Bridge trisections of knotted surfaces in $S^4$

Abstract

We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and describe a set of moves that suffice to pass between two diagrams for the same surface. Using these decompositions, we define a new complexity measure: the bridge number of a knotted surface. In addition, we classify bridge trisections with low complexity, we relate bridge trisections to the fundamental groups of knotted surface complements, and we prove that there exist knotted surfaces with arbitrarily large bridge number.

46 pages, many figures

References in corpus (2)

• Classification of trisections and the Generalized Property R Conjecture
• Genus two trisections are standard

Cited by in corpus (30)

• Bridge trisections of knotted surfaces in 4--manifolds
• Genus two trisections are standard
• Calculating the homology and intersection form of a 4-manifold from a trisection diagram
• Isotopies of surfaces in 4-manifolds via banded unlink diagrams
• Trisections of surface complements and the Price twist
• Bridge trisections in ${\mathbb{CP}}^2$ and the Thom conjecture (with Corrigendum)
• Doubly pointed trisection diagrams and surgery on 2-knots
• Singular branched covers of four-manifolds
• Trisections and spun 4-manifolds
• Kirby-Thompson distance for trisections of knotted surfaces
• Symplectic 4-manifolds admit Weinstein trisections
• Filling braided links with trisected surfaces
• The bridge number of surface links and kei colorings
• The Dihedral Genus of a Knot
• Diagrams of *-Trisections
• Bridge trisections and classical knotted surface theory
• Stein trisections and homotopy 4-balls
• Invariants of knotted surfaces from link homology and bridge trisections
• Pseudo-trisections of four-manifolds with boundary
• Bounding the Kirby-Thompson invariant of spun knots
• Trisections and Ozsvath-Szabo cobordism invariants
• Shadows of 2-knots and complexity
• Bridge trisections and Seifert solids
• Thin Position through the lens of trisections of 4-manifolds
• Nonsmooth manifold decompositions
• Infinitely many standard trisection diagrams for Gluck twisting
• Bridge numbers and meridional ranks of knotted surfaces and welded knots
• Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds
• Flattening knotted surfaces
• Band diagrams of immersed surfaces in 4-manifolds