The relative ℒ-invariant of a compact 4-manifold

Kirby–Thompson distance for trisections of knotted surfaces

“…For later use, we observe the following. In 𝐵 4 , 𝑁(Σ) is homeomorphic to Σ × 𝐷 2 . The intersection 𝑁(Σ) with 𝜕𝐵 4 is a solid torus 𝑉 ′ = (𝜕Σ) × 𝐷 2 ⊂ 𝜕𝐵 4 .…”

“…Consider a curve 𝑠 of 𝑝 1 12 as we traverse the loop Λ defined by the geodesics 𝛼(1), 𝛼 (13), 𝛼(3), 𝛼(23), and 𝛼 (2). By the hypothesis of the case, the curve 𝑠 must be moved as we traverse one of the edges of the loop.…”

“…They show that when $$\mathcal {L}(X)=0$$, then $$X$$ is diffeomorphic to the connected sum of the 4‐sphere with some number (possibly zero) of copies of $$S^1 \times S^3$$, $$S^2 \times S^2$$, and $$\mathbb {C}P^2$$. This was extended to be an invariant for smooth 4‐manifolds with boundary in [2], where it was shown that if the invariant takes the value zero on a rational homology ball, then the rational homology ball is a 4‐ball $$B^4$$. In this paper, we adapt the definition to apply to smooth surfaces $$S$$ properly embedded in $$S^4$$ or $$B^4$$.…”

“…Inspired by Hempel's distance for Heegaard splittings [7], Kirby and Thompson [10] recently defined a nonnegative integer-valued invariant (𝑋) of a smooth 4-manifold 𝑋. They show that when (𝑋) = 0, then 𝑋 is diffeomorphic to the connected sum of the 4-sphere with some number (possibly zero) of copies of 𝑆 1 × 𝑆 3 , 𝑆 2 × 𝑆 2 , and ℂ𝑃 2 . This was extended to be an invariant for smooth 4-manifolds with boundary in [2], where it was shown that if the invariant takes the value zero on a rational homology ball, then the rational homology ball is a 4-ball 𝐵 4 .…”

Trisection diagrams and twists of 4-manifolds

“…There are also moves relating relative trisections inducing different open book decompositions, but we will not discuss them in this paper. See [5] and [2] for more details. A key feature of relative trisections is that two such decompositions can be glued together to form a closed (trisected) 4-manifold.…”