 − 2z 4 a−14 + 2z − 3z 3 + 3z 5 a−13 + 3z 2 − 4z 4 + 3z 6 a−12 + 2z 5 + 2z 7 a−11 + 2 − 6z 2 + 2z 4 − z 6 + z 8 a−10 − 2z − 5z 3 + 8z 5 − 3z 7 a−9 + 1 + 3z 2 + 3z 4 − 3z 6 + z 8 a−8 + z + z 3 − 3z 5 + z 7 a−7 − 2 − 7z 2 + 5z 4 − z 6 a−6 . By the Kauffman bound, we have −max-dega F(a,z) (L6 ) − 1 = tb(L6 ) = TB(L6 ) (= 5). Thus L6 realizes its maximal Thurston–Bennequin number. In the case of L7 , we know that tb(L7 ) = 5 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L7 as follows. F(a,z) (L7 ) = za−13 + z 2 a−12 − z − z 3 a−11 − 2z 2 − z 4 a−10 + z − 3z 3 + z 5 a−9 − 3 − 7z 2 + 5z 4 − z 6 a−8 + 3z − 4z 3 + z 5 a−7 − 4 − 10z 2 + 6z 4 − z 6 a−6 . By the Kauffman bound, we have −max-dega F(a,z) (L7 ) − 1 = tb(L7 ) = TB(L7 ) (= 5). Therefore L7 realizes its maximal Thurston–Bennequin number. From the above calculation, we know that all cycles of G realize their maximal Thurston–Bennequin numbers and the set of the invariants is equal to {−1, −1, −1, 0, 1, 5, 5}. Next we consider a Legendrian graph, denote it by Gn , obtained from G as shown in Fig. 8. We denote the cycles that correspond to L1 , L2 , . . . , L7 by Ln,1 , Ln,2 , . . . , Ln,7 respectively. Note that Li = Ln,i (i = 2, 4, 6). In the cases of Ln,1 and Ln,3 , we know that tb(Ln,1 ) = tb(Ln,3 ) = 2n − 1. On the other hand, F(a,z) (Ln,i ) = (za−5 − (1 − z 3 )a−4 + za−3 − (2 − z 2 )a−2 )n by using the product formula of the Kauffman polynomial under connected sum [6, Lemma 2.8]. Then by the Kauffman bound, we have −max-dega F(a,z) (Ln,i ) − 1 = tb(Ln,i ) = TB(Ln,i ) (= 2n − 1)
(i = 1, 3).
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Fig. 9. The Whitehead link.
In the cases of Ln,5 and Ln,7 , we know that tb(Ln,5 ) = 2n + 1 and tb(Ln,7 ) = 2n + 5. On the other hand, F(a,z) (Ln,i ) = F(a,z) (Li )(za−5 − (1 − z 3 )a−4 + za−3 − (2 − z 2 )a−2 )n (i = 5, 7). Then by the Kauffman bound, we have −max-dega F(a,z) (Ln,5 ) − 1 = tb(Ln,5 ) = TB(Ln,5 ) (= 2n + 1), −max-dega F(a,z) (Ln,7 ) − 1 = tb(Ln,7 ) = TB(Ln,7 ) (= 2n + 5). From the above, we know that the set of the maximal Thurston–Bennequin numbers of cycles of Gn is equal to {2n − 1, −1, 2n − 1, 0, 2n + 1, 5, 2n + 5}. Since such a set is an invariant of spatial graphs, we know that Gn1 is not ambient isotopic to Gn2 if n1 = n2 . This completes the proof. 2 Remark 3.1. It is easily seen that the knots L5 , L6 and L7 are two-bridge knots. Masaharu Ishikawa has informed me that L4 is the knot 10148 [8]. 4. Non-realizable case Definition 4.1. We say that a spatial graph G is mTB-realizable if G is ambient isotopic to a Legendrian graph such that all its cycles realize their maximal Thurston–Bennequin numbers. In this section, we consider which spatial graphs are mTB-realizable. In general, a spatial graph is not necessarily mTB-realizable. In the case of embeddings of K4 , the spatial graph in Fig. 6 is mTB-realizable, however, the planer spatial graph in Fig. 5(b) is not mTB-realizable because of a result in [12]. We say that a Legendrian knot is trivial if it is topologically the unknot and has tb = −1. The trivial Legendrian knot is the only Legendrian knot that has the maximal Thurston–Bennequin number of the unknot among all Legendrian knots which are topologically isotopic to the unknot. Remark 4.2. Eliashberg and Fraser showed all Legendrian knots representing the unknot are determined by their classical invariants, the Thurston–Bennequin numbers and the rotation numbers denoted by rot [2]. Bennequin’s inequality implies that tb(J) + |rot(J)| ≤ −1 for any Legendrian link J which is ambient isotopic to the unknot [1]. So if L is a trivial Legendrian knot, then L satisfies that rot(L) = 0. The Whitehead link is a link with a diagram as in Fig. 9. In [9], Mohnke proved the following. We also give here the proof. Proposition 4.3. The Whitehead link cannot be represented by a Legendrian link that only consists of trivial Legendrian knots. Proof. The Kauffman polynomial of the Whitehead link is computed as follows. F(a,z) (LW h ) = z −1 − 2z + z 3 a−1 − 1 + z 2 − z 4 + z −1 − 4z + 3z 3 a + z 4 a2 − 2z − 2z 3 a3 + z 2 a4 .
