On knotted real projective planes

Abstract

Let [Formula: see text] be a hyperbolic transformation. Let B be a new band attaching to L such that [Formula: see text] is also a hyperbolic transformation. In this paper, we will study the relationship between the realizing surfaces [Formula: see text] and [Formula: see text]. If B is a noncoherent band to both L and [Formula: see text] such that [Formula: see text] is defined, then [Formula: see text] and [Formula: see text] are ambient isotopic, where RP2 is one of the standard real projective planes. We will study the triviality of [Formula: see text] because as an application, RP2 can untangle some knotted sphere [Formula: see text] with suitable conditions, when it is attached to [Formula: see text] by the connected sum.