Doubly pointed trisection diagrams and surgery on 2-knots

Trisection diagrams and twists of 4-manifolds

“…Here, D a is the annular diagram consisting of two parallel b, c arcs and an a arc that differs by a positive Dehn twist. The result is also true if we replace D a with either of D b or D c , or their mirrors (Remark 5.6 [GM18]). Here and in the sequel, we draw a grey arc to indicate arbitrary curves or arcs as in [GM18].…”

“…For expositions of relative trisections and manipulations of their diagrams, see [GM18], [CGPC18a] [CGPC18b], and [C15]. Trisection diagrams can be quite complicated in general, but some standard 4-manifolds admit diagrams of low genus.…”

“…Its mirror image gives a diagram for ν(P + ). Gay and Meier [GM18] studied the special case of surgery along 2-knots in detail. To perform a Gluck twist, no boundary stabilization is needed, and for our purposes the relevant result is the following (Theorem C Part 2):…”

“…Trisections of 4-manifolds were introduced by Gay and Kirby in 2012 as a 4dimensional analogue of Heegaard splittings. Recently, they have been used to prove classical results [LC18] [LC19], as well as understand embedded surfaces via bridge trisection diagrams [MZ18] [GM18]. We extend these ideas, using trisections to understand certain surgeries on 4-manifolds.…”

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].…”