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Fig. 10. The Whitehead link as cycles.
Fig. 11. The handcuff graph and its embeddings.
By the Kauffman bound, we know that TB(LW h ) ≤ −5. On the other hand, if the Whitehead link would be represented by a Legendrian link, denoted by JW h , that only consists of trivial Legendrian knots, then we have tb(JW h ) = −2 by the definition of tb and the condition that the Whitehead link has linking number zero. This is a contradiction. 2 By this proposition, we know that the Whitehead link does not admit a Legendrian embedding such that all its components realize their maximal Thurston–Bennequin numbers. By using this proposition, we have the following. Proposition 4.4. If a finite graph G contains two cycles that have no common edges and vertices, then there is a spatial embedding G of G such that G is not mTB-realizable. Proof. Let G be a graph as in the assumption. Then we can have a spatial embedding G of G such that it contains the Whitehead link denoted by SW h , as in Fig. 10. Suppose there exists a Legendrian graph Γ such that Γ is ambient isotopic to G and all its cycles realize their maximal Thurston–Bennequin numbers. Notice that the isotopy class of SW h , as cycles, does not change by the ambient isotopy for the spatial graphs. Thus, by the assumption, there exist cycles of Γ that are ambient isotopic to SW h . This contradicts Proposition 4.3. 2 Example 4.5. Let G be the handcuff graph. (See Fig. 11(a).) First we consider a Legendrian embedding G0 of G as in Fig. 11(b). Since G0 has two cycles both of which are topologically the unknot and satisfy that tb = −1, the cycles realize their maximal Thurston–Bennequin numbers and then we know that G0 is mTB-realizable. Next we consider a spatial embedding of G which contains the Whitehead link as in Fig. 11(c). By Proposition 4.4, we know that this spatial graph is not mTB-realizable. Definition 4.6. Let L be a link consisting of r components Li (i = 1, 2, ..., r). A link L is completely splittable if there exist mutually disjoint 3-balls Bi (i = 1, 2, ..., r) in R3 such that Li ⊂ Bi .
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Fig. 12. Modifications of crossings.
Fig. 13. Modifications of cusps.
