Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON THE LAGRANGIAN FILLABILITY OF ALMOST POSITIVE LINKS

  • Tagami, Keiji (Department of Mathematics Faculty of Science and Technology Tokyo University of Science)
  • Received : 2018.06.19
  • Accepted : 2018.09.21
  • Published : 2019.05.01
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Abstract

In this paper, we prove that a link which has an almost positive diagram with a certain condition is Lagrangian fillable.

Keywords

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FIGURE 1. A 0-handle attaching (left). A 1-handle attaching (right).

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FIGURE 2. An exact Legendrian cobordism from Λ'' to Λ. In the third line, we use a 1-handle attaching. In the fourth line,we use a Legendrian isotopy. In this picture, for simplicity,we suppose that there is no crossing oriented upward. In the case where there are some crossings oriented upward, we can construct a Legendrian cobordism similarly.

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FIGURE 3. Bunching deformation

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FIGURE 5. Two almost positive diagrams of 10145, which is an almost positive knot. The left diagram satis es (P1). The right diagram satisfies (P2).

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FIGURE 6. Deformation near the negative crossing

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FIGURE 7

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FIGURE 8. The proof of Theorem 1.6 for s = 2. A box in this picture represents a positive crossing. The front projection ∆ of the Legendrian link Λ is the left. By using the Legen-drian version of the Reidemeister move II, we obtain the right diagram ∆'.

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FIGURE 9. An almost positive knot introduced by Stoimenow [40, Example 6.1].

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FIGURE 10. The knot Kn has an almost positive diagram satisfying (P2) (right). The crossing p is the negative crossing.

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FIGURE 4. (color online) Sketch for the proof of Lemma 3.1. The picture (i) is D. In pictures (ii)-(ix), we draw a crossing derived from a crossing of D by a rectangle.

TABLE 1. The Lagrangian llability of non-alternating knots with up to 10 crossings. For example, 819 or its mirror is Lagrangian fillable. Neither 820 nor its mirror is Lagrangian fillable. To prove "Yes*", we find front projections with maximal Thurston-Bennequin numbers and use Theorem 2.1. To find such diagrams, we refer to [9] and [11].

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