Knotting of algebraic curves in CP2

Abstract

For any k ≥ 3, I construct infinitely many pairwise smoothly nonisotopic smooth surfaces F ⊂ CP 2 homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve and having abelian fundamental group π 1 (CP 2 F ). This gives an answer to Problem 4.110 in the Kirby list [K].

Knotted surfaces in 4-manifolds and stabilizations

“…Assume it is oriented according to some local orientation (a local orientation that, in this case, 7 Viewing the standard embedding of RP 2 in S 4 with Euler number −2 (resp. +2) as the image of the fixed point set under the complex involution on CP 2 (resp.…”

“…The surgery in S 4 we will discuss here descends from the equivariant surgery on E(1). 8 The authors do this only for 2 half-twists on the left. In this case the branched cover of the surface is E(1) 2,q .…”

“…[3,10,7,8,9,14,15,17,21,22,23,24,26]. Notably, the phenomenon of infinite exotic knottings is unique to dimension 4.…”

“…1 We will examine the analogous question for surface stabilizations: how many are needed to make exotic knottings smoothly isotopic? We analyze in detail various constructions of exotically knotted and exotically embedded surfaces 2 , which produced non-orientable surfaces in S 4 in the pioneering works of Finashin, Kreck and Viro [10] and Finashin [9], and orientable surfaces (often topologically isotopic to symplectic/complex curves) in many other 4-manifolds in the works of Fintushel and Stern [14,15], Kim and Ruberman [21,22,23,24], Finashin [7,8], Mark [26], Hoffman and Sunukjian [17].…”

Topological triviality of smoothly knotted surfaces in $4$-manifolds

“…Recently, S. Finashin [4] and the first-named author [15] have used variations of rim surgery to find smoothly knotted surfaces whose complements have (nontrivial) cyclic fundamental groups. It is interesting to ask whether these surfaces are topologically nontrivial.…”