In the case when the union of all cycles of a spatial graph represents a completely splittable link, we have the following. Proposition 4.7. Let G be a spatial graph that has a completely splittable link as the union of all its cycles. Then G is mTB-realizable. Proof. Let G be a spatial graph such that the union of all its cycles represents a completely splittable link. We denote the cycles of G by C1 , C2 , . . . , Cs . Using Reidemeister moves, a diagram of G can be transformed so that diagrams D1 , D2 , . . . , Ds of C1 , C2 , . . . , Cs satisfy that Di ∩ Dj = ∅ (i = j). We denote the resulted diagram by DG . Moreover, we can transform the resulted spatial graph into a spatial graph so that all its cycles are (piecewise smooth) Legendrian knots realizing their maximal Thurston–Bennequin numbers by ambient isotopy. More precisely, since any front diagram of a Legendrian knot (realizing the maximal Thurston–Bennequin number) is changed into a diagram of a knot by rounding the cusps and making the strand with smaller slope overcross at each double point as shown in the introduction, we can transform DG so that all diagrams of the cycles are transformed into (a disjoint union of) such diagrams by Reidemeister moves, and then we obtain a desired spatial graph by making cusps at each local maximum and local minimum (with respect to the x-axis) of the resultant diagrams of the cycles. Now we denote the rest part, the closure of G \ (C1 ∪ C2 , . . . , ∪ Cs ), by E. We may assume that E lies in general position with respect to the x-axis. Then we modify each crossing whose overcrossing has positive slope so that the overcrossing has negative slope as follows. If, at the crossing, both of the overcrossing and the undercrossing belong to E, then we modify it as Fig. 12(a) or 12(b). If, at the crossing, the overcrossing (resp. undercrossing) only belongs to E, then we modify it as Fig. 12(a) (resp. Fig. 12(b)). Finally we transform each local maximum and local minimum of the resulted diagram of E as in Fig. 13. Then we obtain a front diagram of a spatial graph, which is ambient isotopic to G, such that each cycle realizes its maximal Thurston–Bennequin number. (We describe this process in Fig. 14 by using a spatial embedding of the handcuff graph.) 2 Acknowledgement This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grantin-Aid for Young Scientists (B), 2011–2014 (23740046).
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Fig. 14. Modification of a diagram of a spatial embedding of the handcuff graph.
References [1] Y. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, in: Topological Methods in Modern Mathematics, Publish or Perish, 1993. [2] Y. Eliashberg, M. Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2) (2009) 77–127. [3] J. Epstein, D. Fuchs, On the invariants of Legendrian mirror torus links, in: Symplectic and Contact Topology: Interactions and Perspectives, Toronto, ON/Montreal, QC, 2001, in: Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, pp. 103–115. [4] D. Fuchs, S. Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (5) (1997) 1025–1053. [5] E. Giroux, Contact geometry: from dimension three to higher dimensions, in: Proceedings of the International Congress of Mathematicians, vol. II, Higher Ed. Press, Beijing, 2002, pp. 405–414. [6] L.H. Kauffman, An invariant of regular isotopy, Trans. Am. Math. Soc. 318 (1990) 417–471. [7] T. Kálmán, Maximal Thurston–Bennequin number of +adequate links, Proc. Am. Math. Soc. 136 (2008) 2969–2977. [8] A. Kawauchi, A Survey of Knot Theory, Birkhäuser Verlag, Basel, 1996, translated and revised from the 1990 Japanese original by the author. [9] K. Mohnke, Legendrian links of topological unknots, in: Proceedings of the Conference on Topology, Geometry and Algebra: Interactions and New Directions, in: Contemp. Math., vol. 279, 2000. [10] L. Ng, Maximal Thurston–Bennequin number of two-bridge links, Algebr. Geom. Topol. 1 (2001) 427–434. [11] L. Ng, A Legendrian Thurston–Bennequin bound from Khovanov homology, Algebr. Geom. Topol. 5 (2005) 1637–1653. [12] D. O’Donnol, E. Pavelescu, On Legendrian graphs, Algebr. Geom. Topol. 12 (3) (2012) 1273–1299. [13] J. Swiatkowski, On the isotopy of Legendrian knots, Ann. Glob. Anal. Geom. 10 (1992) 195–207. [14] S. Tabachnikov, Estimates for the Bennequin number of Legendrian links from state models for knot polynomials, Math. Res. Lett. 4 (1) (1997) 143–156. [15] T. Tanaka, Maximal Bennequin numbers and Kauffman polynomials of positive links, Proc. Am. Math. Soc. 127 (11) (1999) 3427–3432. [16] T. Tanaka, Maximal Thurston–Bennequin numbers of alternating links, Topol. Appl. 153 (14) (2006) 2476–2483. [17] S. Yamada, An invariant of spatial graphs, J. Graph Theory 13 (1989) 531–551. [18] S. Yamada, A topological invariant of spatial regular graphs, in: A. Kawauchi (Ed.), Knots 90, Proc. Int. Conf. on Knot Theory and Related Topics, Walter de Gruyter, Berlin, 1992, pp. 447–454.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On the maximal Thurston–Bennequin number of knots and links in spatial graphs Toshifumi Tanaka Department of Mathematics Education, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193 Gifu, Japan
a r t i c l e
i n f o
Article history: Received 17 May 2014 Received in revised form 19 November 2014 Accepted 19 November 2014 Available online xxxx MSC: 57M27 57M50
a b s t r a c t It is known that there does not exist a Legendrian embedding of K4 such that all its cycles realize the maximal Thurston–Bennequin numbers that only consist of odd numbers. This paper represents an infinite family of spatial graphs that are Legendrian embeddings of K4 such that, for each of the Legendrian embeddings, all its cycles but one realize the maximal Thurston–Bennequin numbers that consist of odd numbers and its one cycle realizes the maximal Thurston–Bennequin number equal to zero. © 2014 Elsevier B.V. All rights reserved.
