On L-space knots obtained from unknotting arcs in alternating diagrams (Q2329062)

A rational homology 3-sphere \(Y\) is an \textit{L-space} if \(\dim_{\mathbb{F}_2}\widehat{HF}(Y)=|H_1(Y;\mathbb{Z})|\), where \(\widehat{HF}\) denotes the hat flavor of Heegaard-Floer homology. One of the numerous sources of examples for L-spaces consists of taking the two-fold branched cover \(\Sigma_2(K)\) of an alternating knot \(K\) [\textit{P. Ozsváth} and \textit{Z. Szabó}, Adv. Math. 194, No. 1, 1--33 (2005; Zbl 1076.57013)]. A knot \(K \subset S^3\) is an \(\textit{L-space knot}\) if it admits a positive Dehn surgery that is an L-space. L-space knots (which include torus knots) are known to be fibered and strongly-quasipositive, however, at the time of writing, the class of L-space knots remains poorly understood. The article under review studies the following way of producing L-space knots. Let \(D\) be a (reduced) diagram of an alternating knot \(K\) with unknotting number one. By specifying an unknotting crossing \(c\) in \(D\), Montesinos' trick implies that for some knot \(K_{D,c}\), one has \[S^3_{\frac{d}{2}}(K_{D,c})=\Sigma_2(K)\] for some \(d\) with \(|d|=\operatorname{det}(K)\). Since \(K\) is alternating, \(K_{D,c}\) is an L-space knot. This article studies the set \(\mathcal{D}\) of all \(K_{D,c}\) obtained in this way. The main results provide a characterisation in terms of \(D\) of when \(K_{D,c} \in \mathcal{D}\) is a satellite knot (Theorem 1.4), a torus knot (Proposition 1.5) and a hyperbolic knot (Corollary 1.6). In Proposition 1.7, the authors also note that given \(n>0\), there are only finitely many knots in \(\mathcal{D}\) with 3-genus less than \(n\); conjecturally there are only finitely many L-space knots with a given genus [\textit{K. L. Baker} and \textit{K. Motegi}, Commun. Anal. Geom. 27, No. 4, 743--790 (2019; Zbl 1432.57002)].