A bypass in the middle

Abstract

We provide a topological characterization for a family of bypasses with a fixed attaching arc to be contractible. This characterization is formulated in terms of the existence of a bypass that is disjoint from the given family away from the attaching region. As an application, we provide new proofs of several h-principles in overtwisted contact 3-manifolds.

Figures (9)

• Figure 1: In the second row it is depicted a Legendrian together with the boundary of two bypasses (in green and orange) attached to its standard neighborhood. These bypasses are different because they destabilize the Legendrian into the two Chekanov-Eliashberg $m(5_2)$ knots depicted in the first row which are not Legendrian isotopic.
• Figure 2: Overtwisted disk. The dividing set is depicted in red and the characteristic foliation in gray.
• Figure 3: A positive bypass, where we can see in light blue the attaching arc, in black the Honda arc, in red the dividing set and finally in gray the characteristic foliation.
• Figure 4: The effect of a bypass attachment.
• Figure 5: Trivial bypass attachment

Theorems & Definitions (25)

• Theorem 1.1: Bypass in the Middle
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 2.1: Giroux
• Theorem 2.2: Giroux Realization Theorem GirouxSurfaces
• Lemma 2.3: Pivot Lemma