Keywords: Knot Spatial graph Legendrian link Legendrian graph
1. Introduction Legendrian graphs are spatial graphs in R3 with the standard contact structure, that are everywhere tangent to contact 2-planes. Legendrian graphs have appeared in the study of contact manifolds [2,5]. In a recent paper, O’Donnol and Pavelescu [12] have investigated Legendrian graphs and shown that a graph admits a Legendrian embedding with all its cycles trivial unknots if and only if it does not contain K4 as a minor. There seems to be no attempt to classify spatial graphs via Legendrian link theory yet. In this paper, we study a spatial graph by considering the set of the maximal Thurston–Bennequin numbers of its cycles, derived from a classical invariant of Legendrian knots and links. A contact structure on the 3-space R3 = {(x, y, z) | x, y, z ∈ R} is a global differential 1-form ξ such that ξ ∧ dξ = 0 everywhere on R3 . We say that a contact structure on R3 is standard if it is given by the E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2014.11.011 0166-8641/© 2014 Elsevier B.V. All rights reserved.
T. Tanaka / Topology and its Applications 180 (2015) 132–141
133
Fig. 1. A front diagram.
differential 1-form dz − ydx. The 3-space endowed with a contact structure dz − ydx is called the standard contact 3-space. A Legendrian link is a smooth embedding of disjoint circles in the standard contact 3-space such that its tangent vector lies in the contact 2-plane, which is the kernel of the standard contact structure, at each point. The front diagram of a Legendrian link is its projection onto the (x, z)-plane. Generically, the only singularities of a front diagram are cusps and transverse double points [13]. (See Fig. 1(a).) We assume that all front diagrams are generic. For example, Fig. 1(b) shows a generic front diagram of a Legendrian knot which is ambient isotopic to the figure eight knot. We obtain a link diagram of the same topological type from a front diagram by rounding the cusps and making the strand with smaller slope overcross at each double point as in Fig. 1(c). For a front diagram F of an oriented Legendrian link, let c(F ) and w(F ) be the number of left cusps of F and the writhe of a link diagram obtained from F as above. The Thurston–Bennequin number is defined as tb(F ) = w(F ) − c(F ). A Legendrian isotopy between Legendrian links J0 and J1 is an ambient isotopy between J0 and J1 with each level Legendrian. The Thurston–Bennequin number is known to be a Legendrian isotopy invariant of Legendrian links. For an oriented link L, we denote by TB(L) the maximal value of tb over all Legendrian links which are ambient isotopic to L. The link invariant TB(L) is called the maximal Thurston–Bennequin number of L. Let L be an oriented link and D a diagram of L. The Kauffman polynomial [6] F(a,z) (L) ∈ Z[a± , z ± ] is defined as a−w(D) (a,z) (D), where (a,z) (D) is a regular isotopy invariant with properties as follows. (i) (ii) (iii)
(a,z) ()
(a,z) (
(a,z) (
= 1; )=a )+
(a,z) (
(a,z) (
) and
) = z(
) = a−1
(a,z) (
(a,z) (
)+
(a,z) (
(a,z) (
);
)).
