On the support genus of Legendrian knots
- PDF / 928,447 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 54 Downloads / 186 Views
Archiv der Mathematik
On the support genus of Legendrian knots Sinem Onaran
Abstract. In this paper, we show that any topological knot or link in S 1 × S 2 sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in S 1 × S 2 has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces L(p, 1). Mathematics Subject Classification. Primary 57R17; Secondary 57M50. Keywords. Contact structures, Legendrian knots, Open book decompositions, Planar open book decompositions.
1. Introduction. In 2002, Giroux showed that given any contact structure ξ on a 3-manifold M , there is an open book decomposition of the 3-manifold such that the contact structure is transverse to the binding of the open book and it can be isotoped arbitrarily close to the pages. In this case, we say the contact 3-manifold (M, ξ) is supported by this open book decomposition. Further, Giroux showed that a contact 3-manifold is Stein fillable (and hence tight) if and only if there is an open book decomposition for the contact 3-manifold whose monodromy can be written as a product of positive Dehn twists, [5]. The purpose of this note is to construct the simplest open book decomposition of lens spaces whose monodromy is a product of positive Dehn twists and which contains a given knot or link on its page. The first result explicitly constructs such open book decomposition for knots and links in S 1 × S 2 . Theorem 1.1. Any topological knot or link in S 1 × S 2 sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. This work was completed with the support of the BAGEP award of the Science Academy, Turkey.
S. Onaran
Arch. Math.
Similar to the support genus invariant of a contact structure defined in [3], a new invariant of a Legendrian knot L in a contact 3-manifold (M, ξ) is defined using open book decompositions in [7]. The invariant is called the support genus sg(L) of L and it is defined as the minimal genus of a page of an open book decomposition of M supporting ξ such that L sits on a page of the open book and the framings given by ξ and the page agree. In [7], it was shown that while any null-homologous knot with an overtwisted complement has sg(L) = 0, not all Legendrian knots have. Moreover, it was shown that any knot or link in the 3-sphere S 3 sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. This guarantees a Legendrian representative of the given knot or link having support genus zero in the standard tight contact S 3 . This paper addresses this theme for lens spaces L(p, 1). According to [2], S 1 ×S 2 has a unique tight and Stein fillable contact structure. Then by [5], the open book decomposition constructed in Theorem 1.1 supports this unique tight and Stein fillable contact structure. As a consequence of Theorem 1.1, we have the following result on the support genus of Legendrian kno
Data Loading...
