Trisections of surface complements and the Price twist

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 , τS(X 4 ), ΣS(X 4 ). This is of particular interest when X 4 = S 4 , as then ΣS(X 4 ) is a homotopy 4-sphere which is not obviously diffeomorphic to S 4 . In this paper, we show how to produce a trisection description of each Price twist on S ⊂ 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.arXiv:1805.00429v3 [math.GT]

Doubly pointed trisection diagrams and surgery on 2-knots

“…In particular, since higher genus surfaces never have 1-bridge trisections, surgery along surfaces of nonzero genus, and along non-orientable surfaces, is necessarily more subtle. Kim and Miller show how to obtain a relative trisection for the exterior of a knotted surface from a (generalised) bridge trisection of the knotted surface [KM18]; they apply their technique to study the Price twist, which can be described as a version of Gluck surgery for knotted projective planes.…”

Trisections obtained by trivially regluing surface-knots

“…For a 2-knot K in X which is in 1bridge position, the decomposition of X − ν(K) into the union of three X i − ν(K)'s is a relative trisection of X − ν(K), where ν(K) is an open tubular neighborhood of K. On the other hand, for a surface-knot S in X which is not a 2-knot, the decomposition of X −ν(S) is never a relative trisection of X −ν(S). Kim and Miller [KM20] introduced a new technique, called a boundary-stabilization, to change the above decomposition of X − ν(S) into a relative trisection.…”

“…In this paper, we call the twist having X the trivial Price twist. Kim and Miller [KM20] described trisections obtained by the Price twist by attaching a relative trisection of ν(P ) to a relative trisection of X − ν(P ) constructed by a boundary-stabilization.…”

Trisections and Ozsvath-Szabo cobordism invariants

“…A new tool in smooth four-manifold topology has recently been introduced under the name of trisected Morse 2-functions (or trisections for short) by Gay and Kirby [GK16]. Recent developments in this area demonstrate rich connections and applications to other aspects of four-manifold topology, including a new approach to studying symplectic manifolds and their embedded submanifolds [LMS20; Lam19; LM18], and to surface knots (embedded in S 4 and other more general 4-manifolds) [MZ17; MZ18] along with associated surgery operations [GM18;KM20].…”