In the late of 1990’s, an upper bound for the maximal Thurston–Bennequin number in terms of the Kauffman polynomial (Theorem 1.1) was given by Fuchs and Tabachnikov [4,14]. We call the bound the Kauffman bound. Let f ∈ Z[x± , y ± ] be a Laurent polynomial and write f = i fi (y)xi where fi (y) are polynomials in y ±1 . We denote the largest exponent of x in f by max-degx f . Theorem 1.1. Let K be an oriented link in R3 . Then −max-dega F(a,z) (L) − 1 ≥ TB(L). It is known that the Kauffman bound is sharp for any positive link and any alternating link [3,10,11,15,16], and Kálmán has shown that the bound is sharp for all +adequate links [7]. (For example, all positive links and alternating links are +adequate.) In general, the Kauffman bound is not necessarily sharp. For example, many negative torus knots do not have the sharpness [11]. A spatial graph of a finite graph G is an embedding of G into R3 . A Legendrian graph is a spatial graph consisting of edges and vertices such that each edge is everywhere tangent to the 2-plane field which is the
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Fig. 2. Reidemeister moves.
kernel of the standard contact structure and non-tangent to each other at the vertices. We say that a spatial ˜ of an embedding of a finite graph G is a Legendrian embedding of G if G ˜ is a Legendrian graph. graph G O’Donnol and Pavelescu have investigated Legendrian graphs and extended classical invariants of Legendrian links to Legendrian graphs [12]. They have shown that if a graph G contains K4 as a minor, then there does not exist a Legendrian embedding of G such that all its cycles realize the maximal Thurston–Bennequin numbers that only consist of odd numbers. Then we consider the following problem which naturally arises from their result. Problem. Does there exist a Legendrian embedding of a graph G which contains K4 as a minor such that all its cycles realize the maximal Thurston–Bennequin numbers? In this paper, we show the following theorem. Theorem 1.2. There exists an infinite family of spatial graphs that are Legendrian embeddings of K4 such that, for each of the Legendrian embeddings, all its cycles but one realize the maximal Thurston–Bennequin numbers that consist of odd numbers and its one cycle realizes the maximal Thurston–Bennequin number equal to zero. In this paper, all knots and links are oriented unless otherwise stated. In Section 2, we shall recall some basic facts about spatial graphs. In Section 3, we shall prove Theorem 1.2. In Section 4, we shall consider which spatial graphs are ambient isotopic to Legendrian graphs such that all their cycles realize their maximal Thurston–Bennequin numbers. 2. Diagrams of spatial graphs In this section, we recall some basic definitions about spatial graphs. Two spatial graphs G1 and G2 are ambient isotopic if there exists an isotopy ht : R3 → R3 such that h0 = id and h1 (G1 ) = G2 . A diagram of a spatial graph is an image of an embedded graph under a projection to the 2-plane such that its singularities are a finite number of double points of edges equipped with over–under information. It is well-known that two diagrams represent isotopic embeddings of a graph if and only if they are related by a finite sequence of Reidemeister moves [17,18,8] depicted in Fig. 2.
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Fig. 3. A front diagram of the unknot with tb = −1.
Fig. 4. The standard smoothing of cycles.
Fig. 5. A cycle of a Legendrian embedding of K4 .
A subgraph of a spatial graph of a finite graph G is a spatial graph which is the image of a spatial embedding restricted on a subgraph of G. A cycle of a spatial graph G of a finite graph G is a subgraph of G which corresponds to a cycle of G. A spatial graph is said to contain a link if the link appears as cycles of G. A planar spatial graph is a spatial graph which is ambient isotopic to an embedding in the 2-plane. Note that any planar spatial graph only contains the unknot as its cycle. 3. Legendrian embeddings of K4 First we consider a trivial Legendrian embedding of K4 . By using the Kauffman bound, we know that the maximal Thurston–Bennequin number of the unknot is equal to −1 since the unknot has a front diagram with tb = −1 as in Fig. 3. A cycle in a Legendrian graph is a piecewise smooth Legendrian knot, that is, a simple closed curve that is everywhere tangent to the contact planes, but has finitely many points, at the vertices, where it may not be smooth. For a given cycle K of a Legendrian graph G, O’Donnol and Pavelescu have defined the standard smoothing of K as shown in Fig. 4 to obtain a (smooth) Legendrian knot, denoted by Kst . Then the Thurston–Bennequin number of K is defined as tb(K) = tb(Kst ). (See [12] for the detail.) The complete graph on 4 vertices K4 is described in Fig. 5(a). We take a Legendrian embedding of K4 as in Fig. 5(b). Then the cycle shown in Fig. 5(c) is the unknot and has the Thurston–Bennequin number −2. So it does not realize its maximal Thurston–Bennequin number. Proof of Theorem 1.2. We consider the following Legendrian graph (Fig. 6), denoted by G. Then we have seven cycles, denoted by L1 , L2 , . . . , L7 , of G with the front diagrams as in Fig. 7.
