#### Volume 20, issue 1 (2020)

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Trisections of surface complements and the Price twist

### Seungwon Kim and Maggie Miller

Algebraic & Geometric Topology 20 (2020) 343–373
##### Abstract

Given a real projective plane $S$ embedded in a $4$–manifold ${X}^{4}$ with Euler number $2$ or $-2$, the Price twist is a surgery operation on $\nu \left(S\right)$ yielding (up to) three different $4$–manifolds: ${X}^{4}$, ${\tau }_{S}\left({X}^{4}\right)$ and ${\Sigma }_{S}\left({X}^{4}\right)$. This is of particular interest when ${X}^{4}={S}^{4}$, as then ${\Sigma }_{S}\left({X}^{4}\right)$ is a homotopy $4$–sphere which is not obviously diffeomorphic to ${S}^{4}$. Here we show how to produce a trisection description of each Price twist on $S\subset {X}^{4}$ by producing a relative trisection of ${X}^{4}\setminus \nu \left(S\right)$. Moreover, we show how to produce a trisection description of general surface complements in $4$–manifolds.

##### Keywords
trisection, knotted surface, Price twist, surgery
##### Mathematical Subject Classification 2010
Primary: 57M50, 57R65
##### Publication
Received: 19 September 2018
Revised: 15 February 2019
Accepted: 16 May 2019
Published: 23 February 2020
##### Authors
 Seungwon Kim National Institute for Mathematical Sciences Daejeon South Korea Center for Geometry and Physics Institute for Basic Science (IBS) Pohang Republic of Korea Maggie Miller Department of Mathematics Princeton University Princeton, NJ United States http://www.math.princeton.edu/~maggiem