A new proof of Property R has appeared by Jamie Conway and Bülent Tosun. A slight shortcut of their argument may be summarized as:

If $K \subset S^3$ is a knot, and 0-framed surgery $S^3_0(K)\cong S^1\times S^2$, then (essentially) using the Floer exact triangle, one
may show that $S^3_1(K)$ is an L-space. An *L-space* is a rational homology 3-sphere $Y$ for which $HF^{red}(Y)=0$. $S^3$ and the Poincaré homology sphere are the only known irreducible L-spaces. This follows from the argument in Proposition 1.2 of Akbulut-Karakurt.

Theorem 1.2 of

*Ozsváth, Peter; Szabó, Zoltán*, **On knot Floer homology and lens space surgeries**, Topology 44, No. 6, 1281-1300 (2005). ZBL1077.57012.

implies that knot Floer homology of $K$ in each grading is rank 1.

- Now by the main result of

*Ni, Yi*, **Knot Floer homology detects fibred knots**, Invent. Math. 170, No. 3, 577-608 (2007); erratum ibid. 177, No. 1, 235-238 (2009). ZBL1138.57031.

$K$ is a fibered knot.

- Hence 0-framed surgery is fibered. But $S^1\times S^2$ fibers in only one way, and hence the fibering must be genus 0, which implies that $K$ is the unknot.

This proof still makes use of much of Gabai's theory, including his construction of taut foliations, which Ni's proof relies on. However, it eliminates the combinatorial topology technology of Scharlemann cycles which previous proofs relied heavily on.

A simplified version of Yi Ni's theorem was proved by Andras Juhasz using sutured Floer homology.

*Juhász, András*, **Floer homology and surface decompositions**, Geom. Topol. 12, No. 1, 299-350 (2008). ZBL1167.57005.

1more comment