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Fig. 6. The Legendrian graph G.
Fig. 7. The cycles of G.
Notice that L1 , L2 and L3 are the unknot topologically and satisfy that TB(Li ) = tb(Li ) = −1 (i = 1, 2, 3). Thus they realize their maximal Thurston–Bennequin numbers. In the case of L4 , we know that tb(L4 ) = 0 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L4 as follows. F(a,z) (L4 ) = 2z − 3z 3 + z 5 a−9 + 2z 2 − 5z 4 + 2z 6 a−8 − z − z 3 + 4z 5 − 2z 7 a−7 + 3 − 6z 2 + 2z 4 − z 6 + z 8 a−6 − 5z − 9z 3 − 7z 5 − 3z 7 a−5 + 5 − 11z 2 + 10z 4 − 3z 6 + z 8 a−4 + −3z + 6z 3 − 2z 5 + z 7 a−3 + 1 + 3z 2 + 3z 4 a−2 − z − z 3 a−1 . By the Kauffman bound, we know that −max-dega F(a,z) (L4 ) − 1 = tb(L4 ) = TB(L4 ) (= 0). Thus L4 realizes its maximal Thurston–Bennequin number. In the case of L5 , we know that tb(L5 ) = 1 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L5 as follows. F(a,z) (L5 ) = − 2z − z 3 a−7 + 1 − 2z 2 + z 4 a−6 + −2z + 2z 3 a−5 + 1 − z 2 + z 4 a−4 + z 3 a−3 − 1 − z 2 a−2 .
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Fig. 8. The Legendrian graph Gn .
By the Kauffman bound, we know that −max-dega F(a,z) (L5 ) − 1 = tb(L5 ) = TB(L5 ) (= 1). Thus L5 realizes its maximal Thurston–Bennequin number. In the case of L6 , we know that tb(L6 ) = 5 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L6 as follows. F(a,z) (L6 ) = − z − z 3 a−15 − z 2 − 2z 4 a−14 + 2z − 3z 3 + 3z 5 a−13 + 3z 2 − 4z 4 + 3z 6 a−12 + 2z 5 + 2z 7 a−11 + 2 − 6z 2 + 2z 4 − z 6 + z 8 a−10 − 2z − 5z 3 + 8z 5 − 3z 7 a−9 + 1 + 3z 2 + 3z 4 − 3z 6 + z 8 a−8 + z + z 3 − 3z 5 + z 7 a−7 − 2 − 7z 2 + 5z 4 − z 6 a−6 . By the Kauffman bound, we have −max-dega F(a,z) (L6 ) − 1 = tb(L6 ) = TB(L6 ) (= 5). Thus L6 realizes its maximal Thurston–Bennequin number. In the case of L7 , we know that tb(L7 ) = 5 from the front diagram in Fig. 7. Then we have the Kauffman polynomial of L7 as follows. F(a,z) (L7 ) = za−13 + z 2 a−12 − z − z 3 a−11 − 2z 2 − z 4 a−10 + z − 3z 3 + z 5 a−9 − 3 − 7z 2 + 5z 4 − z 6 a−8 + 3z − 4z 3 + z 5 a−7 − 4 − 10z 2 + 6z 4 − z 6 a−6 . By the Kauffman bound, we have −max-dega F(a,z) (L7 ) − 1 = tb(L7 ) = TB(L7 ) (= 5). Therefore L7 realizes its maximal Thurston–Bennequin number. From the above calculation, we know that all cycles of G realize their maximal Thurston–Bennequin numbers and the set of the invariants is equal to {−1, −1, −1, 0, 1, 5, 5}. Next we consider a Legendrian graph, denote it by Gn , obtained from G as shown in Fig. 8. We denote the cycles that correspond to L1 , L2 , . . . , L7 by Ln,1 , Ln,2 , . . . , Ln,7 respectively. Note that Li = Ln,i (i = 2, 4, 6). In the cases of Ln,1 and Ln,3 , we know that tb(Ln,1 ) = tb(Ln,3 ) = 2n − 1. On the other hand, F(a,z) (Ln,i ) = (za−5 − (1 − z 3 )a−4 + za−3 − (2 − z 2 )a−2 )n by using the product formula of the Kauffman polynomial under connected sum [6, Lemma 2.8]. Then by the Kauffman bound, we have −max-dega F(a,z) (Ln,i ) − 1 = tb(Ln,i ) = TB(Ln,i ) (= 2n − 1)
(i = 1, 3).
