Trisections of surface complements and the Price twist

Kim, Seungwon; Miller, Maggie

doi:10.2140/agt.2020.20.343

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Trisections of surface complements and the Price twist

Seungwon Kim and Maggie Miller

Algebraic & Geometric Topology 20 (2020) 343–373

DOI: 10.2140/agt.2020.20.343

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 $ν (S)$ yielding (up to) three different $4$ –manifolds: $X^{4}$ , $τ_{S} (X^{4})$ and $Σ_{S} (X^{4})$ . This is of particular interest when $X^{4} = S^{4}$ , as then $Σ_{S} (X^{4})$ 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 $X^{4} ∖ ν (S)$ . Moreover, we show how to produce a trisection description of general surface complements in $4$ –manifolds.

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Keywords

trisection, knotted surface, Price twist, surgery

Mathematical Subject Classification 2010

Primary: 57M50, 57R65

References

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

Volume 20, issue 1 (2020)