ETD

• exotic 4-manifolds,
• Kirby calculus,
• knot concordance,
• knot traces,
• RBG links,
• slice knots,
• torus surgery,
• trace embedding,

One of the main open problems in low-dimensional topology is the smooth 4-dimensional Poincaré conjecture. The existence of exotic smooth structures on #nCP^2 is also open.

One approach to distinguishing smooth structures is to find a knot that is H-slice in one but not the other. In the late 2000s, Freedman, Gompf, Morrison and Walker searched for H-slice knots in the Cappell-Shaneson spheres and tried to obstruct their sliceness using Rasmussen's s invariant. Unlike invariants such as tau, s is only known to satisfy the relevant adjunction inequality for the standard #nCP^2.

After Yasui disproved the Akbulut-Kirby conjecture, in 2021 Manolescu and Piccirillo revived the FGMW strategy by constructing a candidate exotic #nCP^2 around the 0-trace of a knot K'. They sought to replace the 0-trace of an H-slice knot K in #nCP^2 with that of a non-H-slice knot K'. To realise arbitrary 0-surgery homeomorphisms they introduced RBG links. They produced a family of promising pairs (K,K'), where s obstructs the H-sliceness of K', while that of K remained open.

A year later, Nakamura proved that every such K in the MP family is not H-slice. More generally, he showed that s cannot obstruct H-sliceness for any promising knot K' arising from a class of RBG links. Nevertheless, his results neither exclude using s for RBG links outside this class, nor rule out that the MP construction can produce an exotic #nCP^2 detectable by other techniques.

In 2025 Nakamura adapted the MP construction using torus surgery on knot traces. He defined a family of torus surgeries on the 0-trace X_0(C) of the Conway knot and showed that, if any resulting manifold embeds in S^4, replacing it with X_0(C) yields an exotic 4-sphere.

Before reviewing the MPN project, we introduce Kirby calculus and concordance invariants. To define tau and s, we include the construction of knot Floer and Khovanov homologies. This illustrates how differently the two theories lend themselves to extension from the standard #nCP^2 to possible exotic definite 4-manifolds.