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Fig. 9. The Whitehead link.
In the cases of Ln,5 and Ln,7 , we know that tb(Ln,5 ) = 2n + 1 and tb(Ln,7 ) = 2n + 5. On the other hand, F(a,z) (Ln,i ) = F(a,z) (Li )(za−5 − (1 − z 3 )a−4 + za−3 − (2 − z 2 )a−2 )n (i = 5, 7). Then by the Kauffman bound, we have −max-dega F(a,z) (Ln,5 ) − 1 = tb(Ln,5 ) = TB(Ln,5 ) (= 2n + 1), −max-dega F(a,z) (Ln,7 ) − 1 = tb(Ln,7 ) = TB(Ln,7 ) (= 2n + 5). From the above, we know that the set of the maximal Thurston–Bennequin numbers of cycles of Gn is equal to {2n − 1, −1, 2n − 1, 0, 2n + 1, 5, 2n + 5}. Since such a set is an invariant of spatial graphs, we know that Gn1 is not ambient isotopic to Gn2 if n1 = n2 . This completes the proof. 2 Remark 3.1. It is easily seen that the knots L5 , L6 and L7 are two-bridge knots. Masaharu Ishikawa has informed me that L4 is the knot 10148 [8]. 4. Non-realizable case Definition 4.1. We say that a spatial graph G is mTB-realizable if G is ambient isotopic to a Legendrian graph such that all its cycles realize their maximal Thurston–Bennequin numbers. In this section, we consider which spatial graphs are mTB-realizable. In general, a spatial graph is not necessarily mTB-realizable. In the case of embeddings of K4 , the spatial graph in Fig. 6 is mTB-realizable, however, the planer spatial graph in Fig. 5(b) is not mTB-realizable because of a result in [12]. We say that a Legendrian knot is trivial if it is topologically the unknot and has tb = −1. The trivial Legendrian knot is the only Legendrian knot that has the maximal Thurston–Bennequin number of the unknot among all Legendrian knots which are topologically isotopic to the unknot. Remark 4.2. Eliashberg and Fraser showed all Legendrian knots representing the unknot are determined by their classical invariants, the Thurston–Bennequin numbers and the rotation numbers denoted by rot [2]. Bennequin’s inequality implies that tb(J) + |rot(J)| ≤ −1 for any Legendrian link J which is ambient isotopic to the unknot [1]. So if L is a trivial Legendrian knot, then L satisfies that rot(L) = 0. The Whitehead link is a link with a diagram as in Fig. 9. In [9], Mohnke proved the following. We also give here the proof. Proposition 4.3. The Whitehead link cannot be represented by a Legendrian link that only consists of trivial Legendrian knots. Proof. The Kauffman polynomial of the Whitehead link is computed as follows. F(a,z) (LW h ) = z −1 − 2z + z 3 a−1 − 1 + z 2 − z 4 + z −1 − 4z + 3z 3 a + z 4 a2 − 2z − 2z 3 a3 + z 2 a4 .
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Fig. 10. The Whitehead link as cycles.
Fig. 11. The handcuff graph and its embeddings.
By the Kauffman bound, we know that TB(LW h ) ≤ −5. On the other hand, if the Whitehead link would be represented by a Legendrian link, denoted by JW h , that only consists of trivial Legendrian knots, then we have tb(JW h ) = −2 by the definition of tb and the condition that the Whitehead link has linking number zero. This is a contradiction. 2 By this proposition, we know that the Whitehead link does not admit a Legendrian embedding such that all its components realize their maximal Thurston–Bennequin numbers. By using this proposition, we have the following. Proposition 4.4. If a finite graph G contains two cycles that have no common edges and vertices, then there is a spatial embedding G of G such that G is not mTB-realizable. Proof. Let G be a graph as in the assumption. Then we can have a spatial embedding G of G such that it contains the Whitehead link denoted by SW h , as in Fig. 10. Suppose there exists a Legendrian graph Γ such that Γ is ambient isotopic to G and all its cycles realize their maximal Thurston–Bennequin numbers. Notice that the isotopy class of SW h , as cycles, does not change by the ambient isotopy for the spatial graphs. Thus, by the assumption, there exist cycles of Γ that are ambient isotopic to SW h . This contradicts Proposition 4.3. 2 Example 4.5. Let G be the handcuff graph. (See Fig. 11(a).) First we consider a Legendrian embedding G0 of G as in Fig. 11(b). Since G0 has two cycles both of which are topologically the unknot and satisfy that tb = −1, the cycles realize their maximal Thurston–Bennequin numbers and then we know that G0 is mTB-realizable. Next we consider a spatial embedding of G which contains the Whitehead link as in Fig. 11(c). By Proposition 4.4, we know that this spatial graph is not mTB-realizable. Definition 4.6. Let L be a link consisting of r components Li (i = 1, 2, ..., r). A link L is completely splittable if there exist mutually disjoint 3-balls Bi (i = 1, 2, ..., r) in R3 such that Li ⊂ Bi .
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Fig. 12. Modifications of crossings.
Fig. 13. Modifications of cusps.
In the case when the union of all cycles of a spatial graph represents a completely splittable link, we have the following. Proposition 4.7. Let G be a spatial graph that has a completely splittable link as the union of all its cycles. Then G is mTB-realizable. Proof. Let G be a spatial graph such that the union of all its cycles represents a completely splittable link. We denote the cycles of G by C1 , C2 , . . . , Cs . Using Reidemeister moves, a diagram of G can be transformed so that diagrams D1 , D2 , . . . , Ds of C1 , C2 , . . . , Cs satisfy that Di ∩ Dj = ∅ (i = j). We denote the resulted diagram by DG . Moreover, we can transform the resulted spatial graph into a spatial graph so that all its cycles are (piecewise smooth) Legendrian knots realizing their maximal Thurston–Bennequin numbers by ambient isotopy. More precisely, since any front diagram of a Legendrian knot (realizing the maximal Thurston–Bennequin number) is changed into a diagram of a knot by rounding the cusps and making the strand with smaller slope overcross at each double point as shown in the introduction, we can transform DG so that all diagrams of the cycles are transformed into (a disjoint union of) such diagrams by Reidemeister moves, and then we obtain a desired spatial graph by making cusps at each local maximum and local minimum (with respect to the x-axis) of the resultant diagrams of the cycles. Now we denote the rest part, the closure of G \ (C1 ∪ C2 , . . . , ∪ Cs ), by E. We may assume that E lies in general position with respect to the x-axis. Then we modify each crossing whose overcrossing has positive slope so that the overcrossing has negative slope as follows. If, at the crossing, both of the overcrossing and the undercrossing belong to E, then we modify it as Fig. 12(a) or 12(b). If, at the crossing, the overcrossing (resp. undercrossing) only belongs to E, then we modify it as Fig. 12(a) (resp. Fig. 12(b)). Finally we transform each local maximum and local minimum of the resulted diagram of E as in Fig. 13. Then we obtain a front diagram of a spatial graph, which is ambient isotopic to G, such that each cycle realizes its maximal Thurston–Bennequin number. (We describe this process in Fig. 14 by using a spatial embedding of the handcuff graph.) 2 Acknowledgement This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grantin-Aid for Young Scientists (B), 2011–2014 (23740046).
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Fig. 14. Modification of a diagram of a spatial embedding of the handcuff graph.
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